1. First foundations of mathematics
1.1. Introduction to the foundations of mathematics
What is mathematics
Mathematics is the study of systems of elementary objects, whose only considered
nature is to be exact, unambiguous (two objects are equal or different, related or not;
an operation gives an exact result...). Mathematics as a whole can be seen as
«the science of all possible worlds» of this kind (of exact objects).
Mathematical systems are conceived as «existing» independently of our usual world
or any particular sensation, but their study requires some form of representation.
Diverse ways can be used, that may be equivalent (giving the same results) but with
diverse degrees of relevance (efficiency) that may depend on purposes. Ideas may
first appear as more or less visual intuitions which may be expressed by drawing or
animations, then their articulations may be expressed in words or formulas for careful
checking, processing and communication. To be freed from the limits or biases of a
specific form of representation, is a matter of developing other forms of representation,
and exercise to translate concepts between them. The mathematical
adventure is full of plays of conversions between forms of representation,
which may initiate us to articulations between mathematical systems themselves.
Theories
Mathematics is split into diverse branches according to the kind of systems
being considered. These frameworks of any mathematical work may either
remain implicit (with fuzzy limits), or formally specified as theories. Each
theory is the study of a supposedly fixed system that is its world of
objects, called its model. But each model of a theory may be just one of its
possible interpretations, among other equally legitimate models. For example,
roughly speaking, all sheets of paper are systems of material points, models
of the same theory of Euclidean plane geometry, but independent of each other.
The word «theory» may take different meanings between mathematical and
nonmathematical uses (in ordinary language and other sciences).
A first distinction is by nature (general kind of objects); the other distinction,
by intent (realism vs. formalism) will be discussed later.
Nonmathematical theories describe roughly or qualitatively some systems
or aspects of the world (fields of observation) which escape simple exact description.
For example, usual descriptions of chemistry involve drastic approximations, recollecting
from observations some seemingly arbitrary effects whose deduction from
quantum physics is usually out of reach of direct calculations. The lack of clear
distinction of objects and of their properties induces risks of mistakes when
approaching them and trying to infer some properties from others, such as to
infer some global properties of a system from likely, fuzzy properties of its parts.
Pure mathematical theories, only describing exact systems, can be
protected from the risk to be «false», by use of properly rigorous methods (formal rules)
designed to ensure the preservation of exact conformity of theories to their models.
In between both, applied mathematical theories, such as theories of physics
are also mathematical theories but the mathematical systems they
describe are meant as idealized (simplified) versions of aspects of given
realworld systems while neglecting other aspects; depending on its accuracy,
this idealization (reduction to mathematics) also allows for correct deductions
within accepted margins of error.
Foundations and developments
Any mathematical theory, which describes its model(s), is made of a content
and is itself described by a logical framework. The content of a theory is made
of components which are pieces of description (concepts and information,
described in 1.3). A theory starts with a choice of foundation made of a
logical framework and an initial version of its content (hopefully rather small, or at
least simply describable). The components of this initial version are qualified as
primitive.
The study of the theory progresses by choosing some of its possible
developments : new components resulting from its current content, and
that can be added to it to form its next content. These different contents, having
the same meaning (describing the essentially same models), play the role of
different presentations of the same theory.
Any other possible development (not yet chosen) can still be added later, as the
part of the foundation that could generate it remains. Thus, the totality of
possible developments of a theory, independent of the order chosen to process
them, already forms a kind of «reality» that these developments explore.
To express the properties of its models, the content of a theory includes a list of
statements, which are formulas meant as true when interpreted in any model.
Primitive statements are called axioms. Further statements called theorems
are added by development to the content, under the condition that they are proven
(deduced) from previous ones : this ensures them to be true in all models,
provided that previous ones were. Theorems can then be used in further
developments in the same way as axioms.
A theory is consistent if its theorems will never contradict each other.
Inconsistent theories cannot have any model, as the same statement cannot be
true and false on the same system.
The Completeness Theorem (1.9,
4.6) will show that the
range of all possible theorems precisely reflects the more interesting reality
of the diversity of models, which indeed exist for any consistent theory.
Other kinds of developments (definitions and constructions) which add other
components beyond statements, will be described in 1.5, 4.8 and 4.9.
There are possible hierarchies between theories, where some can
play a foundational role for others. For instance, the foundations of
several theories may have a common part forming a simpler theory,
whose developments are applicable to all.
A fundamental work
is to develop, from a simple initial basis, a convenient body of knowledge
to serve as a more complete "foundation", endowed with efficient tools
opening more direct ways to further interesting developments.
Platonism vs Formalism
Mathematics, or each theory, may be approached in two ways (as further discussed
in 1.9):
 The Platonic or realistic view, considers the mathematical realm
or some particular described systems, as preexisting realities to be explored (or
remembered, according to Plato). This is the approach of intuition which by imagining
things, smells their order before formalizing them.
 A formalistic or logicist view focuses on language, rigor
(syntactic rules) and dynamical aspects of a theory, starting
from its formal foundation, and following the rules of development.
Many philosophers of
mathematics carry obsolete conceptions of such views as forming a multiplicity
of opposite beliefs (candidate truths) on the real nature of mathematics. But after examination,
just remain these two necessary and complementary views,
with diverse shares of relevance depending on topics :
By its limited abilities, human thought cannot directly operate in a fully realistic way over
infinite systems (or finite ones with unlimited size), but requires
some kind of logic for extrapolation, roughly equivalent to formal reasonings
developed from some foundations ; this work of formalization can prevent
possible errors of intuition. Moreover, mathematical objects cannot form any
completed totality, but only a forever temporary, expanding realm, whose precise
form is an appearance relative to a choice of formalization.
But beyond its inconvenience for expressing proofs, a purely
formalistic view cannot hold either because the clarity and selfsufficiency
of any possible foundation (starting position with formal development rules), remain
relative: any starting point had to be chosen somehow arbitrarily, taken from and
motivated by a larger perspective over mathematical realities; it must be defined in some
intuitive, presumably meaningful way, implicitly admitting its own foundation, since any
try to specify the latter would lead to a path of endless regression, whose realistic
preexistence would need to be admitted.
The cycle of foundations
Despite the simplicity of nature of mathematical objects, the general
foundation of all mathematics turns out to be quite complex (though
not as bad as a physics theory of everything). Indeed, it is itself a
mathematical study, thus a branch of mathematics, called mathematical
logic. Like any other branch, it is full of definitions and theorems
about systems of objects. But as its object is the general form
of theories and systems they may describe, it provides the general
framework of all branches of mathematics... including itself.
And to provide the framework or foundation of each considered
foundation (unlike ordinary mathematical works that go forward from
an assumed foundation), it does not look like a precise starting point,
but a sort of wide cycle composed of easier and harder steps. Still this
cycle of foundations truly plays a foundational role for mathematics, providing
rigorous frameworks and many useful concepts to diverse branches of
mathematics (tools, inspirations and answers to diverse philosophical questions).
(This is similar to dictionaries defining each word by other words, or to
another science of finite systems: computer programming. Indeed computers
can be simply used, knowing what you do but not why it works; their working
is based on software that was written in some language, then compiled by
other software, and on the hardware and processor whose design and production
were computer assisted. And this is much better than at the birth of this field.)
It is dominated by two theories:

Set theory describes the universe of «all mathematical
objects», from the simplest to the most complex such as infinite
systems (in a finite language). It can roughly be seen as one theory,
but in details it will have an endless diversity of possible variants
(indeed differing from each other).

Model theory is the study of theories (their formalisms as systems
of symbols), and systems (possible models of theories). Proof theory
completes this by describing formal systems of rules of proofs. While these
are usually meant as general topics (admitting variants of concepts), the
combination of both can be specified into precise versions (mathematical
theories) called logical frameworks, each giving a precise format of
expression for a wide range of possible theories, and a format in which all
proofs in any of these theories can in principle be expressed. There is an
essentially unique main logical framework called firstorder logic,
by which the concepts of theory, theorem (as provable statement) and
consistency of each theory, find their natural mathematical definitions;
but other logical frameworks are sometimes needed too.
Each one is the natural framework to formalize the other: each set theory is formalized
as a theory described by model theory; the latter better comes as a development from
set theory (defining theories and systems as complex objects) than directly as a theory.
Both connections must be considered separately: both roles of set theory, as a
basis and an object of study for model theory, must be distinguished. But these
formalizations will take a long work to complete.
1.2. Variables, sets, functions and operations
Starting mathematics is a matter of introducing some simple concepts from the founding
cycle, which may seem as selfsufficient as possible (while they cannot be absolutely so).
A usual and natural solution is to start with a set theory not fully formalized as an
axiomatic theory. This will be briefly done in 1.2, intuitively explaining the concepts of
set and function. Then 1.3 will introduce the main picture of foundations (model theory)
by which set theory can be formalized, with its main subtleties (paradoxes).
Constants
A constant symbol is a symbol seen as denoting a unique object,
called its value. Examples: 3, ⌀, ℕ. Those of English
language usually take the form of proper names and names with «the»
(singular without complement).
Free and bound variables
A variable symbol (or a variable), is a symbol which, instead
having an a priori definite value, comes with the concept of possible
values, or possible interpretations as taking a particular value. Each possibility
gives it a role of constant. These possible values may as well
be infinitely many, or only one or even none.
It can be understood as limited by a box, whose inside has multiple
versions in parallel, articulating different viewpoints over it:
 The variable is called fixed when seen "from inside", which means it has a given value,
and is thus usable as a constant.
 It is called bound when seen from
the «outside» where the diversity of its possible values is
considered fully known, gathered and
processed as a whole.
 It is called free to describe a coexistence of both statuses (views over it):
a local view seeing it as fixed, and an external view giving the context of its variations.
More precisely with respect to given theories, fixing a variable
means taking a free variable in a theory and more lengthily ignoring its
variability, therefore simulating the use of the other theory obtained by holding
this symbol as a constant.
The diverse «internal viewpoints», corresponding to each possible
value seen as fixed, may be thought of as abstract «locations» in the
mathematical universe, while the succession of views over a symbol
(qualifying it as a constant, a free variable or a bound variable), can be
seen as a first expression of the flow of time in mathematics: a
variable is bound when all the diverse "parallel locations inside
the box" (possible values) are past. All these places and
times are themselves purely abstract, mathematical entities.
Ranges and sets
The range of a variable, is the meaning it takes when seen
as bound: it is the «knowledge» of the considered totality of its possible or
authorized values (seen in bulk: unordered, ignoring their context),
that are called the elements of this range. This «knowledge» is
an abstract entity that can actually process (encompass) infinities of
objects, unlike human thought. Any range of a variable is called a set.
A variable has a range when it can be bound, i.e. when an encompassing view over all its
possible values is given. Not all variables of set theory will have a range. A variable without
a range can still be free, which is no more an intermediate status between fixed and bound,
but means it can take some values or some other values with no claim of exhausitivity.
Cantor defined a set as a «gathering M of definite and separate
objects of our intuition or our thought (which are called the "elements" of M) into a whole».
He explained to Dedekind : «If the totality of elements of a multiplicity can be
thought of... as "existing together", so that they can be gathered
into "one thing", I call it a consistent multiplicity or a "set".» (We expressed this
"multiplicity" as that of values of a variable).
He described the opposite case as an «inconsistent multiplicity»
where «admitting a coexistence of all its elements leads to a
contradiction». But noncontradiction cannot suffice to
generally define sets: the consistency of a statement does not
imply its truth (i.e. its negation may be true but unprovable);
facts of noncontradiction are often themselves unprovable (incompleteness theorem);
and two separately consistent coexistences might contradict each
other (Irresistible
force paradox / Omnipotence
paradox).
A variable is said to range over a set, when it is bound
with this set as its range. Any number of variables can be
introduced ranging over a given set, independently of each other and of
other variables.
Systematically renaming a bound variable in all its box, into
another symbol not used in the same context (same box), with the
same range, does not change the meaning of the whole. In practice,
the same letter can represent several separate bound variables (with
separate boxes), that can take different values without conflict, as
no two of them are anywhere free together to compare their values.
The common language does this continuously, using very few variable
symbols («he», «she», «it»...)
Functions
A function is an object f made of the following data:
 A set called the domain of f, denoted Dom
f
 For each element x of Dom f,
an object written f(x), called the image of x
by f or value of f at x.
In other words, it is an entity behaving as a variable whose value
is determined by that of another variable called its argument with
range Dom f : whenever its argument is fixed (gets a name, here
"x", and a value in Dom f), f becomes also fixed, written
f(x). This actually amounts to conceiving a variable f where
the "possible views" on it as fixed, are treated as objects x conceptually
distinct from the resulting values of f.
As we shall see later, such an entity (dependent variable) f would not be
(viewable as) a definite object of set theory if its argument had no range, i.e.
could not be bound (it would only be a metaobject, or object of model theory,
that we shall call a functor in 1.4)
Operations
The notion of operation generalizes that of function, by
admitting a finite list of arguments (variables with given
respective ranges) instead of one. So, an operation gives a result (a value)
when all its arguments are fixed. The number n of arguments
of an operation is called its arity ; the operation is
called nary. It is called nullary if n=0
(it is a constant), unary if n=1 (it is a function),
binary if n=2, ternary if n=3...
Nullary operations are useless as their role is already played by their
unique value; 2.3 will show how to construct those with arity
> 1 by means of functions.
The value of a binary operation f on its fixed arguments
named x and y (i.e. its value when its arguments are
assigned the fixed values of x and y), is denoted
f(x,y).
Generally, instead of names, the arguments are pictured by places around
the symbol, namely the left and right spaces in parenthesis, to be filled by
any expression giving them desired values.
An urelement (pure element) is an object not playing any other role
than that of element: it is neither a set nor a function nor an operation.
1.3. Form of theories: notions, objects and metaobjects
The variability of the model
Each theory describes its model as a fixed system.
But from the larger viewpoint of model theory, this is a mere «choice» of one possible
model (interpretation) in a wide (usually infinite) range of other existing, equally legitimate
models of the same theory. Now this fixation of the model, like the fixation of any variable,
is but the elementary act of picking any possibility, ignoring any issue of how to specify
an example in this range. Actually these «choice» and «existence» of models can be quite
abstract. In details, the proof
of the Completeness theorem will effectively «specify» a model of any consistent
theory for the general case, but its construction will involve an infinity of steps, where
each step depends on an infinite knowledge. Regardless this difficulty, the attitude of
implicitly fixing a model when formally studying any mathematical theory, remains the
normal way of interpreting it (except somehow for set theory as explained later).
Notions and objects
Each theory has its own list of notions, usually designated
by common names, formally serving as the kinds of variables it can use
; each model interprets each notion as a set that is the common
range of all variables of this kind. For example, Euclidean geometry
has the notions of «point», «straight line», «circle» and more, and is usually
expressed using a different style of variable symbol for each. The
objects of a theory in a model, are all possible values
of its variables of all kinds (the elements of all its notions) in this model.
Onemodel theory
Any discussion on several theories T and systems M that may be
models of those T, takes place in model theory, with its
notions of «theory» and «system» that are the respective kinds of the variables
T and M. But when focusing on one theory with a fixed model, the
variables T and M now fixed disappear from the list of variables.
Their kinds, the notions of theory and model, disappear from the notions list too.
This reduces the framework, from model theory, to that of onemodel theory.
A model of onemodel theory, is a system [T,M] which combines
a theory T with a model M of T.
The diversity of logical frameworks
The role of a logical
framework, as a precise version of (one)model theory with its associated
proof theory, is to describe : The admissible forms of contents for theories ;
 In particular, the syntactic structures of possible statements and other
expressions, which can be called their "grammar" ;
 The meaning of these contents and expressions on the models ;
 The rules of development of theories.
Here are those we shall see,
roughly ordered from the poorest to the most expressive
(though the order depends on the ways to relate them):
 Boolean algebra, also called propositional calculus (1.6);
 Algebra;
 Firstorder logic;
 Duality (for geometry) and the tensor formalism for linear algebra;
 Secondorder logic (5.1, 5.2);
 Higherorder logic (5.2);
 Set theory.
We shall first describe the main two of them in parallel. Firstorder logic is the
most common version of model theory, describing firstorder theories we shall
also call here generic theories. Set theory, which can encompass all
other theories, can also encompass logical frameworks and thus serve itself as
the ultimate logical framework as will be explained in 1.B.
Most frameworks manage notions as types (usually in finite number for
each theory) classifying both variables and objects. Notions are called types if
each object belongs to only one of them, which is then also called the type of
the variables that can name it. For example, an object of Euclidean
geometry may be either a point or a straight line, but the same
object cannot be both a point and a straight line. But set theory will need
more notions beyond types: classes, which will be introduced in 1.7.
Examples of notions from various theories
Theory 
Kinds of objects (notions) 
Generic theories 
Urelements classified by types to play different roles 
Set theory 
Elements, sets, functions, operations,
relations, tuples... 
Model theory 
Theories, systems and their components
(listed below) 
Onemodel theory

Objects, symbols, types or other notions, Booleans,
structures (operators, predicates), expressions (terms, formulas)... 
Arithmetic 
Natural numbers 
Linear Algebra 
Vectors, scalars... 
Geometry 
Points, straight lines, circles... 
Metaobjects
The notions of a onemodel theory T_{1}, normally
interpreted in [T,M], classify the components of T
(«type», «symbol», «formula»...), and those of M («object»,
and the means to interpret components and expressions of T there).
But the same notions (even if from a different logical framework) can be interpreted in
[T_{1}, [T,M]], by putting the prefix meta on them.
By its notion of «object», onemodel theory distinguishes the
objects of T in M among its own objects in [T,M],
that are the metaobjects. The above rule of use of the meta prefix
would let every object be a metaobject; but we will make a vocabulary
exception by only calling metaobject those which are not objects:
symbols, types or other notions, Booleans, structures, expressions...
Set theory only knows the ranges of some of its own variables, seen
as objects (sets). But, seen by onemodel theory, every variable of
a theory has a range among notions, which are metaobjects only.
Components of theories
In a given logical framework, the content of a theory consists in 3
lists of components of the following kinds, where those of each of the latter
two kinds are finite systems using those of the previous kind.
 A list of abstract types, names that will designate types in each system;
 A language (vocabulary): list of structure
symbols, names of the structures forming the system (1.4).
 A list of axioms chosen among expressible statements with
this language (1.9).
Settheoretical interpretations
Any generic theory can be interpreted (inserted, translated) in set theory
by converting its components into components of set theory. This is the usual
view of ordinary mathematics, studying many systems as «sets with relations or
operations such that...», with possible connections between these systems.
Let us introduce both the generic interpretations applicable to any generic
theory, and other ones usually preferred for particular theories.
Any interpretation converts each abstract type into a symbol (name) designating
a set called interpreted type (serving as the range of variables of that type,
whose use is otherwise left intact). This symbol is usually a fixed variable in the
generic case, but can be accepted as constant symbol of set theory in special
cases such as numbers systems (ℕ, ℝ...).
In generic interpretations, all objects (elements of interpreted types) are
urelements, but other kinds of interpretations called standard
by convention for specific theories may do otherwise.
For example, standard interpretations of geometry represent points by
urelements, but represent straight lines by sets of points.
Generic interpretations will also convert structure symbols into fixed variables
(while standard ones may define them using the language of set theory).
Any choice of fixed values of all types and structure symbols, defines a choice of
system. Once the language is seen
as a set (in particular if it is finite), models become objects of set theory, owing
their multiplicity to the variability of types and structure symbols. This integrates
all needed theories into the same set theory, while gathering representatives
of all their considered models inside a common model of set theory. This is
why models of set theory are called universes. When adopting set
theory as our conceptual framework, this concept of "interpretation" becomes
synonymous with the choice (designation) of a model.
1.4. Structures of mathematical systems
The structures, interpreting each structure symbol from a given language over
a list of types (or notions), form a described system by relating the objects of some given
types, giving their roles to the objects of each type with respect to those of other types.
According to these roles, objects may be thought of as complex objects, in
spite of have otherwise no nature like urelements.
Firstorder structures
The kinds of structures (and thus the kinds of structure symbols) allowed in firstorder theories, thus
called firstorder structures, will be classified into operators and predicates. They are described
as operations designated by structure symbols in a set theoretical interpretation. More powerful
structures called secondorder
structures will be introduced in 5.1, coming from set theoretical tools or as
packs of an additional type with firstorder structures.
An operator is an operation between interpreted types.
On the side of the theory before interpretation, each operator symbol
comes with its symbol type made of
 its list of arguments (variable symbols figured as places
around the operator symbol instead of names),
 for each argument, its abstract type, whose value as a set will be the
range of this argument in any interpretation;
 its type of results, type which will contain all results of the operation
it will designate in any interpretation with given values of its arguments.
In a theory with only one type, this data is reduced to the arity.
The constant symbols (or constants) of a theory
are its nullary operator symbols (having no argument).
Unary operators (that are functions) will be called here functors
(*).
The list of types is completed by the Boolean type, interpreted as the set
of two elements we shall denote 1 for «true» and 0 for «false». A variable of this
type (outside the theory) is called a Boolean variable.
A paraoperator is a generalized operator allowing the
Boolean type among its types of arguments and results.
A (logical) connective is a paraoperator with only Boolean arguments
and values.
A predicate is a paraoperator with Boolean values, and at
least one argument but no Boolean argument.
As will be formalized in 2.6.,
any nary operator f may be reduced to the (n+1)ary predicate
(y = f(x_{1},...,x_{n})), true for a
unique value of y for any
chosen values of x_{1},...,x_{n}.
Structures of set theory
Formalizing set theory, means describing it as a theory with its notions,
structures and axioms. We shall admit 3 primitive notions : elements (all
objects), sets and functions. Their main primitive structures are introduced
below. Most other primitive symbols and axioms will be presented in 1.7,1.8, 2.1
and 2.2, in a dedicated logical framework, convertible into firstorder logic by a
procedure also described in 2.1. Still more primitive components will be needed
and added later (2.3, 2.6, 2.7, 4.3). Optional ones, such as the axiom of
choice (2.12), will open a diversity of possible set theories.
This view of set theory as described by (one)model theory, relates the terminologies
of both theories in a different way than when interpreting generic theories in set theory.
As the set theoretical notions (sets, functions...) need to keep their natural names when
defined by this formalization, it would become incorrect to keep that terminology
for their use in the sense of the previous link (where notions were "sets" and
operators were "operations"). To avoid confusion, let us here only use the model theoretical
notions as our conceptual framework, ignoring their set theoretical interpretations.
We shall describe in 1.7 and 1.B how both links can be put together, and
how both ways to conceive the same theories (describing them by model
theory or using a set theoretical interpretation) can be reconciled.
One aspect of the role of sets is given by the binary predicate ∈ of belonging :
for any element x and any set E, we say that x is in E
(or x belongs to E, or x is an element of E, or
E contains x) and write x ∈ E, to mean that
x is a possible value of the variables with range E.
Functions f play their role by two operators: the domain
functor Dom, and the function evaluator, binary operator
that is implicit in the notation f(x), with arguments
f and x, giving the value of any function f
at any element x of Dom f.
About ZFC set theory
The ZermeloFraenkel set theory (ZF, or ZFC with the axiom of choice)
is a generic theory with only one type «set», one structure symbol ∈ ,
and axioms. It implicitly assumes that every object is a set, and thus a set
of sets and so on, built over the empty set.
As a rather simply expressible
but very powerful set theory for an enlarged founding cycle, it can be a good
choice indeed for specialists of mathematical logic to conveniently prove diverse
difficult foundational theorems, such as the unprovability of some statements,
while giving them a scope that is arguably among the best conceivable ones.
But despite the habit of authors of basic math courses to conceive their
presentation of set theory as a popularized or implicit version of ZF(C), it is
actually not an ideal reference for a start of mathematics for beginners:
 It cannot be selfcontained as it must assume the framework of model theory to
make sense.
 Its axioms, usually just admitted (as either intuitive,
obvious, necessary or just historically selected for their consistency and the
convenience of their consequences), would actually deserve more subtle and
complex justifications, which cannot find place at a starting point.
 Ordinary mathematics, using many objects usually not seen as sets,
are only inelegantly developed from this basis. As the roles of all needed
objects can anyway be indirectly played by sets, they did not require another
formalization, but remained cases of discrepancy between the
«theory» and the practice of mathematics.
The complexity and weirdness of these needed developments
do not disturb specialists just because once known possible, they
can simply be taken for granted.
Formalizing types and structures as objects of onemodel theory
To formalize onemodel theory through the use of the meta prefix,
both metanotions of "types" and "structures" are given their roles by
metastructures as follows.
Since onemodel theory assumes a fixed model, it only needs one
metatype of "types" to play both roles of abstracts types (in the theory)
and interpreted types (components of the model), respectively given by
two metafunctors: one from variables to types, and one from objects
to types. Indeed the more general notion of «set of objects» is not used
and can be ignored.
But the metanotion of structure will have to remain distinct from the language,
because more structures beyond those named in the language will be involved (1.5).
Structures will get their roles as operations, from metastructures similar to
the function evaluator (see 3.13.2 for clues), while the
language (set of structure symbols) will be interpreted there by a metafunctor from
structure symbols to structures.
However, this mere formalization would leave
undetermined the range of this notion of structure. Trying to conceive this range as that of
«all operations between interpreted types» would leave unknown the source
of knowledge of such a totality. This idea of totality will be formalized in set
theory as the powerset (2.7), but its meaning will still depend on the
universe where it is interpreted, far from our present concern for onemodel theory.
1.5. Expressions and definable structures
Terms and formulas
Given the first two layers of a theory (a list of types and a language), an
expression (that is either a term or a formula), is a finite system of occurrences
of symbols, where an occurrence of a symbol in an expression is a place where
that symbol is written (for example the term « x+x » has two occurrences
of x and one of +).
Each expression comes in the context of a given list of available free variables.
In expressions of firstorder theories and set theory, symbols of the following kinds may occur.
 Variables of each type:
 Free variables, from the list of available ones ;
 Bound variables, whose occurrences are contained by binders (see 1.8) ;
 Paraoperator
symbols:
 Structure symbols from the language (operators and predicates) ;
 One equality symbol per type (predicate with 2 arguments of
the same type) abusively all written = ;
 Logical connectives (1.4, listed in 1.6) ;

The conditional operator may
be introduced for abbreviation (2.6).
 Binders (1.8):
 Quantifiers
∀ and ∃ (1.10) are the only primitive binders of firstorder logic ;
 More binders will be introduced in set theory.
Any expression will give (define) a value (either an object or a Boolean)
for each possible data of
 A system interpreting the given types and structure symbols;
 Fixed values of available free variables in this system.
In firstorder logic, let us call logical symbols the quantifiers and symbols of
paraoperators outside the language (equality, connectives and conditional operator):
their list and their meaning in each system are determined by the logical framework and
the given types list, which is why they are not listed as components of individual theories.
Let us sketch a more precise description (the case of expressions with only free
variables and operator symbols, called algebraic terms, will be formalized in
set theory in 4.1 for only one type).
Each expression contains a special occurrence of a symbol called its root,
while each other occurrence is the root of a unique subexpression
(another expression which we may call the subexpression of that occurrence).
The type of an expression, which will be the type of its values, is given
by the type of result of
its root. Expressions with Boolean type are formulas; others, whose type belongs
to the given types list, are terms (their values will be objects).
Expressions are built successively, in parallel between different lists of available free variables.
The first and simplest ones are made of just one symbol (as root, having a value by
itself) : constants and variables are the first terms; the
Boolean constants 1 and 0 are the simplest formulas.
The next expressions are then successively built as made of the
following data:
 A choice of root, occurrence of either a paraoperator symbol (beyond constants
we already mentioned) or a binder;
 If the root is a binder: a choice of variable symbol, to be bound by it;
 A list of previously built expressions, whose format (number and
types of entries) is determined by the root : for a
paraoperator symbol, this format is given by its list of its arguments.
Display conventions
The display of this list of subexpressions directly attached to the root requires a
choice of convention. For a paraoperator symbol other than constants :
 Most binary paraoperator symbols are displayed as one
character between (separating) both arguments, such as in x+y
 Symbols with higher arities can be similarly displayed by
several characters separating the entries, such as in the addition
x+y+z of 3 numbers.
 Functionlike displays, such as +(x,y) instead of
x+y, are more usual for arities other than 2 ;
parenthesis may be omitted when arities are known (Polish notation).

A few symbols «appear» only implicitly by their special way of putting
their arguments together : multiplication in xy, exponentiation in
x^{n}.

Parenthesis can be part of the notation of a
symbol (function evaluator, tuples...).
Parenthesis can also be used to distinguish (separate) the subexpressions, thus
distinguish the root of each expression from other occurring symbols. For example
the root of (x+y)^{n} is the exponentiation operator.
Variable structures
Usually, only few objects are named by the constants in a given language.
Any other objects can be named by a fixed variable, whose status
depends on the choice of theory to see it:
 An ordinary variable symbol, usable by expressions which by a binder
can let it range over all objects of its type;
 A new constant symbol, to be added to the language, forming another
theory with a richer language;
The difference vanishes in generic interpretations which turn
constant symbols into variables (whose values define different models).
By similarity to constants which are particular structures (nullary operators),
the concept of variable can be generalized to that of variable structure.
But those beyond nullary operations (ordinary variables) escape the above list
of allowed symbols in expressions. Still some specific kinds of use of variable
structure symbols can be justified as abbreviations (indirect descriptions)
of the use of legitimate expressions. The main case of this is explained below,
forming a development
of the theory ; more possible uses will be introduced in 1.10 (the view of a
use as an abbreviation of other works amounts to using a developed version of
the logical framework).
Structures defined by expressions
In any theory, one can legitimately introduce a symbol meant either as a variable
structure (operator or predicate) or a Boolean variable (nullary predicate, not a
"structure"), as abbreviation of, thus defined by, the following data :
 An expression (terms define operators, while formulas define predicates and Booleans);
 Among its available free variables, a selection of those which will be bound
by this definition in the role of arguments of the intended structure;
the rest of them, which remain free, are called parameters;
 Each of its possible values as a structure or a Boolean comes by
fixing the values of parameters.
The variability of this structure is the abbreviation of the variability of all its parameters.
In set theory, any function f is synonymous with the functor defined by the term
f(x) with argument x and parameter f (but the domain of this
functor is Dom f instead of a type).
The terms without argument define simple objects (nullary operators) ; the one made
of a variable of a given type, seen as parameter, suffices to give all objects of its type.
Now let us declare (the range of) the metanotion of "structure" in onemodel theory,
and thus those of "operator" and "predicate", as having to include at least
all those reachable in this way: defined by any expression with any possible
values of parameters. The minimal version of such a metanotion can be
formalized as a role given to the data of an expression with values of its
parameters. As this involves the infinite set of all expressions, it is usually
inaccessible by the described theory itself : no single expression can suffice.
Still when interpreting this in set theory, more operations between interpreted
types (undefinable ones) usually exist in the universe. Among the few
exceptions, the full set theoretical range of a variable structure with all
arguments ranging over finite sets (as interpreted types) with given size
limits, can be reached by one expression whose size depends on these limits.
Invariant structures
An invariant structure of a given system (interpreted language), is a
structure defined from its language without parameters (thus a constant one).
This distinction of invariant structures from other structures, generalizes
the distinction between constants and variables, both to cases of
nonzero arity, and to what can be defined by expressions instead of
directly named in the language. Indeed any structure named by a symbol
in the language is directly defined by it without parameter, and thus invariant.
As will be further discussed in 4.8, theories can be developed by definitions,
which consists in naming another invariant structure by a new symbol added to
the language. Among aspects of the preserved meaning of the theory,
are the metanotions of structure, invariant structure, and the range of theorems
expressible with the previous language.
1.6. Logical connectives
We defined earlier the concept of
logical connective. Let us now list the main useful ones, beyond both nullary ones
(Boolean constants) 1 and 0. (To this will be added
the conditional connective in 2.1).
Tautologies
Their properties will be expressed by tautologies, which are formulas only
involving connectives and Boolean
variables (here written A, B, C), and true for all possible
combinations of values of these variables.
So, they also give necessarily true formulas when replacing these variables by any defining formulas using any
language and interpreted in any systems. Such definitions of Boolean variables by
formulas of a theory may restrict their ranges of possible values depending on each other.
Tautologies form the rules of Boolean algebra, an algebraic theory describing
operations on the Boolean type, naturally interpreted as the pair of elements 0 and 1 but also
admitting more sophisticated interpretations beyond the scope of this chapter.
The binary connective of equality between Booleans is written ⇔ and
called equivalence: A ⇔ B is read «A
is equivalent to B».
Negation
The only useful unary connective is the negation ¬, that
exchanges Booleans (¬A is read «not A»):
It is often denoted by barring the root of its argument, forming
with it another symbol with the same format:
x ≠ y
x ∉ E (A ⇎ B)

⇔ ¬(x = y)
⇔ ¬(x ∈ E) ⇔ (A ⇔ ¬B)

(x is not equal to y)
(x is not an element of E)
(inequivalence) 
Conjunctions, disjunctions
The conjunction ∧ means «and», being true only when both
arguments are true;
The disjunction ∨ means «or», being true except when both
arguments are false.
Each of them is :
Idempotent
(A ∧ A) ⇔ A
(A ∨ A) ⇔ A

Commutative
(B∧A) ⇔ (A∧B)
(B∨A) ⇔ (A∨B)

Associative
((A∧B)∧C) ⇔ (A∧(B∧C))
((A∨B)∨C) ⇔ (A∨(B∨C))

Distributive over the other
(A ∧ (B∨C)) ⇔ ((A∧B)
∨ (A∧C)) (A ∨ (B∧C)) ⇔ ((A∨B)
∧ (A∨C))

This similarity (symmetry) of their properties comes from the fact they are
exchanged by negation:
(A ∨ B)
⇎ (¬A ∧ ¬B)
(A ∧ B) ⇎ (¬A ∨ ¬B)
The inequivalence is
also called exclusive or because (A
⇎ B) ⇔ ((A ∨ B) ∧ ¬(A
∧ B)).
Chains of conjunctions such as (A ∧ B ∧ C),
abbreviate any formula with more parenthesis such as ((A ∧ B)
∧ C), all equivalent by
associativity ; similarly for chains of disjunctions such as (A
∨ B ∨ C).
Asserting (declaring as true) a conjunction of formulas amounts to successively
asserting all these formulas.
Implication
The binary connective of implication ⇒ is defined as (A ⇒ B) ⇔ ((¬A)
∨ B). It can be read «A implies B», «A
is a sufficient condition for B», or «B is a
necessary condition for A». Being true except when A
is true and B is false, it expresses the truth of B
when A is true, but no more gives information on B
when A is false (as it is then true).
Moreover,
(A ⇒ B) ⇎
(A ⇒ B) ⇔ 
(A ∧ ¬B) (¬B ⇒ ¬A) 
The formula ¬B ⇒ ¬A is called the contrapositive
of A ⇒ B.
The equivalence can also be redefined as
(A ⇔ B) ⇔ ((A ⇒ B) ∧ (B ⇒ A)).
Thus in a given theory, a proof of A ⇔ B can be
formed of a proof of the first implication (A ⇒ B), then a
proof of the second one (B ⇒ A), called the converse of (A
⇒ B).
The formula A ∧ (A ⇒ B) is equivalent to A ∧ B
but will be written A ∴ B, which reads «A therefore B», to
indicate that it is deduced from the truths of A and A ⇒ B.
Negations turn the associativity and distributivity of ∧ and ∨, into various
tautologies involving implications:
(A ⇒ (B ⇒ C)) ⇔ ((A ∧ B) ⇒ C)
(A ⇒ (B ∨ C))
⇔ ((A ⇒ B) ∨ C)
(A ⇒ (B ∧ C)) ⇔ ((A ⇒ B) ∧ (A ⇒ C))
((A ∨ B) ⇒ C) ⇔
((A ⇒ C) ∧ (B ⇒ C))
((A ⇒ B) ⇒ C) ⇔
((A ∨ C) ∧ (B ⇒ C))
(A ∧ (B ⇒ C)) ⇔
((A ⇒ B) ⇒ (A ∧ C))
Finally, (A ⇒ B)
⇒ ((A∧C) ⇒ (B∧C))
(A ⇒ B) ⇒
((A∨C) ⇒ (B∨C)).
Chains of implications and equivalences
In a different kind of abbreviation, any chain of formulas
linked by ⇔ and/or ⇒ will mean the chain of conjunctions of these
implications or equivalences between adjacent formulas:
(A ⇒ B
⇒ C) ⇔ ((A ⇒ B) ∧ (B ⇒ C)) ⇒
(A ⇒ C)
(A ⇔ B ⇔ C) ⇔ ((A ⇔ B) ∧ (B
⇔ C)) ⇒ (A ⇔ C)
0 ⇒ A ⇒ A ⇒ 1
(¬A) ⇔ (A ⇒ 0) ⇔ (A ⇔ 0)
(A ∧ 1) ⇔ A ⇔ (A ∨ 0) ⇔ (1 ⇒ A) ⇔ (A
⇔ 1)
(A ∧ B) ⇒ A ⇒ (A ∨ B)
1.7. Classes in set theory
In any system, a class is a unary
predicate A seen as the set of objects where A is true, that is «the
class of all x such that A(x)».
In a set theoretical universe,
each set E is synonymous with the class of the x such that x∈E
(defined by the formula x∈E with argument x and parameter E).
However, this involves two different interpretations of the notion of set,
that need to be distinguished as follows.
Standard universes and metasets
Interpreting (inserting)
set theory into itself, involves articulating two interpretations (models) of set theory, which
will be distinguished by giving the meta prefix to the one used as framework. Aside generic
interpretations, set theory has a standard kind of interpretation into itself, where each
set is interpreted by the class (metaset) of its elements (the synonymous object and
metaobject are now equal), and each function is interpreted by its synonymous metafunction.
This way, any set will be a class, while any class is a metaset of objects. But some
metasets of objects are not classes (no formula with parameters can define them);
and some classes are not sets, such as the class of all sets (see Russell's
paradox in 1.8), and the universe (class of all objects, defined by 1).
Definiteness classes
Set theory accepts all objects as «elements» that can belong to sets and be
operated by functions (to avoid endless further divisions between sets of
elements, sets of sets, sets of functions, mixed sets...). This might be formalized
keeping 3 types (elements, sets and functions), where each set would have a
copy as element, identified by a functor from sets to elements, and the same for
functions. But beyond these types, our set theory will anyway need more
notions as domains of its structures, which can only be conveniently
formalized as classes. So, the notions
of set and function will also be classes named by predicate symbols:
Set = «is a set»
Fnc = «is a function»
In firstorder logic, any expression is ensured to take a definite value, for
every data of a model and values of all free variables there (by virtue of its
syntactic correction, that is implicit in the concept of «expression»).
But in set theory, this may still depend on the values of free variables.
So, an expression A (and any structure defined from it) will be called
definite, if it actually takes a value for the given values of its free variables
(seen as arguments and parameters of any structure it defines). This condition
is itself an everywhere definite
predicate, expressed by a formula dA with the same free variables.
Choosing one of these as argument, the class it defines is the metadomain,
called class of definiteness, of the unary structure defined by A.
Expressions should be only used where they are definite, which will be done
rather naturally. The definiteness condition of (x ∈ E) is Set(E).
That of the function evaluator f(x) is Fnc(f) ∧
x ∈ Dom f.
But the definiteness of the last formula needs a justification, given below.
Extended definiteness
A theory with partially definite structures, like set theory, can be formalized (translated) as
a theory with one type and everywhere definite structures, keeping intact all expressions
and their values wherever they are definite : models are translated one way by giving
arbitrary values to indefinite structures (e.g. a constant value), and in the way back by
ignoring those values. Thus, an expression with an indefinite subexpression may be
declared definite if its final value does not depend on these extra values.
In particular for any formulas A and B, we shall regard the formulas
A ∧ B and A ⇒ B as definite if A
is false, with respective values 0 and 1, even if B is not definite. So,
let us give them the same definiteness condition
dA ∧ (A ⇒ dB)
(breaking, for A ∧ B, the symmetry between A
and B, that needs not be restored). This formula is made definite by the
same rule provided that dA and dB were definite. This way, both formulas
A ∧ (B ∧ C)
(A ∧ B) ∧ C
have the same definiteness condition (dA ∧
(A ⇒ (dB ∧ (B ⇒ dC)))).
Classes will be defined by everywhere definite predicates, easily expressible by
the above rule as follows.
Any predicate A can be extended beyond its
domain of definiteness, in the form dA ∧ A (giving 0), or
dA ⇒ A (giving 1).
For any class A and any unary predicate B definite in all A, the class defined
by A∧B is called the subclass of A
defined by B.
1.8. Binders in set theory
The syntax of binders
This last kind of symbol can form
an expression by taking a variable symbol, say here x, and an
expression F which may use x as a free variable (in addition to the free
variables that are available outside), to give a value depending on the unary structure
defined by F with argument x. Thus, it separates the «inside»
subexpression F having x among its free variables, from the «outside»
where x is bound. But in most cases (in most theories...),
binders cannot keep the full information on this unary structure, which is too complex
to be recorded as an object as we shall see below.
We shall first review both main binders of set theory : the setbuilder and the function
definer. Then 1.10 will present both main quantifiers. Finally 2.1 and 2.2 will
give axioms to complete this formalization of the notions of sets and functions in set theory.
The syntax differs between firstorder logic and set theory, which manage the
ranges of variables differently. In firstorder logic, ranges are types,
implicit data of quantifiers. But the ranges of binders of set theory are sets
which, as objects, are designated by an additional argument of the binder
(a space for a term not using the variable being bound).
Setbuilder
For any unary predicate A definite on all elements of a set E, the
subclass
of E defined by A is a set : it is the range of a
variable x introduced as ranging over E, so that it
can be bound, from which we select the values satisfying
A(x). It is thus a subset of E, written
{x∈E  A(x)} (set of all x
in E such that A(x)): for all y,
y ∈ {x∈E  A(x)}
⇔ (y ∈ E ∧ A(y))
This combination of characters { ∈  } forms the notation of a binder
named the setbuilder: {x∈E  A(x)}
binds x with range E on the formula A.
Russell's paradox
If the universe (class of all elements) was a set then, using it, the set
builder could turn all other classes, such as the class of all sets, into sets.
But this is impossible as can be proven using the setbuilder itself :
Theorem. For any set E there is a set F
such that F ∉ E. So, no set E can contains all sets.
Proof. F={x∈E  Set(x) ∧ x ∉ x}
⇒ (F ∈ F ⇔ (F ∈ E ∧ F ∉ F))
⇒ (F ∉ F ∴ F ∉ E). ∎
This will oblige us to keep the distinctions between sets and
classes.
The function definer
The function definer ( ∋ ↦ ) binds a variable on a term,
following the syntax E ∋ x ↦ t(x),
where  x is the variable,
 E is its range,
 the notation
"t(x)" stands for any term, here abbreviated (to describe
the general case) using the functor symbol t defined by this term with
argument x (and possible parameters here kept implicit).
Being definite if t(x) is definite for all x in E,
it takes then the functor t and restricts its domain (definiteness class) to the
set E, to give a function with domain E. So it converts functors into
functions, reversing the action of the function evaluator (with the Dom functor)
that converted (interpreted) functions into their role (meaning) as functors
whose definiteness classes were sets.
The shorter notation x ↦ t(x) may be used when
E is determined by the context, or in a meta description to designate a
functor by specifying the argument x of its defining term.
Relations
A relation is a role playing object of set theory similar to an operation
but with Boolean values : it acts as a predicate whose definiteness
classes (ranges of arguments) are sets (so, predicates could be
described as relations between interpreted types). Now unary relations
(functions with Boolean values), will be represented as subsets S
of their domain E, using the setbuilder in the role of definer, and ∈
in the role of evaluator interpreting S as the predicate
x ↦ (x ∈ S).
This role of S still differs from the intended unary relation, as it ignores its domain
E but is definite in the whole universe, giving 0 outside E.
This lack of operator Dom does not matter, as E is usually known
from the context (as an available variable).
As the function definer (resp. the setbuilder) records the whole
structure defined by the given expression on the given set, it
suffices to define any other binder of set theory
on the same expression with the
same domain, as made of a unary structure applied to its
result (that is a function, resp. a set).
1.9. Axioms and proofs
Statements
An expression is ground if its list of available free variables is empty (all its variables are
bound), so that its value only depends on the system where it is interpreted.
In firstorder logic, a statement is a ground formula. Thus, it will have a definite Boolean
value (true or false) only depending on the choice of a system that interprets its
language.
The axioms list of a theory is a set of statements, meant to be all true in some
given system(s) called models of the theory (as discussed below).
Provability
A proof of a statement A in a theory T, is a finite
model of proof theory, having A as "conclusion" and involving a finite list of axioms
among those of T.
As explained later (and as precisely studied by specialists) the concept of proof can be fully
formalized (a proof theory can be precisely written), which can take the form of a proof verification
algorithm (only requiring an amount of computing resources related to the size of a given proof). But most
mathematical works are only partially formalized : semiformal proofs are used with just enough
rigor to give the feeling that a full formalization is possible, without actually writing it ; an intuitive
vision of a problem may seem clearer than a formal reasoning. Likewise, this work will present proofs
naturally without explicit full formalization : sometimes using natural language articulations,
proofs will mainly be written as successions of statements each visibly true thanks to
previous ones, premises, axioms, known theorems and rules of proof, especially those of
quantifiers (1.10).
Yet without giving details of proof theory, let us review some general properties.
We say that A is provable in T, or a theorem of T, and
write T ⊢ A if a proof of A in T exists. This is the halting condition of a proof
searching algorithm, processed by an abstract "computer" with unlimited (infinite)
available time and resources, trying every possible piece of proof until it eventually finds
a valid proof of the given statement.
In practice,
we only qualify as theorems the statements known as such, i.e. for which a proof is known.
But synonyms for "theorem" are traditionally used according to their importance: a theorem
is more important than a proposition; either of them may be deduced from an
otherwise less important lemma, and easily implies an also less important corollary.
Any good proof theory needs of course to be sound, which means only "proving" always true
statements : provability implies truth in every model (where all axioms are true).
Logical validity
For a given language L, a statement A is called logically valid, which is written
⊢ A, if it is provable with L, without using any axiom (thus a theorem of any theory containing
L, regardless of axioms). Then A is true in every system interpreting L, thanks to the
soundness of the logical framework. The simplest logically valid statements are the tautologies (whose Boolean variables are replaced
by statements); others will be given in 1.10.
A proof of A using some axioms can also be seen as a proof of (conjunction of these axioms
⇒ A) without axiom, thus making this implication logically valid.
Refutability and consistency
A refutation of A in T, is a proof of ¬A. If one exists
(T ⊢ ¬A), the statement A is called refutable (in T).
A statement is called decidable (in T) if it is provable or refutable.
A contradiction
of a theory T is a proof of 0 in T. If one exists (T ⊢ 0), the theory T is called contradictory or inconsistent ; otherwise it is called consistent.
If a statement is both provable and
refutable in T then all are, because it means that T is inconsistent, independently
of the chosen statement:
⊢ (A ∧ ¬A) ⇔ 0
((T ⊢ A) ∧ (T ⊢ B))
⇔ (T ⊢ A∧B)
((T ⊢ A) ∧ (T ⊢ ¬A)) ⇔ (T ⊢ 0).
In an inconsistent theory, every statement is provable. Its framework being sound, such a theory has no model.
A refutation of A in T can be equivalently seen as a contradiction of the theory T∧A obtained
by adding A to the axioms of T.
Realistic vs. axiomatic theories in mathematics and other sciences
Apart from the distinction of nature
(mathematical vs. nonmathematical), theories
may also differ by the intention of their use, between realism and formalism.
An axiomatic theory is a theory given formally with an axioms list that means
to define its range of models, as the class of all systems, interpreting the given
language, where all axioms are true (rejecting those where some axiom is false).
This makes automatic the truth of axioms in any model.
Nonrealistic theories outside mathematics (not called "axiomatic" by lack of mathematical form)
would be works of fiction describing imaginary or possible future systems.
A realistic theory is a theory involved to describe either a fixed system or
the systems from a range, seen as given from some independent reality. Its given axioms
are statements which, for some reason, are considered known as true on all these systems.
Such a theory is true if all its axioms are indeed true there. In other words, these
systems are models, qualified as standard for contrast with other, unintended
models of that theory taken axiomatically.
This truth will usually be ensured for realistic theories of pure mathematics : arithmetic
and set theory (though the realistic meaning of set theory will not always be clear
depending on considered axioms). These theories will also admit nonstandard
models, making an effective difference between their realistic and axiomatic meanings.
Outside pure mathematics, the truth of realistic theories may be dubious (questionable):
nonmathematical statements over nonmathematical systems may be ambiguous (illdefined),
while the truth of theories of applied mathematics may be approximative, or speculative as the
intended "real" systems may be unknown (contingent among other possible ones). There, a theory
is called falsifiable
if, in principle, the case of its falsity can be discovered by comparing its
predictions (theorems) with observations. For example, biology is relative to a huge number of
random choices silently accumulated by Nature on Earth during billions of years ; it
has lots of "axioms" which are falsifiable and require a lot of empirical testing.
Euclidean geometry was first conceived as a realistic theory of applied mathematics
(for its role of first theory of physics),
then became understood as an axiomatic theory of pure mathematics among diverse other,
equally legitimate geometries in a mathematical sense; while the real physical
space is more accurately described by the nonEuclidean geometry of General Relativity.
1.10. Quantifiers
A quantifier is a binder taking a unary predicate (formula) and giving a
Boolean value.
In set theory, the full syntax for a quantifier Q binding a
variable x with range E on a unary predicate A,
is
Qx∈E, A(x)
where A(x) abbreviates the formula defining A,
whose free variables are x and possible parameters.
A shorter notation puts the range as an index (Q_{E}x,
A(x)), or deletes it altogether (Qx, A(x))
when it may be kept implicit (unimportant, or fixed by the context, such
as a type in a generic theory).
The two main quantifiers (from which others will be defined later) are:
 The universal quantifier ∀, read as «For all x (in...),... »).

The existential quantifier ∃, read as «There exists x (in...) such that... »
The universal quantifier of set theory can be seen as defined from the set builder:
(∀x∈E, A(x)) ⇔ {x∈E
 A(x)} = E.
The one of firstorder logic can be defined in set theoretical interpretations,
seeing A as a function and its Boolean values as objects:
(∀x, A(x)) ⇔ A = (x ↦ 1)
Anyway (∀x, 1) is always true.
∃ can be defined from ∀ with the same range :
(∃x, A(x))
⇎ (∀x, ¬A(x)).
Thus (∃x, A(x)) ⇔
A ≠ (x ↦ 0).
With classes,
(∃_{C } x, A(x)) 
⇔ (∃x, C(x) ∧ A(x))
⇔ ∃_{C∧A} x, 1 
(∀_{C } x, A(x)) 
⇔ (∀x, C(x) ⇒ A(x)) 
∀x, C(x)

⇔ ∃_{C } y, x=y

Inclusion between classes
A class A is said to be included in a class
B when ∀x, A(x) ⇒ B(x). Then
A is a
subclass of B, as ∀x, A(x) ⇔ (B(x)
∧ A(x)). Conversely, any subclass of B is
included in B.
The inclusion of A in B
implies for any predicate C (in cases of definiteness):
(∀_{B}
x, C(x)) ⇒ (∀_{A} x,
C(x))
(∃_{A} x, C(x)) ⇒ (∃_{B}
x, C(x))
(∃_{C }x, A(x)) ⇒ (∃_{C
}x, B(x))
(∀_{C }x, A(x)) ⇒ (∀_{C
}x, B(x))
Rules of proofs for quantifiers on a unary predicate
Just like expressions were described by allowing to take already made expressions
to form new ones, the concept of proof may be formalized by using already known proofs
to form new ones. Here are some intuitively introduced "rules", still without claiming to
fully formalize proofs.
Existential Introduction. If we have terms t,
t′,… and a proof of (A(t) ∨ A(t′)
∨ …), then ∃x, A(x).
Existential Elimination. If ∃x, A(x),
then we can introduce a new free variable z with the
hypothesis A(z) (the consequences will be
true without restricting the generality).
These rules express the meaning of ∃ : going from some term to ∃ then from
∃ to z, amounts to letting z represent that term
(without knowing which, but they can be gathered into one by the conditional operator). They give
the same meaning to ∃_{C } as to its 2 above
equivalent formulas, bypassing (making implicit) the extended
definiteness rule for (C ∧ A) by focusing on the
case when C(x) is true and thus A(x)
is definite.
While ∃ appeared as the designation of an object, ∀ appears as a
deduction rule: ∀_{C }x, A(x)
means that its negation ∃_{C }x, ¬A(x)
leads to a contradiction.
Universal Introduction. If from the mere hypothesis C(x)
on a new free variable x we could deduce A(x),
then ∀_{C }x, A(x).
Universal Elimination. If ∀_{C }x,
A(x) and t is a term satisfying C(t),
then A(t).
Introducing then eliminating ∀ is like replacing x by t
in the initial proof.
These rules can be used to justify the following logically valid formulas
((∀_{C }x, A(x))
∧ (∀_{C }x, A(x) ⇒ B(x)))
⇒ (∀_{C }x, B(x))
((∃_{C }x, A(x))
∧ (∀_{C }x, A(x) ⇒ B(x)))
⇒ (∃_{C }x, B(x))
((∀_{C }x, A(x))
∧ (∃_{C }x, B(x))) ⇒
(∃_{C }x, A(x) ∧
B(x))
(∀_{C }x,
A(x)∨B(x)) ⇒ ((∀_{C }x,
A(x)) ∨ (∃_{C }x,
B(x)))
(∃_{A} x, ∀_{B}
y, R(x,y)) ⇒ (∀_{B} y,
∃_{A} x, R(x,y))
∀x, (∀y, R(x,y)) ⇒
R(x,x) ⇒ (∃y, R(x,y))
The formula (∃_{C }x, A(x)) ∧
(∀_{C }x, A(x) ⇒ B(x))
will be abbreviated as (∃_{C }x, A(x) ∴
B(x)) while it implies but is not generally equivalent to
(∃_{C }x, A(x) ∧ B(x)).
Completeness of firstorder logic
Beyond the required quality of soundness of the proof theoretical part of a logical framework,
more remarkable is its converse quality of completeness : that for any axiomatic theory
it describes, any statement that is true in all models is provable. In other words, any unprovable
statement is false somewhere, and any irrefutable statement is true somewhere. Thus, any
consistent theory has existing models, but often a diversity of them, as any undecidable
statement is true in some and false in others. Adding some chosen undecidable statements
to axioms leads to different consistent theories which can «disagree» without conflict, all
truly describing different existing systems. Firstorder logic was found complete
as expressed by the completeness
theorem, which was originally Gödel's
thesis : from a suitably formalized concept of «proof», the resulting class of (potential)
theorems of any firstorder theory was found to be the "perfect" one, coinciding with the
class of universally true statements (true in all models). This quality of firstorder logic,
comes in addition with its ability to express all mathematics : any more powerful
logical framework can anyway be developed from set theory (or more directly its theories
can be interpreted in set theory), itself translatable as a firstorder theory.
Hence its central importance in the foundations of mathematics : its
completeness resolves much of a priori divergence between
Platonism and formalism
for firstorder axiomatic theories, while giving proper mathematical definiteness to the a priori
intuitive concepts of "proof", "theorem" and "consistency".
Such a proof theory for firstorder logic is essentially unique: the equivalence between
any two sound and complete proof theories as concerns the existence
of a proof in any theory T, concretely
appears by the possibility to translate any "proof" for the one into a "proof" for the other.
Among the possible formalizations of proofs, are formulas V(p,s) of
firstorder arithmetic (the firstorder
theory describing the system ℕ of natural numbers with four symbols 0, 1, +, ⋅ with axioms
listed in 4.3 and 4.4), meaning that the number p encodes a valid proof of a
statement encoded by the number s, depending on unary predicate symbols
t,l,a which encode the classes of components of T
(its types, symbols and axioms ; the induction axioms of arithmetic would need to
admit the use of symbols t,l,a if these symbols
were not definable, while they usually are since we only consider definable theories).
The resulting provability formula (∃p, V(p,s)) is
independent of the chosen sound and complete way to define
V, in the sense that all such provability formulas are provably equivalent to each other
in firstorder arithmetic. More formally, for any two such arithmetical formulas V, V' encoding
proof verification in the same theory,
(Arithmetic + symbols and axioms for t,l,a) ⊢ ∀s, (∃p,
V(p,s)) ⇔ (∃p,
V'(p,s))
However to complete the definition of provability, once written as a formula of this kind,
remains the issue of interpreting the ∃p realistically,
in the standard model of arithmetic ℕ, intuitively described as the system
of only all actually finite natural numbers, as will be clarified in Part 4. This use of
the standard ℕ can be intuitively understood as the use of actual infinity, needed by
lack of predictable limits on the needed (minimal) sizes of proofs for given statements.
Indeed as will be
explained in 1.A, incompleteness theorems will undermine this definiteness in both
ways : establishing both the unpredictability of the size of needed proofs,
and the irreducible undecidability of some provability predicates in any algorithmic
formalization of arithmetic (while the realistic view sees them all as definite).
The proof of the completeness theorem, which requires the axiom of infinity (existence
of ℕ) will consist in building a model of any consistent firstorder axiomatic theory, as follows
(details in 4.6). The (infinite) set of all ground terms with operation symbols derived from the theory
(those of its language plus others coming from its existence axioms), is turned into a model
by progressively defining each predicate symbol over each combination of values of its
arguments there, by a rule designed to keep consistency.
This construction is nonalgorithmic : it is made of an infinity of steps, each of which involving
an infinite knowledge (whether the given predicate on given arguments, seen as a candidate
additional axiom, preserves consistency with previously accepted ones).
Actually, most foundational theories such as set theories,
do not have any algorithmically definable model.
1.11. Secondorder universal quantifiers
Let us call secondorder quantifier, a quantifier binding a variable structure symbol
over the range of all structures of its symbol
type, may this be conceived as the
range of all definable ones (with all possible defining expressions whose free variables may have
any list of parameters beyond the given arguments) or as the full set theoretical
range, that is the range of all such structures which exist in the universe, relating the given interpreted types.
The use of such a quantifier (and thus of variable structure symbols) is not allowed in firstorder logic,
but belongs to some other logical frameworks instead, such as secondorder logic (part 5).
Still while keeping firstorder logic as the main framework of a given theory,
some secondorder quantifiers may be used to describe some of its
meta level aspects in the following ways (which will be
involved in the formalization of set theory in 2.1 and 2.2). Let T be
a firstorder theory, T' its extension by a structure symbol s
(without further axiom) and F a ground formula of T' (in firstorder logic)
also denoted F(s) when seen as a formula of T using the
variable structure symbol s in secondorder logic.
Secondorder Universal Introduction. If T'⊢F then T
entails the secondorder statement (∀s, F(s)).
This holds for any model and the full set theoretical range of s, independently of
the universe in which models and structures s are searched for.
Secondorder Universal Elimination. Once a secondorder statement
(∀s, F(s)) is accepted in a theory T,
it is manifested in firstorder logic as a schema of statements,
that is an infinite list of statements of the form (∀parameters, F(s)) for each
possible replacement of s by a defining expression with parameters.
Applying secondorder universal elimination after secondorder universal introduction,
means deducing from T a schema of theorems, each one indeed deducible
in firstorder logic by the proof obtained from the original proof by replacing s by its definition.
In secondorder logic, a new binder B can be defined by an expression here abbreviated
as F(A) using a symbol A of variable unary structure whose argument will be
bound by B:
∀A, (Bx, A(x)) = F(A)
By secondorder universal elimination, this comes down to a schema of definitions in firstorder logic :
for each expression defining A, it defines (Bx, A(x)) like a structure symbol,
by the expression F(A) whose available free variables are the parameters
of F plus those of A.
Axioms of equality
In firstorder logic with given types and a given language, some ground formulas involving
= are logically valid for the range of interpretations keeping = as the = predicate
of set theory, but no more for the larger range of interpretations letting
it become any other predicate. A possible list of axioms of equality, is a list of
some of such formulas which suffice to imply all others in this context. One such list
consists in the following 2 axioms per type, where the latter is meant as an axiom schema
by secondorder universal elimination of the variable unary predicate A:
 ∀x, x = x (reflexivity)

∀A,∀x,∀y, (x = y ∧ A(y)) ⇒ A(x).
Variables x and y can also serve among parameters in definitions of A.
This can be understood by reordering quantifiers as (∀x,
∀y, ∀A), or as deduced from cases only using other free variables a, b, by
adapting an above logically valid formula as ∀a, ∀b,
(∀x, ∀y, R(a, b, x,y)) ⇒ R(a, b, a,b).
Diverse definitions of A give diverse formulas (assuming reflexivity):
Formula
3. ∀x,∀y, x = y ⇒ y = x
4. ∀x, ∀y, ∀z, (x = y ∧ y = z) ⇒
x = z
5. ∀f, ∀x, ∀y, x = y ⇒ f(x) = f(y)
6. ∀A, ∀x, ∀y, x = y ⇒ (A(x) ⇔ A(y))
7. ∀x, ∀y, ∀z, (x = y ∧ z = y) ⇒
z = x 
Name
Symmetry Transitivity

A(u) used
y = u u = z
f(u) = f(y)
¬A(u) z = u 
We shall abbreviate (x = y
∧ y = z) as x = y = z.
5. is an axiom schema with f ranging among functors between any two types.
6. can also be deduced from symmetry.
Remark. (1.∧7.) ⇒ 3., then 3. ⇒ (4. ⇔ 7) so that (1.∧7.) ⇔ (1.∧3.∧4.).
Another possible list of axioms of equality consists in formulas 1. to 5. where f and
A range over the mere symbols of the language, each taken once per argument :
the full scheme of 2. is implied by successive deductions for each occurrence of symbol
in A. This will be further justified in 2.11 (equivalence
relations).
Introducing a variable x defined by a term t by writing (x = t ⇒ ...),
in other words putting the axiom x = t, can be
seen as justified by the above rules in this way : t = t ∴
∃x, (x = t ∴ ...).
Philosophical aspects of the foundations of
mathematics
Let us complete our initiation to the foundations of mathematics
by a more philosophical aspect : how, while independent of our time, the mathematical realm has a flow of its
own time, first in model theory (here), then in set theory (next page and in complements to part 2).
This will further clarify the difference between sets and classes, and bring a full justification for the set generation principle.
1.A. Time in model theory
The time order between interpretations of expressions
In a model, interpretations of expressions depend on each other, thus
come as calculated after each other. This time order follows the order from
subexpressions to expressions containing them.
For example, to make sense of the formula xy+x=3, the free variables
x and y must take values first; then, xy takes a value, obtained by
multiplying them. From this, xy+x takes a value, and then the
whole formula (xy+x=3) takes a Boolean value, depending on those
of x and y. Finally, taking for example the ground formula ∀x,
∃y, xy+x=3, its Boolean value (which is false
in the world of real numbers), «is calculated from» those taken by
the previous formula for all possible values of x and y,
and therefore comes after them.
Interpretations of a finite list of expressions of a theory in a model may be involved in another expression,
either simply by taking them as its subexpressions, or formally describing these expressions as objects.
This encompassing expression is interpreted after them all, but may still belong to the same theory.
But the interpretations of infinitely many expressions (such as all terms, handled as values of a variable)
can be handled by a single one not of the same theory, but only of another theory at a meta level over it
(onemodel theory).
The metaphor of the usual time
I can speak of «what I told about at that time»: it has a sense if
that past saying had one, as I got that meaning and I remember it.
But mentioning «what I mean», would not itself inform on what it is,
as it might be anything, and becomes absurd in a phrase that
modifies or contradicts this meaning («the opposite of what I'm
saying»). Mentioning «what I will mention tomorrow», even if I knew
what I will say, would not suffice to already provide its meaning
either: in case I will mention «what I told about yesterday» (thus
now) it would make a vicious circle; but even if the form of my
future saying ensured that its meaning will exist tomorrow, this would still
not provide it today. I might try to speculate on it, but the actual
meaning of future statements will only arise once actually expressed
in context.
By lack of interest to describe phrases without their
meaning, we may focus on past expressions, while just "living" the present
ones and ignoring future ones. So, my current universe
of the past that I can describe today, includes the one I could describe yesterday,
but also my yesterday's comments about it and their meaning. I can thus describe
today things outside the universe I could describe yesterday. Meanwhile I neither
learned to speak Martian nor acquired a new transcendental intelligence, but
the same language applies to a broader universe with new objects. As
these new objects are of the same kinds as the old ones, my universe
of today may look similar to that of yesterday; but from one
universe to another, the same expressions can take different
meanings.
Like historians, mathematical theories can only «at every given time» describe a
system of past mathematical objects ; set theory itself describes a universe of «all
mathematical objects», which is at any time the current «everything», made of our past.
The act of interpreting expressions there, itself «happens» forming a mathematical
present outside this universe (beyond this past). Then, describing our previous act
of description, means expanding its scope by something else beyond this «everything».
The infinite time between models
As a onemodel
theory T' describes a theory T with a model M,
the components (notions and structures)
of the model [T,M] of T', actually fall into 3
categories:
 The components of T and its developments as a formal
system (abstract types, structure symbols, expressions, axioms,
proofs from axioms), that aim to describe the model but remain
outside it and independent of it.
 The components of M (interpretations of types and
structure symbols)
 The interpretation (attribution of values) of all expressions
of T in M, for any values of their free
variables.
This last part of [T,M] is a mathematical construction
determined by the combination of both systems T and M
but it is not directly contained in them : it is built after them.
So, the model [T,M] of T', encompassing T with the
interpretation of all its formulas in the present model M of past objects,
is the next universe of the past, coming once the infinity of all current interpretations
in M of formulas of T becomes past.
Or can it be otherwise ? Would it be possible for a theory T
to express or simulate the notion of its own formulas and
compute their values ?
Truth undefinability
Some theories T (which we assume welldefined), such as set theory (from which model theory can be developed)
can describe themselves: they can describe in each model M a system structured as a copy of the same theory with a
notion of "its formulas". Equivalently (via some developments), it can describe in M
a system ℕ behaving as a system of natural numbers.
But then in such a theory, "truth is undefinable": the
metapredicate of interpretation (truth) of all its ground formulas in any given model, cannot coincide with any
invariant unary predicate (definition interpreted in the same model) over the copies of these
formulas in the class of all described "ground formulas".
A first proof of this Truth Undefinability theorem
is given below. A more rigorous
proof of a stronger version will be given in Part 5.
The Berry paradox
This famous paradox is the idea of "defining" a natural number n
as, for example "the smallest number not definable in less than twenty words".
This would define in 10 words a number... not definable in less than
20 words. But this does not bring a contradiction in mathematics
because it is not a mathematical definition. Making it more precise, provides
a simple proof of the truth undefinability theorem: Let H be the set of formulas of T with one free
variable intended to range over this ℕ, and shorter than (for example)
1000 occurrences of symbols (taken from the finite list of symbols
of T, logical symbols and variables).
Consider the formula of T' with one free variable n
ranging over ℕ, expressed as
∀F∈H, F(n) ⇒
(∃k<n, F(k))
This formula cannot be false on more than one number per formula in
H, which are only finitely many (an explicit bound of their number
can be found). Thus it must be true on some numbers.
If it was equivalent to some formula B∈H, we would get
∀n∈ℕ, B(n) ⇔ (∀F∈H,
F(n) ⇒ (∃k<n, F(k)))
⇒ (∃k<n, B(k))
contradicting the existence of a smallest n on which B is true.
The number 1000 was picked in case translating this formula into
T was complicated, ending up in a big formula B, but still in
H. If it was so complicated that 1000 symbols didn't suffice, we could
try this reasoning starting from a higher number. Since the existence of an
equivalent formula in H would anyway lead to a contradiction,
no number we might pick can ever suffice to find one. This shows the
impossibility to translate such formulas of T' into equivalent formulas of
T, by any method much more efficient than the kind of mere
enumeration suggested above.
This infinite time between theories, will develop as an endless
hierarchy of infinities.
On the incompleteness theorem
(to be completed)
The size (complexity) of a proof, reflects the number of elements
that need to be described in an attempted construction of a counterexample to a given
theorem, to observe the necessary failure of this construction (while the choice of formalization
only affects the size of proofs in more moderate proportions); this is not a priori
predictable from, say, the size of the statement, for reasons which will be developed later.
As will be shown
by the incompleteness
theorem, unclarities remain about provability and existence of models:
 The proven equivalence between consistency and existence of a model, does not inform
whether these are both true, or both false:
 Provability, if true, can stay
practically unknowable: if a proof exists, nothing says how to find it; it may be
too big to be ever given.
 The consistency of a theory may be true but unprovable: completeness only ensures
that, in this case (and it may be undecidable whether it is the case or not), there also exists
universes in which, equivalently, this theory is inconsistent and it has no models.
For any two welldefined theories (in an algorithmic sense)
T, T' which encompass arithmetic (such as set theories),
no
suffice to make these formulas decidable for every value
of s, as soon as they aim to encode provability in a theory which also contains arithmetic.
The time of proving
If no proof of a statement could be found within given
limited resources, it may still be a theorem whose proofs could not be found as they may be any
longer. This is often unpredictable, for deep theoretical reasons which will appear from the study of the
incompleteness theorem and related ones such as Gödel's
speedup theorem :  It is unlikely to be predicted in advance: intuitively,
a reliable prediction of existence of proof of A would be a small proof of existence of a
larger proof of A, and thus look like a small proof of A itself (though this implication
between provability properties cannot be itself a theorem);
 No amount of vain search can justify to give up, as a theorem may require
indescribably large proofs (thus beyond the resources of any computer or even all the information that our visible universe may contain): by lack of inverse method (construction of proofs from the absence of counter
examples), no limit can be described for the size of the smallest needed proofs for given theorems
(or even logically valid statements beyond tautologies) that would only depend on the size of the
statement, beyond the simplest cases.
Set theory as a unified framework
Structure definers in diverse theories
Let us call structure definer any binder B which, used on
diverse expressions A, faithfully records the unary structure defined
by A on some range E (type or class defined by an argument
here implicit), i.e. its result S = (Bx, A(x)) can
restore this structure by an evaluator V (symbol or expression) :
V(S, x) = A(x) for all x in E.
Admitting the use of negation and the possibility to interpret
Booleans by objects (in a range with at least 2 objects, which is often
the case), Russell's paradox shows that adding both following requirements on
a structure definer in a theory would lead to contradiction :
 All such S belong to E
 V can occur in the expression A and use x anyhow in
its arguments, namely V(x, x) is allowed, which makes
sense as 1. ensures the definiteness of any V(S, S).
Let us list the remaining options. Set theory rejects 1. but keeps 2. But since 1.
is rejected, keeping 2. may be or not be an issue depending on further details.
As will be explained in 4.9, extending a generic theory (whose ranges of binders
were the types) by a new type K given as the set of all structures
defined by a fixed expression A for all combinations of values of its
parameters, forms a legitimate development of the theory (a construction).
Indeed a binder on a variable structure symbol S with such a range K
abbreviates a successive use of binders on all the parameters of A
which replaces S. Here A and the system
interpreting it come first, then the range K of the resulting S
and their interpretations by V are created outside
them : A has no subterm with type K, thus does
not use V (which has an argument of type K).
The notion of structure in firstorder logic (as a onemodel theory) has this similarity
with the notion of set in set theory : for each given symbol type beyond constants,
the class of all structures of that type is usually not a set, calling "sets" such
ranges K (of structures defined by a fixed expression with variable
parameters), or subclasses of these.
The fully developed
theory with the infinity of such new types constructed for all possible expressions A,
can become similar to set theory by gathering to a single type U all constructed types
K of variable structures of the same symbol type (structures over the same sets),
interpreted by the same symbol V (which could be already used by A).
This merely packs into V different structures without conflict
since they come from different types K of the first argument.
This remains innocent (rewriting what can be done without it) as long as in
the new theory, the binders of type U stay restricted to one of these
"sets" K (or covered by finitely many of them, which are actually
included in another one).
In set theory, the ranges of binders
are the sets. Thus, beyond the simplifying advantage of removing types,
set theory will get more power when accepting more classes as sets.
Other theories, which we shall ignore in the rest of this work, follow more
daring options:
 Keeping 1. and rejecting 2. will be shown consistent by
Skolem's Paradox (4.6)
but would be quite unnatural.

Even weirder is NF
("New Foundations", so called as it was new when first published in 1937),
combining 1. with a lighter version of 2. restricting the possible syntax of
A to forbid occurrences of (x∈x) or any way to define it.
 The most extreme is lambda
calculus, that keeps both points but
tolerates the resulting contradiction by ignoring Boolean logic with its
concept of "contradiction". This "theory" does not describe any object but
only its own terms, seen as computable functions. As a computation system,
its contradictions are computations which keep running, never giving any
results.
The unified framework of theories
Attempts to formalize onemodel theory in firstorder logic cannot completely specify
the metanotions of «expressions» and «proofs». Indeed as will be explained in 4.7
(Nonstandard
models of Arithmetic), any firstorder theory aiming to describe finite systems
without size limit (such as expressions and proofs) inside its model (as classes
included in a type), will still admit in some models some pseudofinite ones,
which are infinite systems it mistakes as «finite» though sees them larger than any size
it can describe (as the latter is an infinity of properties which it cannot express as a
whole to detect the contradiction ; these systems will also be called nonstandard
as «truly finite» will be the particular meaning of «standard» when qualifying kinds of
systems which should normally be finite).
To fill this gap will require a secondorder universal quantifier
(1.10), whose meaning is best expressed (in appearance though not really
completely) after insertion
in set theory (whose concept of finiteness will be defined in 4.5). As this insertion
turns its components into free variables whose values define its model [T,
M], their variability removes its main difference with model theory
(the other difference is that model theory can also describe theories without models).
This view of model theory as developed from set theory, will be exposed in
Parts 3 and 4, completing the grand tour of the foundations of mathematics
after the formalization
of set theory in a logical framework.
Given a theory T so described, let T_{0} be the external
theory, also inserted in set theory, which looks like a copy of T as any
component k
of T_{0} has a copy as an object serving as a component of T.
In some proper formalization, T_{0} can be defined from T as
made of the k such that («k» ∈ T) is true, where the notation
«k» abbreviates a term of set theory designating k
as an object, and the truth of this formula means that the value of this term
in the universe belongs to T.
This forms a convenient unified framework for describing theories interpreted
in models, encompassing both previous ones (settheoretical and modeltheoretical):
all works of the theory T_{0} (expressions, proofs and other
developments), have copies as objects formally described by the model theoretical
development of set theory as works of the theory T. In the same
universe, any system M described as a model of T is indirectly
also a (settheoretical) model of T_{0}.
This powerful framework is bound to the following limits :
 Starting from an arbitrary theory T_{0} with infinitely many
components, a corresponding T cannot be directly defined, as a definition
must be expressible by finite means and thus cannot depend on an infinity of
metaobjects. Any infinite list of components of T_{0} must be
defined by some rules, for getting T as defined by the same rules.
 The risk for a universe to be nonstandard in the sense of containing
pseudofinite systems (which formalizations of set theory in firstorder logic
cannot prevent, but only a condition of standardness of interpretation
can), leads to the following possible discrepances between
T_{0} and T :
 T_{0} only copies the standard components of
T, ignoring nonstandard ones that usually exist for infinite theories
in a nonstandard universe. Then a model of T_{0}
in this universe may fail to be a model of T by not fitting some
nonstandard components (axioms) of T.
 By the incompleteness theorem, T may be inconsistent while
T_{0} is consistent, either due to nonstandard axioms of
T, or to a nonstandard contradiction between standard axioms, which
may exist even if these theories are identical (having a finite initial content).
As such a contradiction means that T has no model in this universe
(nonstandard «proofs» are only valid in the universe containing them),
models of T_{0} may either have the above
failure (in the infinite case), or only exist outside this universe.
So understood, the conditions of use of this unified framework
of theories, are usually accepted as legitimate assumptions,
by focusing on welldescribed theories (though no welldescribed set
theory can be the "ultimate" one as mentioned below), interpreted in standard
universes whose existence is admitted on philosophical grounds;
this will be further discussed in philosophical pages.
Set theory as a unified framework of itself
The above
unified framework is applicable to the case of set theory itself, thus
expanding the tools of interpretation
of set theory into itself already mentioned in 1.7. Namely,
in the above unified framework, the theory T_{0}
describing M and idealized as an object T, will be set theory
itself. Taking it as an identical copy of the set theory serving as framework,
amounts to taking the same set theory interpreted by two universes.
A kind of theoretical difference between both uses of set theory will turn out to be
irreducible (by the incompleteness theorem): for any given (invariant) formalization
of set theory, the existence of a model of it (universe), or equivalently its consistency,
formalized as a set theoretical statement with the meta interpretation, cannot be
logically deduced (a theorem) from the same axioms. This statement, and thus
also the stronger statement of the existence of a standard universe,
thus forms an additional axiom of the set theory so used as framework.
Zeno's Paradox
Achilles runs after a turtle; whenever he crosses the distance to
it, the turtle takes a new length ahead.
Seen from a height, a vehicle gone on a horizontal road approaches
the horizon.
Particles are sent in accelerators closer and closer to the speed of
light.
Can they reach their ends ?
Each example can be seen in two ways:
 the «closed» view, sees a reachable end;
 the «open» view ignores this end, but only sees the movement
towards it, never reaching it.
In each example, a physical measure of the «cost» to
approach and eventually reach the targeted end, decides its «true»
interpretation, according to whether this cost would be finite or
infinite, which may differ from the first guess of a naive
observer.
But the world of mathematics, free from all physical costs and
where objects only play conventional roles, can accept both
interpretations.
Each generic theory is «closed», as it can see its model (the ranges
of its variables) as a whole (that is a set in its set theoretical
formulation): by its use of binders over types (or classes), it
«reaches the end» of its model, and thus sees it as «closed». But
any possible framework for it (onemodel theory and/or set theory)
escapes this whole.
As explained above, set theory has multiple possible models : from
the study of a given universe of sets, we can switch to that of a
larger one with more sets (that we called metasets), and new
functions between the new sets.
As this can be repeated endlessly, we need an «open» theory
integrating each universe described by a theory, as a part (past) of
a later universe, forming an endless sequence of growing realities,
with no view of any definite totality. This role of an open theory
will be played by set theory itself, with the way its expressions only bind
variables on sets (1.8).
Set theory and
foundations of mathematics
Next : 2. Set theory (continued)