Index of special words and notations


Words
Explained in
Comment
Mathematics ; mathematical logic
1.1

theory
1.1, 1.3, 1.A

system
1.1, 1.3, 1.4,
3.1, 3.2
= a set with structures, interpreting given types and symbols.
Could not call this its officially name of structure which I left to its other standard meaning which sounds better.
model
1.1, 1.A
= interpretation of a theory.
See also (Mod, Tru) in 4.8
content / components of a theory
1.1, 1.3

foundation, primitive
1.1
Ref : wikipedia Primitive notion
development
1.1, 1.3, 4.8, 4.9
Did not find this concept made explicit elsewhere.
axiom
1.1, 1.3, 1.5,
1.10, 1.A

statement
1.1, 1.9 The syntactic difference (1.9) making them not particular formulas seems original, the special syntax of binders in set theory (1.8) being absent from traditional presentations (ZF).
set theory
1.1-1.4, 1.7,
1.8, 1.10...

model theory
1.1, 1.3-1.7,
1.9...

theorem, proof, provable,
proof theory, (in)consistent
1.1, 1.6, 1.9,
1.F, 4.6, 5.7
see also "refutation"
constant (symbol)
1.2, 1.4
value (of a variable)
1.2
variable (symbol)
1.2, 1.5, 4.1
fixed variable
1.2, 1.3

free variable
bound variable
1.2, 1.8
Ref: wikipedia
range ; range over
1.2
element
1.2, 1.7

pure element 1.2, 1.7 elsewhere called urelement or atom, while the latter word has another standard meaning
set 1.2, 1.7, 1.E

function, argument, domain 1.2, 1.10

value of a function
1.2
functor
1.2, 1.4,
1.D, 1.E
Not found this named elsewhere. Not the concept of functor in category theory (particular case, not named in this work but implicitly used such as in actions of categories).
operation
1.2, 1.10, 2.3

relation
1.4, 1.10, 2.3

arity ; nullary ; unary; binary...
1.2
See also: tuple, operation, relation
notion
1.3, 1.7
here precisely defined in a way not seen elsewhere
object of a theory
1.3 to not confuse with objects of a category
one-model theory
1.3 Did not see it named nor conceived elsewhere.
logical framework
1.1, 1.3, 1.A This concept is of course crucial but not named elsewhere this way, maybe not any other way, but specific ones are named by the word "logic" with adjectives.
first-order logic
generic theory
1.3 see also: model theory
[T,M] 1.3
only used in 1.B and 1.7; invented notation (how else would you write that?)
meta-(notions of one-model theory)
meta-object
1.3
meta-(notions of set theory) 1.7

type (abstract type,
interpreted type)
1.3 Hardly ever mentioned in the math literature, where it is also called sort, but that latter word does not require them to be disjoint (see "order-sorted logic")
language
1.3, 3.1
= list of structure symbols
structures
1.3, 1.4
here only first-order structures : operators and predicates
second-order structures
(1.3) 5.1
interpretation (of a theory into set theory) 1.3
designates a model in the universe
generic interpretation
1.3 I coined that name
standard
1.3, 1.D, 4.7

universe
1.3, 1.D, 1.F, 5.3

operator, predicate
1.4

Boolean
1.4
Logicians use this name in some ways but elsewhere call it propositional, while in ordinary math "proposition" is synonymous with theorem, which is different.
para-operator
1.4
I coined that name
(logical) connective
1.4, 1.6

1.4

f(x) evaluator
Dom
1.2, 1.4
I coined the name "evaluator" as I am not aware of any standard name for this.
ZF
1.4
Only commented here, not described nor used. See also 1.D (specification) and 5.B
ground, expression, formula
1.5
This definition of "expression" as "term or formula" just seems to be an implicit standard.
term, occurrence, root 1.5, 4.1

binder 1.5, 1.8 I coined it, for what seems to be elsewhere called "Variable-binding operator" as I give "operator" another meaning and cannot see how else to do.
equality =
1.5, 1.6

variable structure
1.5

definition, abbreviation
1.5, 4.8

parameter
1.5
see also : constructions
invariant 1.5, 3.1, 3.3,
3.6, 4.8, 5.1

tautology
logically valid formula
1.6

⇔ , ¬ , ≠ , ∉, ⇎
1.6
equivalent, not (negation), different, not in, inequivalent (exclusive or)
∧ , ∨   1.6
conjunction, disjunction
implication ⇒
necessary condition
sufficient condition
contrapositive, converse
1.6

1.6
Therefore. This notation seems standard but I only found it in lists of html symbols, not in math texts.
Wikipedia says it only serves as natural language abbreviation, while I use it in formulas as synonym of ∧ (but not of ⇒).
chain of conjunctions
chain of disjunctions
chain of implications
1.6
Did not see these officially declared elsewhere though they are so much needed.

refutation, refutable, decidable,
lemma, proposition, corollary
1.6
see also "theorem"


















axioms of equality
1.9
References : wikibooks - Encyclopedia of math. Used in 4.6 (proof of the completeness theorem)