Philosophical proof of consistency of the Zermelo-Fraenkel set
It is known that no theory can prove its own consistency. Thus we
cannot attempt to provide any formal proof of consistency of an
axiomatic set theory such as ZF in the absolute. The only possible
ways to justify its consistency are thus non-rigorous, philosophical
In other texts of this site, we have already justified
some of the axioms which are equivalent to some of the ZF axioms:
in the first part we
expressed a "Set
Generation Principle" from which
several set theory axioms can be deduced; in metamathematical
complements, we gave a philosophical justification of the validity
of this principle, as well as a general framework to make sense of set theory axioms and
informally assess their validity (and thus their consistency).
But the powerset
axiom cannot be justified in this way,
especially when combined with the axiom of infinity. This is a big
problem, and will finally be the main source of doubt on the
consistency of ZF. We shall discuss it below. But first let us focus
on the rest:
Why the naive justification of the axiom schema of replacement
The naive approach giving the feeling of consistency of this axiom
schema is misleading, and let us explain why.
This naive approach to these axioms consists in interpreting them as
just saying "any function has an image set". We did justify that any
function has an image set, however the axioms of replacement say
much more. The difference is in how functions are defined. They are
defined in different ways.
First we can say about the difference, that the "functions" accepted
in ZF are any class of oriented pairs, only restricting its domain
to a set - and these "classes" of oriented pairs are even more
general than our definition of a "class" since they accept open
quantifiers in their expression, which we have forbidden ; while we
only accepted functions that were already present as objects in the
universe. The difference between both has the same nature as the
difference between sets and classes : which classes should be
accepted as objects (sets) ? Which classes of oriented pairs should
be accepted as objects (functions) ? The step from a function to its
image set can be accepted as automatic, as we justified it. The
problem is why would any class of pairs satisfying the criteria of
the replacement axioms (its domain is a set and each of its elements
has a unique image), be accepted as a function (in other words, when
does this class of oriented pairs form a set) ?
If this class of oriented pairs was fixed, then indeed we could
consider it as a set, not in this universe, but in the next
universes. So, there are two problems.
One problem is that, ifever this class is indeed fixed, then it only
forms a set in a next universe, not the one we started with.
The second problem is that, as this process does not keep the
universe but adds more and more objects to the universe, the very
definitions of these functions may be differently interpreted from a
universe to the other, so that these classes have indeed no reasons
to be fixed : once added the given functions and their image sets to
the universe, we end up in a new universe where this function we
added might no more fit with the definition by which it was
introduced. In fact, the new class of oriented pairs defined by the
formula may not just have got more more elements, but also lost some
that no more satisfy the formula due to open quantifiers whose value
may change between universes. Thus the addition to the universe of
an object that was reported missing by an instance of such an axiom
(with given values of the parameters), may not help to obtain a
universe where this instance of the axiom becomes true. Thus it is
not a step forwards to the construction of a universe satisfying ZF.
So, can we finally find a universe where these axioms are formally
How to really justify this axiom schema
To simplify, let us present this in its equivalent form of the axiom
schema of collection or of boundedness (see Wikipedia article on
"Axiom schema of replacement" expressing them and reporting their
The justification of consistency goes as follows (and is inspired by
the proof of equivalence with the axiom
schema of reflection):
Let us add to the formalism of set theory a new schema of
operators named RF depending on an arbitrary formula F,
called "Operators of Divine Revelation", defined as follows.
First, let us assume that this operator has the divine power of
interpreting formulas "in the absolute". The problem with
interpreting formulas is with open quantifiers: a formula starting
by an existential quantifier may be false in this universe but
true in some larger universe where an element satisfying its
subformula can be found. While from the viewpoint of a given
universe it can be unknowable whether a larger universe may exist
with an element satisfying the formulas, we may assume that God
knows the answer anyway and has the power to reveal such an
element disregarding whether any constructive process is currently
available for it.
In the case of formulas with several open quantifiers, their
absolute meaning can be
starting with the
innermost open quantifiers (about the existence of an object
satisfying a bounded formula, with thus clear meaning), down to
the outermost quantifiers (whose absolute meaning is seen while
interpreting subformulas in absolute ways themselves).
Now we shall express the action of the operators RF,
by the axioms : for all formula F with at least 1 free variable,
∀ (parameters) , (∃ x, F(x, parameters)) ⇒ (∃ x ∈
RF(parameters), F(x, parameters)).
But we must specify here : these do not intend to be mere axioms in
a formal system, but to express an "absolute definition" of what
these operators do: all open quantifiers in these formula are
assumed to take their absolute value. This defines the actions of
these operators "in the absolute", independently of any specific
So in the absolute, how are these axioms justified ? By the fact
that when there exists an object satisfying a formula, then there
is a smallest rank in the cumulative hierarchy, of such objects
(see wikipedia article "Von Neumann universe"). Those of smallest
rank form a set.
(No sorry, as pointed out among concrete
examples, there is no such a divine revelation available for
finding lovers, what a pity...)
After this, and only after this, let us ask the question : what
specific universe can we take ? Because there cannot be any
absolute, ultimate universe; but we need to pick one for our
formulas to make sense. Our operators of divine revelation could
be defined independently from any choice of a universe, as they
did not depend on the universe, they did not have any specific
domain: they only took an absolute value for every specific values
of the parameters.
The answer is : we can pick up the smallest universe which is
stable by these operators. It is constructed by a simple recursion (yes, the union of a sequence, recursively defined on the simple set
ℕ of natural numbers, thus showing how misleading is the
naive interpretation of the replacement axioms !) listing all these
operators, starting with some universe (such as ℕ)
and repeteadly applying all these operators as well as the
powerset, to enlarge the universe. First let us label these
operators by numbers. For each n from 0 to infinity, enlarge
the universe by applying the first n operators to all objects
of the current universe, as well as taking the powerset.
In this way, every operator is used an infinity of times : the
nth operator is applied once at each step after the nth step, so
as to not let any newly created object escape the property that,
when subsequently used as a value of a parameter, the resulting
"absolutely" existing x will also exist in the universe (as it
will be added at the next step). This way, the value of each
formula in the resulting universe, coincides with its absolute
value that we refered to (thanks to the applications of the
operators to each subformula that make the value of each open
quantifiers in this universe coincide with its absolute sense).
Note that the universe built in this way is obtained as the
union of a sequence of universes parametered by integers, thus
contradicting the idea that the image of every sequence is a set.
The point is that this sequence cannot be defined inside the
language of ZF itself, because it requires to use the divine
revelation operators, which do not belong to the language of the
theory ; they cannot be redefined because, according to the truth undefinability
theorem, even if we admit to give
sense to individual formulas with open quantifiers, there is no
systematic way inside the theory, to express the "true interpretation" of
the whole set of formulas (represented as objects "systems of
symbols") with unlimited complexity.
Can the powerset axiom ever be justified ?
There is a possible intuitive feeling that the powerset axiom does
not make contradiction. Can there be anything to say that may bring
more precise justification to it ?
I'm afraid not, for the following reason.
For any candidate way to see all subsets of ℕ to form an acheived
whole, it is likely that such a view will be more or less equivalent
to a way to explicitly list these subsets. But the very construction
of such a listing will probably also provide a way to label these
subsets by numbers. But the proof of Cantor's theorem then gives an
explicit way to construct another subset of ℕ outside this list,
thus contradicting the exhaustivity of such a list.
Thus we may give up and try to see if there can be an explicit
contruction of not all subsets, but of those that can behave as such
in a given context. For example, in second order arithmetics... we
may get clues on this question by studying the behavior of the
cardinals of sets of constructible subsets of ℕ as depending on the
ordinal order of construction. But sorry I did not study the
existing works on the subject...
However we can still admit the (non-formalizable) intuition of the
validity of the power set axiom, as meaning that for every set E it
is coherent to assume to be in a universe where E appears to have a
power set, although there is no way to fully express in formal ways
that this power set is meant to be the ultimate one independently of
the universe, so that such an intention of ultimateness can as well
be dismissed as a non-mathematical idea.
Such a position is roughly what Solomon Feferman explained in his
the Continuum Hypothesis a definite mathematical problem?"
Can Large Cardinals be justified ?
Large Cardinals, starting with the supposedly "first large cardinal"
that is the Inaccessible Cardinal, can be seen as just an extension
of application (fuller expression) of the idea underlying the axiom
schema of replacement. However that idea here generalized was only
the naive interpretation of this schema, which we saw above to be
flawed. Or at least it does not have a "philosophically rigorous"
justification in the sense that the above justification of the axiom
schema of replacement can be seen as "philosophically rigorous".
We might still admit this idea as sort of intuitive justification
with a more vague status, a similar status of vague justification as
that of the power set axiom.
But as far as I am aware of, there is no "philosophically rigourous"
extention of the above justification, to the case of the
I once saw an article that claimed to give such a justification, in
terms of some version of the Reflection Principle. However I was not
convinced and I consider it to be flawed.
Here is the problem I see with the Inaccessible.
In ZFC, for any cardinal K we can find a larger cardinal with
cofinality K (that is Aleph K). However it does not really help
finding a cardinal equal to its own cofinality, because if we repeat
the operation taking Aleph (Aleph K), Aleph (Aleph (Aleph K)) and so
on up to infinity, and then take the union of them, we end up with a
cardinal with cofinality...only ω (the countable one). We can go
much further, still using the axioms of replacement, and get a
cardinal K' with cofinality K and that keeps the advantage of the
limit cardial we just built, that is, such that any cardinal X<K'
is the cofinality of some cardinal between X and K'.
But this still does not give the Inaccessible, and however we try to
continue the process there is still a gap.
But we can see a bigger trouble in the following consideration : I
once read (I forgot the reference) that in ZF without Choice, it
remains coherent to assume that all ordinals have cofinality ω.
Because while the ordinals which are cardinals (= not in bijection
with any smaller ordinal) do not form a set, we need the axiom of
choice to prove that those who are not limits of smaller cardinals,
are their own cofinality; which can become false in the absence of
axiom of choice.
And the above justification for the consistency of ZF consisted in
building a model of ZF with (externally) countable cofinality:
something definitely not the same as an inaccessible cardinal.
Thus if a cardinal seems to have a big cofinality, it may be just
because we are in a universe that fails to contain a countable
sequence of smaller ordinals that reaches it, while such a sequence
might exist outside.
The situation can be seen as a version of the famous Omnipotence
paradox : "Could an omnipotent being create a stone so heavy
that even he could not lift it?" God cannot both create a gun so
powerful that it can break any shield, and a shield so solid that it
can resist any gun.
So : can God find a cardinal so big that even he cannot find a
sequence (or any function with domain some smaller cardinal) of
smaller ordinals leading to it ? Anyone can form one's own opinion
or lack of opinion on the question.
Here is an argument against the mathematical
provability of God's existence (not just that the question cannot be
formalized): it comes from the following 2 points
But just like Godel's ontological argument, this argument can
be objected based on the debatable choice of the concept of God involved. In particular it differs from
the one I described elsewhere.
However, neither the question whether the above description (while one of the best admitting
a mathematical approach that which might be thought of ) properly matches what we might
like to see as the "true sense" of the word "God", nor whether any "God" so described above
may "really exist" or be a mere theoretical fiction, would suffice to destroy its value as an argument
for the consistency and appropriateness
of ZF as an axiomatic foundation of set theory.
- The consistency
of ZF can be deduced from God's existence
- The incompleteness
theorem shows that the consistency of ZF is mathematically
Then almost the same stuff also proves that the mind is not algorithmic
(by an argument more rigorous than Penrose's, which, I admit, skeptics
have reasons to dismiss). Namely:
- ZF is a VERY strong axiomatic system
- Still, with some effort, a good mathematician's mind is able to justify the consistency
of ZF as very reliable. Thus, according to the incompleteness theorem, any candidate
formal system that claims to account for the abilities of a good mathematician's mind
would have to be an even stronger system.
- However, there would be no point for natural evolution to give such strong
axioms as a basis of the minds proving abilities. Much weaker systems would
largely suffice for all survival purposes, with needed developments having nothing
to do with strength in the hierarchy of formal systems which the incompleteness theorem deals with.
Therefore, natural evolution cannot account for the good mathematician's thinking abilities.
I stumbled on this research article about large cardinals, which may be
interesting, but I had no time to check it : A proposed characterisation
of the intrinsically justified reflection principles by Rupert McCallum
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