Philosophical proof of consistency of the Zermelo-Fraenkel set theory

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 ones.

First axioms

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 other texts, 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 is wrong

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 true ?

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 equivalence).
The justification of soundness and consistency goes as follows. The idea essentially follows that of actual mathematical proofs of theorems: the equivalence of the schema of replancement with the axiom schema of reflection (which I read in Krivine, Theorie des ensembles), which is also the same idea as the deduction of the Lowenheim Skolem theorem from DC.

The idea is to grasp the abstract theoretical possibility, and thus consistency, of a certain concept of God with the following abilities. Neither the question whether this description properly matches what we might like to see as the "true sense" of the word "God", nor whether any "God" so described 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.

First, let us assume that God has the 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 a formula, we may assume that God knows the answer anyway.
In the case of formulas with several open quantifiers, their absolute meaning can be defined recursively, 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 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 a divine answer to the following prayer : "Please God, if in any standard universe You know of, there exist objects satisfying Your divine interpretion of this formula in relation with this object, reveal some to me !" : God has the power to reveal any such existing elements disregarding whether any constructive process is currently available for it (No sorry, as pointed out among concrete examples, there is no such a divine revelation available for finding lovers, what a pity...). Formally, the action of the operators RF should satisfy the axioms : for all formula F with at least 1 free variable,

∀ (parameters) , (∃ x, F(x, parameters)) ⇒ (∃ x ∈ RF(parameters), F(x, parameters)).

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 universe.

(The view of this when taking ZF as a framework, is that when there exists an object satisfying a formula, then there is a smallest rank in the cumulative hierarchy, of such objects; those of smallest rank form a set. See wikipedia article "Von Neumann universe").

We also need (unless it can be seen as already contained there ?) to allow for the operator of image set of a set by a function defined using these divine revelation operators.

Now 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 indexed by ℕ, 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 Skolem's paradox refutes the possibility 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 text "The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem"
If we really consider it a problem, we might try to replace the powerset axiom by something weaker, some kind of induction principle that would still allow for the possibility of interpreting formulas in systems. We could then still apply to that weaker theory the whole above reasoning about the schema axiom of replacement, to get a sort of set theory still rather strong but now fully justified. Quite weaker, you might say, as an example of model is given by the set of hereditarily countable sets. Reference - other

I got the following reply from Joel David Hamkins who co-authored this reference work on set theory without power set:
If Hω1 is the set of hereditarily countable sets, then it satisfies collection and replacement. The reason is that if you have countably many hereditarily countable sets, the set of those sets is also hereditarily countable.
And yes, you can build the constructible universe properly in ZFC without power set. The definition of the levels of the hierarchy Lα is the same as usual, and I believe that everything works fine as expected. And since L has a definable well-ordering of the universe, replacement in L implies collection there, since we can always pick least witnesses.
There are also interesting remarks in p.56 of S. Simpson's book about a set theory ATR0set which is essentially equivalent to the subsystem ATR0 of second-order arithmetic, and in p.57 about the βk-model reflection principle.

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 Inaccessible.
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.

Metaphysical consequences

Lack of mathematical provability of God's existence

Here is an (unserious) argument against the mathematical provability of God's existence (not just that the question cannot be formalized): it comes from the following 2 points
  1. The consistency of ZF can be deduced from God's existence
  2. The incompleteness theorem shows that the consistency of ZF is mathematically unprovable.
But just like Godel's ontological argument, this argument can be objected based on the debatable choice of the concept of God involved (while one of the best mathematical approaches which might be thought of). In particular it differs from the one I described elsewhere.

Non-algorithmicity of the mind (variant of the Penrose-Lucas argument)

moved to a separate page, follow the link


Separating Hierarchy and Replacement by Randall Holmes

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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