But the powerset axiom, which we introduced in the second part, 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:

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 ?

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 R_{F} 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
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 we shall express the action of the operators R_{F},
by the axioms : for all formula F with at least 1 free variable,

∀ (parameters) , (∃ x, F(x, parameters)) ⇒ (∃ x ∈
R_{F}(parameters), F(x, parameters)).

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.

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 text "Is the Continuum Hypothesis a definite mathematical problem?"

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.

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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