# Believing in the axiom of Power Set

I am struggling to find a philosophical reason for believing in the axiom of power set, and I was hoping you can give me some justifications.

I am not looking for answers of the form "it's convenient to use power set axiom" or "why wouldn't it be true?", as my view of the mathematical world tends to be platonist. Just for pure philosophical and logical arguments for this axiom.

Following the exposition in Roman Kossak, Mathematical Logic : On Numbers, Sets, Structures, and Symmetry (Springer, 2018).

If we start with a finite set, with n elements (n little), it is easy to list all its subsets (they are 2^n).

If we use a bigger one (but still finite), like e.g. a chessboard, we can exercise ourselves in selecting different "configurations" of squares : the obvious ones, like that of all black squares, but also every selection of squares (not necessarily adiacent) will do.

Having 64 squares available, we have a quite huge choice of selections.

Each selections is a subset of the set of squares making the chessboard.

Having said that, what is the "philosophical status" of the statement: “There exists the set of all subsets of the chessboard” ?

If we imagine to enlarge the chessboard to a 16 x 16 schema (we have only doubled the size of the side !), the number of possible selections grows up to 2^(256), that is quite huge : around the number of atoms in the universe.

Thus, if the set of all subsets exists, where does it exist?

But here is the key point of the mathematical theory of sets: set theory is the theory of mathematical infinite.

Mathematics lives in an infinite universe. Thus, the limitation discussed above is not consistent with the "infinitistic attitude" of modern mathematics.

Going back to our chessboard, we can freely click here and there to select any set of squares.

It is hard to say that some subsets of the chessboard do exist, but some others don’t. Since none can be excluded, we are inclined to accept that they all somehow "exist", and therefore there must "exist" the set of all of them.

This is the "conceptual" ground of the power set axiom.

Obviously, there is no way to "prove" the axiom; outside mathematics, IMO there is no compelling reason to "believe it", but there is no compelling reason to believe the mathematical theory of sets, either.

From a more "pragmatic" point of view, all "initial" axioms of set theory allows us to assert the existence of very few sets : the emptys set and other sets quite "slim".

To do mathematics, we need the set of naturals, and thus we have to postulate the existence of an infinite set, and that of the reals, and thus we have to postulate the existence of the set of all subsets of the set of naturals.

Again, we need them if we want to use set theory for its primary purpose : to develop the mathematics of infinite.

• Hardly as cut and dried as you make it sound IMO. There's plenty of professional doubt about the powerset axiom. For example mathoverflow.net/questions/133597/… has a number of references. – user4894 Feb 25 '19 at 19:35
• @user4894: I think this is an adequate answer. You do not state axioms so that you can (or have to) believe them, but because they serve certain fundamental functions to make something else possible. Thus, you may doubt the necessity of it as an axiom for the ends it is supposed to serve. But the reality or existence of (purely formal) sets is indeed more a question of belief (read: faith) rather than mathematics (or philosophy). – Philip Klöcking Feb 25 '19 at 19:38
• @user4894 - agreed: there is a minority but interesting "sect" of mathematicians that do not believe in infinity. IMO the issue is that we do not "believe"... math is not a religion. – Mauro ALLEGRANZA Feb 25 '19 at 19:41
• @MauroALLEGRANZA That's a terribly disingenuous remark coming from someone of your stature on this site. I did not say mathematicians doubt infinity, though some do. I said that there is a cottage industry of mathematicians who question the powerset axiom. A common example is constructive mathematics, where we replace the full powerset with the constructive powerset. – user4894 Feb 25 '19 at 19:47
• @user4894: Still, this is the difference between all possible treatments of infinity (needing the PSA) and specific treatments of infinity. A general theory of sets, i.e. about all possible applications of sets, necessitates the PSA, while for particular applications of Set Theory, it does suffice to have less demanding premises, i.e. replace PSA. The answers in the question you linked are quite explicit about it not being necessary for the actual practice, i.e. application of sets, which still can differ from the (general) Set Theory proper. – Philip Klöcking Feb 25 '19 at 19:53

For modern sensibilities, it's maybe worth considering instead the axiom of function sets: given any two sets X and Y, there is a set of all functions from X to Y.

Given the equivalence between the notions "subset of X" and "function from X to the set {true, false}", there is an alternative notion of power set meaning "the set of all functions from X to {true, false}", and the axiom of function sets guarantees the latter.

Given the alternative power set, I presume the usual power set can be constructed via the axiom of unions and the axiom of subsets. Or alternatively, via the axiom of replacement.

(note that, in more general foundations you can replace {true, false} with the relevant "set of truth values")

As for justifying the axiom of power sets directly, the important thing to consider is higher order reasoning.

Suppose you have some type T of thing you like to talk about. For example, "natural number".

As a (presumed) practitioner of classical logic, you probably reason in terms of predicates on T — propositions one can make about the things of type T.

For example, if T is "the natural numbers", you might reason with propositions such as "____ = 17" or "____ is the only solution to 2x + 3 = 7" or "____ is even" or "____ is prime" or

Now, here is the important part: you can engage in higher order reasoning. That is, you aren't just limited to manipulating specific propositions: you can reason about propositions in general. For example, you can make statements like "For any propositions P,Q,R, if P implies Q and Q implies R, then P implies R".

So here's the important principle:

If a mathematician can talk about things of type T, they can also talk about propositions one can make about things of type T

When set theory is used in foundations, the sets are the manifestations (or descriptions or encodings or whatever) of the various types of things we can talk about, and subsets correspond precisely to the predicates we can discuss in this language.

So, in the language of set theory, the principle above is precisely the axiom of power sets.

The power set axiom postulates: For any set X exists a set P(X) which comprises as elements exactly the subsets of X.

Hence the power set axiom acts as a tool to form new sets from existing ones, by fixing a certain defining property.

As one knows, not any defining property is admissible for defining new sets, see the Russell antinomy. Compared to unrestricted use of defining properties the power set axiom is a mild version. It does not introduce new inconsistencies. Hence power sets exist in the context of Zermelo Fraenkel set theory.

But IMO this kind of existence does not follow neither from logical reasons nor from philosophical argumentation.

A deep consequence of the power set axiom is the result: For any set X the cardinality of P(X) is stricly bigger than the cardinality of X. Moreover 20, the cardinality of the power set of the naturals, equals the cardinality of the reals.

Hence accepting the set of reals implies at least the existence of the power set of the naturals.

Whether power sets exist in the Platonic world of forms is a question I cannot answer. Unfortunately, I do not remember, contrary to Platonic anamnesis :-)

After Russell's paradox (and others) the idea of "what a set is?" has changed! from the very broad idea of a set being an extension of a predicate (Frege), to a an entity that is constructed in a controlled step-wise manner in stages, beginning from some particular lower level objects, call them Ur-elements, then construct sets of those, then we go to the next level of constructing sets of those and the ones prior to them and so on... It is this image of a hierarchy that justifies the power set axioms and other axioms of Zermelo set theory.

Now it is obvious that the next level must be at least some subset of the power class of the prior level, like for example the set of all definable subsets of the prior level, so the real question would turn to be about:

How much width we accept per stage of the Hierarchy?

An argument towards full width would definitely motivate the power set axiom. The main problem of the other direction is when we must stop? which appears to be ad-hoc most of the times, if we want to adhere to strict constructibility criterion then we'd end up with thin stages, but this appears to be a rather strict restriction rather than a kind of reality about the matter. If one opt for maximality, then definitely the power set axiom would be the maximal next stage. I don' know why in some sense maximals appears less ad-hoc than particular restrictions.

All in all, of all axioms of ZFC, the power set axiom seem to be the most doubtful, and no easy line of justification, especially of the natural sort you are calling for, is available.

However, there are many technical reasons for its development, like the study of reals, having easier notation, etc.., but that would be pragmatic.

In a mereological foundation of set theory, like that of David Lewis, sets appear as labeling bodies, catalogs, but this is only to make them achieve a hierarchical buildup, so it would be expected for the next stage in the hierarchy to unravel the part-hood of the prior level, and to the full spectrum of it as well! Since this is just part of capturing its mereological properties in the hierarchical buildup, it would be strange to think of hidden parts not been visualized in the hierarchy, since part-hood (which is equivalent to subclass-hood by Lewis's definitions and premises) is already the MAIN primitive in that setting. With Lewis he stresses, through his premises, that subclass is exactly equivalent to part of class, and that this is a basic mereological property of classes! Now going hierarchical through the Epsilon membership relation, it would be expected from the set principle to un-ravel this essential nature of classes in set language; i.e., subset is exactly equivalent to part of set, and so the full girth of powering is to be exposed!