The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By lexical ordering of finite formulas we see that the set of finite formulas is countable. So it is impossible to define all elements of the uncountable power set. The power set axiom seems doubtful. Therefore my question.



What would remain of current mathematics without axiom of power set?
This is a real question.


You're not the only one who find the power set axiom dubious. At the time of the great foundational controversies, Russell and Weyl both expressed a similar view. It is now known, from work of Weyl, Wang, Feferman, the reverse mathematics school, and others, that the vast bulk of mainstream mathematics can be developed without power sets. You can effectively treat objects like $\mathcal{P}(\mathbb{N})$ and $\mathbb{R}$ as proper classes.


You might be interested in looking at my paper on "Mathematical conceptualism" at arxiv:math/0509246. Here's a direct link: <http://arxiv.org/pdf/math/0509246.pdf>


