TITLE: Don't the axioms of set theory implicitly assume numbers?
QUESTION [20 upvotes]: When one writes down the axioms of ZFC, or any other axiomatic theory for that matter, and making statements like "let x, y ..."  doesn't this assume an understanding (and thus existence) of natural numbers implicitly?  (Q1)
How is the reader to interpret statements such as existence of separate symbols, nevermind sets, without an intuitive notion of numbers?
Bourbaki talks about this within the framework of metamathematics, but then declares that the reader can read words, differentiate between different words etc. and that to assume otherwise is idiotic.
Is there an introduction to these circle of ideas & debates somewhere you'd recommend? (Q2)

REPLY [2 votes]: If i may add (although this post is kind of old)
This circularity "problem" (if one wishes to see it as such), appears in (what is refered as) classical mathematics.
Now one should bear in  mind that even these classical mathematics (continuing along the lines of aristotle, euclid, archimedes, leibniz, cantor, hilbert, russel, goedel, etc..), have been cutoff and formalised (or sterilised if you like) to a greater extened that originaly meant.
In any case this is not the main argument.
But i would like to draw attention to the intuitionistic flavor of mathematics (and especially of the LEJ Brouwer path) (see for example LEJ Brouwer, Cambridge Lectures on Intuitionism, most of first lecture plus the appendix on marxists.org).
There Brouwer, aware of the problem, explicitly takes on the issue of mathematics over language or syntax.
Excerpt:

FIRST ACT OF INTUITIONISM
Completely separating mathematics from mathematical language and hence
  from the phenomena of language described by theoretical logic,
  recognising that intuitionistic mathematics is an essentially
  languageless activity of the mind having its origin in the perception
  of a move of time. This perception of a move of time may be described
  as the falling apart of a life moment into two distinct things, one of
  which gives way to the other, but is retained by memory. If the twoity
  thus born is divested of all quality, it passes into the empty form of
  the common substratum of all twoities. And it is this common
  substratum, this empty form, which is the basic intuition of
  mathematics.