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Set theory seems to have the property of being "universal", in the sense that any branch of math can be expressed on its language. Is there any other branch of math with this property? I am asking because set theory is a branch of math. It is predicate calculus plus the binary relation ∈ plus some axioms. I believe that something similar could be said of most branches of mathematics. So, why is set theory so unique? or is it not?

noelia
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    Category theory can express an awfully lot of information about mathematics.

    In fact there are set theories which are built on the language of category theory which are equivalent to $\sf ZFC$ (or weaker variants of it).

     – Asaf Karagila Apr 19 '13 at 14:29
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    Well, if you are willing to climb up one level of abstraction, all of mathematics can be carried out in arithmetic: first formalise everything in ZFC, then formalise the formalisation in PA! – Zhen Lin Apr 19 '13 at 14:53
  • @ZhenLin: Then formalize PA in ZFC again; and repeat ad nauseam. :-) – Asaf Karagila Apr 19 '13 at 15:14
  • @ZhenLin and Asaf: Question: How is it possible to formalize ZFC in PA, if PA only admits $\Pi_m^n$ and $\Sigma_m^n$ formulas, and there are formulas of ZFC (such as the GCH or the existence of indescribable cardinals) that cannot be expressed in that format? – Wolphram jonny Apr 19 '13 at 15:54
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    @julian: You can encode the language and the axioms of $\sf ZFC$ with natural numbers, and you can encode proofs and provability and so on. – Asaf Karagila Apr 19 '13 at 15:59
  • @Asaf Thanks! I guess I will have to post a question about this, because a few of my recent questions seem to be related to my misunderstanding of this issue. – Wolphram jonny Apr 19 '13 at 16:01

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There is a long, fascinating, and often-told story about the nineteenth century project for the rigorization of analysis, and about the re-construction of classical mathematics in terms of natural numbers and sets of natural numbers and sets-of-sets of natural numbers, etc. etc. (and if we are feeling particularly austere we can even re-construct the naturals in a pure set theory which lacks urelements, so everything gets implemented in pure set theory). There are lots of good recountings of the story -- here's a short one with lots of pointers to more: http://plato.stanford.edu/entries/settheory-early/

I mention the history because it explains why set theory has long been thought to have a special "foundational" place in the architecture of mathematics. But does it really? Can category theory (for example) provide an alternative foundation? Now we've got over our wobbles about a hundred-and-twenty years ago when some thought classical mathematics was threatened by paradoxes of the infinite, does mathematics need universal foundations?

BIG questions, too big for here! But here's one line of thought that I've encountered from mathematicians, which perhaps underlies some of the continuing nods to the special place of set theory.

Suppose working on Banach spaces, or algebraic topology, or whatever, I conjecture all widgets are wombats. And then the bright young grad students try to prove or disprove Smith's Conjecture. Young Jane claims to have refuted the conjecture by finding a structure in which there is a widget which isn't a wombat.

Well, what are the rules of the game here? What kit is Jane allowed to use in her structure building? To give her a best shot at refuting the conjecture, she perhaps ideally wants some kind of all-purpose kit that only minimally constrains what she can build -- she wants the mathematical equivalent of a Lego kit where you can pretty much attach anything onto anything, rather than the equivalent of a building kit you can only make toy houses from, or one you can only make toy cars from. (Perhaps Smith's Conjecture works fine for, so to speak, houses and cars.)

What the standard set theory of the iterative hierarchy seems to provide is just such an all-purpose mathematical Lego kit. We start with some things (or if you like, with nothing at all), and then we are allowed to put them together however you like into new things, and then we are allowed to put what we've got together however we like ad libitum, and to keep on going as long as we like. Precisely because the rules for building new sets allow maximising at every step (the idea is at each level we are allowed every possible new combo, and there is no limit to the levels), we really do get an all-purpose structure-building kit. And having such a mathematical Lego kit is just what Jane ideally needs if she is to have untrammelled free rein in coming up with her widget which isn't a wombat.

Or so the story goes, in outline ...

Peter Smith
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