# Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties.

For example, in group theory you deal with groups; in topology, open sets; in analysis, the set of reals, etc.

Is there something more general which englobes all maths? Is this a too reductionist view?

Maybe I should replace set theory with category theory, but the same question holds.

## closed as primarily opinion-based by Michael Hoppe, Claude Leibovici, Yiorgos S. Smyrlis, Lucian, J.R.Mar 4 '14 at 11:14

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• Set theory + logic + various definitions, maybe. – naslundx Mar 4 '14 at 9:28
• See also this post – Mauro ALLEGRANZA Mar 4 '14 at 10:07
• Set theory is much much younger than mathematics. Don't forget that much of concrete mathematics can be carried out in full higher-order arithmetic, which is far weaker than ZFC set theory. Only rather abstract subfields of mathematics need more than finitely many power-sets of the naturals, and even then there are alternative foundational systems like type theories that can be rather different from set theories yet support the same mathematics. – user21820 Aug 10 '18 at 17:09

I disagree that most branches of mathematics are just an application of set theory and logic. The fact that most areas of mathematics use set related notions and employ logic does not mean they are applications of these areas.

For instance, would you say that English Literature = English Words + English Grammar? After all, every piece of English literature uses English words and follows (more or less) the English grammar.

There are many views on what mathematics is. One school of thought on the subject is called formalism and it (more or less) asserts that mathematics is devoid of any meaning and everything is just formal manipulations on a piece of paper. There is no doubt that this is a factually true statement, in the sense that all our proofs are formal manipulations on a piece of paper that follow some very strict rules. But, there is (to me at least) no doubt that such a point-of-view is devoid of any relevance to the working mathematician. A mathematician never wakes up in the morning and says "let's choose some axioms at random and see what I can prove from them". Instead, a mathematician has a plan. A program. Something to achieve. That something transcends the formalism. There is a reason why we study the algebraic structures we do and not the endless other examples of algebraic systems that follows any set of arbitrary axioms. There is a reason why we study certain objects like $\mathbb R ^n$ or $\ell _p$ and not a trillion (and more) other arbitrary sets that can be concocted in a myriad of nonsensical ways. What gives sense to the mathematics we care about can't be attributed to set theory and logic. Neither set theory nor logic tell us how to find interesting axioms to study.

The answer does not change if you replace set theory by category theory.

• Very well written! – Martin Brandenburg Mar 4 '14 at 16:16

No.

For example, if we spend the time to formalize group theory in a set theory (take ZFC as an example), then by the end of this long and arduous exercise, we probably haven't learned a whole lot of new concepts in group theory. We've just worked out how to implement the old concepts in ZFC. Furthermore, we're probably no better at discovering new group-theoretic ideas than we were when we started the whole enterprise. In fact, we might be worse, since now we've (potentially) allowed ourselves to get stuck in the ZFC way of doing things.

The same critique can be directed at any approach to founding mathematics; since its not the only way of doing things, we have to be careful not to let our creativity be stifled by just focusing on that one particular approach. This does not mean we should avoid foundations; rather it means we should always be open to new foundational ideas.

By the way, the main application of set theory to (basic) group theory is that you can actually prove the existence of entities that we take for granted, like Cartesian products of groups. Their existence follows from the existence of Cartesian products of sets, which is a basic result of set theory. (Actually, some approaches take the existence of a Cartesian product of sets as an axiom).

Anyway, the point is this. Just because we can formalize a lot of mathematics using set theory + logic, this does not mean that mathematics $=$ set theory + logic.

• (+1) but I have to disagree with the claim that the main application of set theory to group theory (basic or not) is, e.g., the existence of products and the like. These are trivialities. An example of an application of set theory to basic group theory would be the existence of certain maximal subgroups by employing Zorn's Lemma. An application to advanced group theory would be any of a multitude of results that construct a certain counter example to a group theory claim by means of, e.g., large cardinals. – Ittay Weiss Mar 4 '14 at 11:05
• I strongly disagree that set theory doesn't provide new insides and ideas to group theory - consider for example the Whitehead problem. (Another candidate for a set theoretical solution of a group theoretical problem is the existence of bad groups which is afaik still open.) – Stefan Mesken Jun 29 '16 at 12:57
• @Stefan, that wasn't what I meant. My claim isn't that set theory doesn't provide new ideas to group theory, but rather that the process of formalizing group theory within ZFC (or any similar such theory) doesn't offer any new insights into group theory. – goblin Jun 29 '16 at 13:07
• @goblin Well, okay. But formalizing constructions is not what set theory is about (even though it was originally motivated by that). Formalizing a certain construction in set theory is like having breakfast - it prepares you to do your job, but it doesn't determine the purpose of your actions. – Stefan Mesken Jun 29 '16 at 13:35
• And by the way: I don't think that people should worry to be 'looked into the setting of ZFC'. Set theory (and other foundations) seem to develop into some healthy pluralism. – Stefan Mesken Jun 29 '16 at 13:38