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Why is it important, that mathematics can be formalized in set theory?

As one can read in the thread Are there areas of mathematics that cannot be formalized in set theory?

Today known mathematical theories can be formalized in set theory (ZFC + suitable additional axioms). Why is this important?

I would say: Because every mathematical theory can be formalized in set theory one can use set theory to answer foundational questions about mathematics. Am I right with my claim?

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    $\begingroup$ Philosophically, it's interesting that mathematics can be formalized at all. It tells us something about the relationship (or lack therefof) of mathematics to the real world. $\endgroup$ – Jack M May 7 '15 at 23:46
  • $\begingroup$ The question supposes that mathematics can be formalized in set theory, which I don't dispute...but it may be hard to define what one means by "mathematics". In some circles of thought I might imagine people objecting to the premise. $\endgroup$ – Christopher A. Wong May 7 '15 at 23:57
  • $\begingroup$ @ChristopherA.Wong: I am no expert in this topic, but I am sure that most mathematical theories can be formalized in set theory... So at least one can ask about the advantages after having formalized a theory... $\endgroup$ – Stephan Kulla May 8 '15 at 0:02
  • $\begingroup$ @JackM: Your comment leads me to another interesting question: Is the universe we are living in part of the von Neumann universe? I mean, most mathematical objects of analyis and algebra can be represented by (complex) structures of sets and physics uses those objects to model the real world. So: Is the real world representable by a (very complex) structure of sets? But I guess this question is not really answerable... ;-) $\endgroup$ – Stephan Kulla May 8 '15 at 0:09
  • $\begingroup$ It is important to know that all of mathematics can be derived from a few underlying principles -- the rules of logic and the essential properties of sets. It cuts all the hand-waving by at least one half. $\endgroup$ – Dan Christensen May 8 '15 at 14:42
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The important part is the fact we can at all formalize mathematics in a uniform method.

This means that believing that one theory is consistent, lets you automatically know that everything you can formalize in terms of that theory is consistent as well.

Set theory itself is not particularly important, it is just useful, since it lets us talk about collections of mathematical objects as first-order citizens of the mathematical universe.

Of course, $\sf ZFC$ itself has limitations imposed by the incompleteness theorems, but we can circumvent them by clever tricks such as adding more axioms that will let us talk about set theory from within set theory. Axioms like large cardinals, or the existence of "nice" models of $\sf ZFC$ within our universe.

There are other approaches to formalizing mathematics. Some people think that we have the natural numbers and nothing more, so we can and should formalize things inside some system of arithmetic. Other people prefer type theory, or categories, or other bases for mathematics. But the important thing is that philosophically all mathematics can be done within a particular framework.

So if someone disagrees on the meaning of the word "mathematics", they might have strong opinions against set theory or other types of foundations for mathematics.

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Depends on the time that you live in! In the 19th century, mostly because of influence by David Hilbert, it was just an interesting question to ask. Imagine living in that time, we have just formalized all of Geometry with some fixed axioms, something similar had happened with Analysis and Arithmetic. So, is there a branch of Mathematics sufficiently "global" where we can formalize all of Mathematics? Then we see that Set Theory works and we try to give it axioms and some concepts are very natural (nowadays it is very common to think of Groups as "sets with a binary operation and some axioms" $(G,*)$ or a Topological Space as a "set with a topology on it" $(X,\tau)$) however some others concepts are not (think of natural numbers where you have that $0\in 1\in 2\in\dots$). Nevertheless Set Theory is still useful to study Mathematics in a meta-theoretical way, only the fact that adding large cardinal axioms helps you measure (in some way) the consistency of logical systems is amazing, and helps us "trust" more in the mathematics we practice everyday. Yet, people still search for different theories where we can formalize all of mathematics, a popular one today is Homotopy Type Theory (or just some type theories) because it allows us to program proof-assistants more easily than set theory axioms, and this kind of software can help a lot of mathematicians to verify that their proofs are correct.

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  • $\begingroup$ Formalizing mathematics in set theory was more or less a 20th century thing. $\endgroup$ – Asaf Karagila May 8 '15 at 5:05
  • $\begingroup$ @AsafKaragila yeah, but the questions were posed at the end of the 19th century. $\endgroup$ – Jonathan Julián Huerta May 8 '15 at 5:18
  • $\begingroup$ Yes. The questions were posed at the last few years of the century. I am sure that Riemann and Gauss weren't that concerned with formalizing their mathematics in set theory. Or with set theory. Gauss in particular. $\endgroup$ – Asaf Karagila May 8 '15 at 5:21

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