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We know that ZFC was formulated to avoid some paradoxes inherent to Cantor's naive set theory, such as Russell's paradox, which inquires about the truth of the existence of the set of all sets.

The notion of set, however, comes prior to any axiomatic set theory: to write ZFC's nine axioms, we are using sets of variables, constants, functions and relations. Those primitive, pre-ZFC sets, don't generate paradoxes simply because we choose never to write paradoxes with them: those sets are solely used to generate ZFC.

The point is: can we avoid ZFC in its entirety by keeping on formulating mathematics with those primitive sets and never touching any paradoxes?

Incognito
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  • The paradoxes show that a statement and its negation can be proved in that system. The fear is that since you know you can generate a paradox, that you may unknowingly use one in your proof (though maybe not one of the known ones) and therefore may be able to prove a statment $\phi$ and its negation $\lnot \phi$. ZFC is a formalization of what rules you can use and seemingly using those rules does not lead to a paradox (though this has not be proved and can't really be proved, but that is another discussion) – Atticus Christensen Mar 19 '15 at 17:47
  • So ZFC was formulated for people not to formulate paradoxes by accident on their proofs? – Incognito Mar 19 '15 at 18:00
  • I think Asaf's first paragraph is a good analogy for what I was trying to say – Atticus Christensen Mar 19 '15 at 19:51
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    I imagine that it is possible to lead a long and distinguished career in mathematics without ever having to dust off the ZFC's. The intuitions of Newton, Leibnitz and Gauss have been more or less confirmed by 20th century set theorists leaving most mathematicians today to forge on with worrying much at all about inadvertently uncovering any Russell's-Paradox-type inconsistencies in the foundations of their various fields. – Dan Christensen Mar 20 '15 at 02:52

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The question reads equivalent to "Will my hard drive always have free space, as long as I don't check that?", and the answer is of course not. If there is a paradox, then you will run into it or at some point be convinced that it is unlikely to exist.

But if we look deeper into your question, then we can look at it as a question asking about where do the axioms of $\sf ZFC$ "live" themselves.

The answer can be looked at from three main point of views:

  1. There is a big universe of sets, and we believe that it satisfies the axioms of $\sf ZFC$ and devoid of paradoxes (at least in the Russell-Cantor-Burali Forti kind of paradoxes).

    Then we formulate some axioms which seem "reasonable" and we want to investigate models of those axioms which are "toy universes" if you will, and in the process learn about truth and false statement in the real universe of sets.

  2. There is no universe of sets. We have the natural numbers, a gift given to us by Brouwer's god. Using clever tricks, like Godel coding, we develop first-order logic using the natural numbers, and we have a formula which states when something is an axiom of $\sf ZFC$. And everything we do, we do in the confines of finding a proof or refutation of statements from these axioms.

    But these are manipulations of natural numbers. There are no sets, there is no magical bean or some figure behind the curtain. We have the natural numbers, and we work there.

    Of course, you might feel obligated to ask, where did the natural numbers come from? Aren't they a product of the universe of sets? Yes, this is one option. Another is to related claim that all we do is write a few strings on paper and manipulate them.

These are philosophical questions and approaches to mathematics, and you have to figure it out on your own. Oh yeah, I promised three approaches.

  1. Everything about infinity is bollocks, and we waste our time in modern set theory and modern mathematics. Okay, this one is not as popular as the others. And it's a good thing too, I guess.
Asaf Karagila
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  • Asaf, for 2 surely if we can write down strokes and call them natural numbers, we can write down sentences and call them formulae without any need for Godel encodings? – Atticus Christensen Mar 19 '15 at 19:56
  • Sure. You can just write things on paper. But if you want to appeal to so many induction arguments later on, especially "induction on the complexity of the formula" and such, then it's easy to do it in a framework that is easy for induction. This is why we have the natural numbers. – Asaf Karagila Mar 19 '15 at 20:15
  • Sure. My point was more on one that natural numbers don't really have a precedence of ontology – Atticus Christensen Mar 19 '15 at 20:19
  • Well. Depends who you ask. If you ask Brouwer, he might say otherwise. And I've met a handful of other people who were finitists, in the sense that the natural numbers exists; but all those infinite sets? That's a nice idea, but immaterial. – Asaf Karagila Mar 19 '15 at 20:21
  • I guess to clarify even more, I mean that natural numbers shouldn't really have precedence of ontology over other finite objects, like formulae if we are thinking about natural numbers as strokes. – Atticus Christensen Mar 19 '15 at 20:41
  • Ah, okay, with that I agree. But I guess we shouldn't think about them in a physical way. Because then how can you differentiate between $2^{1000000000}$ and $3^{1000000000}+4421$? – Asaf Karagila Mar 19 '15 at 20:42