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Is it logically possible that there really is no absolutely consistent formal system, already discovered or yet undiscovered, that can serve as a foundation for mathematics?

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    @EliranH I believe ZF is a formal system. Apr 14 '16 at 14:35
  • Title and body of your posting: they are really two different question. Would you mind updating the answer to clarify what you're interested in?
    – DBK
    Apr 14 '16 at 16:31
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1. Yes, it is logically possible that there is no consistent formal system that can serve as a foundation for mathematics.

This follows immediately from Gödel's second Incompleteness theorem, which proves that that no consistent axiomatic system which includes Peano arithmetic (or stronger) can prove its own consistency.

Assuming that any such foundational theory T is such an effectively generated formal system, i.e an axiomatic theory expressive enough to develop basic properties of the natural numbers, T cannot prove its own consistency. Thus is it logically possible that it is inconsistent.

2. No, it is highly unlikely that there are no consistent foundations for mathematics.

This would mean that every conceivable candidate is inconsistent. But this is not what our mathematical practice tells us.

The assumption that de facto candidates for such a foundational theory of mathematics (ZFC plus some other axioms) are consistent is not a simple convention. Logicians apply ampliative reasoning usually found in the science and think they have some good measure of inductive support to believe that they these candidates are consistent: they have been working with ZFC for almost a century and have never encountered an inconsistency. (Cf. this with the discovery of the inconsistency of Frege's system, which was discovered by Russell while Frege's publication was still in press.)


Note that we might be able to prove the consistency of such a foundational theory relative to the assumed consistency of another theory: If ZFC is consistent, so is ZFC plus some axiom. But this is not what you're after, I guess?

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  • Your first point hits the nail in the head, I'm quite ashamed to not have noticed it myself. Also, thanks for the insightful second point, which is not really about my question, but is an interesting read anyway. Apr 14 '16 at 17:45
  • The second point is not supported. We took forever to find the inconsistencies in earlier systems. They are just not near the surface. And we have proposed all kinds of workarounds that can be verified to solve the problems that we see. But it is not our mathematical experience that systems we trust do not ultimately have paradoxes farther down. It is our experience from Pythagoras forward that they are inconsistent, but can be patched endlessly.
    – user9166
    Apr 14 '16 at 18:00
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    Your first point relies on a mathematical theorem. Can you really conclude that it's logically possible that the whole of math is inconsistent based on a mathematical theorem?
    – Eliran
    Apr 14 '16 at 18:17
  • ZF has been proven (from the "outside") to be Post consistent, by Godel himself no less. The Continuum Hypothesis can be neither proved nor disproved in ZF.
    – Era
    Apr 14 '16 at 18:45
  • Re: point 2, I took it as a given that a foundation for mathematics must be a foundation for all mathematics. Even if ZFC is "foundational" (and it is), we know for sure that it's not a consistent foundation for all mathematics because it is incomplete. Godel's theorems tell us that we will never have such a foundation, only incomplete "foundational" systems.
    – Era
    Apr 14 '16 at 18:48
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Let me first point out that a consistency proof is something carried out within mathematics.

Now, assume PA is your foundation-candidate. Is it consistent? Well, Gödel's second incompleteness theorem shows that PA cannot prove its own consistency. So you must assume some stronger theory, say ZF, in order to prove PA consistent (it is well known that PA's consistency can be proven within ZF).

But, you may now ask, is ZF consistent? Of course, ZF cannot prove its own consistency (Gödel), so again you must assume some stronger theory to prove ZF consistent.

This can go on. In the end, you must assume something. You must take some mathematical theory to be the background theory with which you work in mathematics, and you have to take its consistency on faith (or rather on the fact that no one has ever found a contradiction within it).

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  • Sorry to have misled you, I was talking about absolute consistency. Apr 14 '16 at 17:41
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    @Constantine but what exactly do you mean by that? My point is that some math has to be used in the background. See my comment to the chosen answer.
    – Eliran
    Apr 14 '16 at 18:28
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The current formalist approach, officially retreating from the inconsistencies in ordinary logic back to what is formalizable, kind of proves that if there is no natural formalization of mathematics, we will simply declare one. From the point of view of the other contenders in the argument over the natural foundations of mathematics, this is the current state of affairs. Formalists won by fiat, we simply will have formalization because that is the way it is supposed to be. Then we all went back to being closet Platonists and ignoring the natural holes in human logic.

From a neo-Intuitionist point of view, this is to be expected. If mathematics is the repository of abstract intuitions that are adequately widely shared, instead of being a study of some real logical core at the center of reality, then in order to share it widely, we need to feel good about its internal consistency. Mathematics is meant to codify and collect our logic, so if it extends outside the bounds of our trust in our own logic, we will just circumscribe it for our own comfort.

There may be wars over the best formalism, but in the end we accept that there should be one, and if worse comes to worst, we will simply choose.

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