Constructibility and Mathematical Existence

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There are several minor variations of the definition of 0 , which make no significant difference to its properties. Instead of being considered as a subset of the natural numbers, it is also possible to encode 0 as a subset of formulae of a language, or as a subset of the hereditarily finite sets, or as a real number. The condition about the existence of a Ramsey cardinal implying that 0 exists can be weakened. Chang's conjecture implies the existence of 0.

Donald A. Martin and Leo Harrington have shown that the existence of 0 is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface analytic game has the same Turing degree as 0. Silver showed that the existence of an uncountable set of indiscernibles in the constructible universe is equivalent to the existence of 0. Its existence implies that every uncountable cardinal in the set-theoretic universe V is an indiscernible in L and satisfies all large cardinal axioms that are realized in L such as being totally ineffable.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 4 months ago. Active 3 months ago. Viewed 3k times. But mathematicians would rather have open problems gives them busy work. Does it mean that all research into alternative theories with large cardinals, determinacy, etc. There are also quite a few current examples in real analysis, group theory, model theory, etc. Caicedo May 20 at In a bit more detail: At the end of the day this gets to a question about what the purpose of foundations of mathematics is. And indeed I take the "profane" view of foundations to what I believe is a rather unpopular extreme: I think that even if we go full Platonist, and even if we're fully convinced after deep technical work that a given principle is true of the universe of sets, that's not enough to add it as an axiom to ZFC.

I'm not claiming this matches the actual history of ZFC at all, by the way! I do separately happen to actually support these per the end of the answer but that's a separate point. Re: 1 , I think you underestimate how complicated that is for someone with no logic background. I'm basing this off of numerous conversations I've had with such mathematicians.

Re: 2 , I honestly find that quite distasteful - I really think that one of the key criteria of a foundational theory is that it be maximally accessible. Those seem to be just an ugly way of disguising satisfaction to not look like logic. Replacement is set-builder notation.

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The axiom that guarantees the existence of a set of the first form is separation. The axiom that guarantees the existence of the second form is replacement. This seems like a much less alien justification than an appeal to only somewhat alien transfinite recursion. Timothy Chow. Both of which are arguably foundational axioms just as much as they are set theoretic axioms. The Whitehead problem was considered important. If history had turned out differently and it had been announced that Shelah had solved it by using deep structural facts about sets, rather than that Shelah showed it to be independent of the axioms, then maybe people would have said, "Hmm so those set theorists figured out something useful after all!

But that's different from acceptance by the mathematical community. It makes me think that an axiom is much more likely to gain foundational status if mathematicians use it "unconsciously"; i. What makes an axiom true?

set theory - Why not adopt the constructibility axiom $V=L$? - MathOverflow

Among them are the following, which I prove or sketch in the article: Observation. From the introduction to the article: Let me briefly summarize the position I am defending in this article, which I shall describe more fully section in 4. Joel David Hamkins.

But yes, these facts are proved in ZFC only for countable models, which is viewed as a "toy" multiverse. We have in principle no way to prove things like this for the full actual multiverse, except in analogy with the toy multiverse. For example, in my paper jdh. Now, the first issue: Set theory is not supposed to go so much against its original direction of inquiry based on unlimited set-formation. And then large cardinals ruin this hot take, but that's a deep fact. On the contrary, the geometric continuum seems to be something very distant, inaccessible from the natural numbers and incomensurable with it.

Third point, I think Suslin lines are very antigeometric. But I agree that these may not seem very strong.

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The constructible real numbers seems not to be the intended real numbers, so proving statements about this object is not the same thing as solving a classical mathematical problem. Admiting nonmeasurable parts is another, but this is in ZFC already which is not neutral with respect to geometry and combinatorics according to what I understand.

Sign me up! I'm not sure this phrase has any meaning whatsoever. It is just some sort of religious proclamation and I find the religion declaring that mathematical notions are linguistics constructs no better than the one declaring that the visible universe is just a mathematical object, while, if we talk about personal preferences, the latter holds much more appeal to me. I should have said "decide", not "settle".

Fair enough. May 21 at Andreas Blass. It looks like Mostowski's lemma holds without choice ncatlab. I'm pretty sure it's possible for two different transitive models of ZF without choice to have the same sets of ordinals.

Pace Nielsen. I have no opinion on this, or whether it even makes sense to ask that question but I find it interesting nonetheless.

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Fix an effective list of all arithmetic statements. This is not computable by a Turing machine. Stock photo. Brand new: lowest price The lowest-priced brand-new, unused, unopened, undamaged item in its original packaging where packaging is applicable. He utilizes this system in the analysis of the nature of mathematics, and discusses many recent works in the philosophy of mathematics.

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