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List of Statements Independent of ZFC

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Re: List of Statements Independent of ZFC

#101

Post by mirimir »

drdeca wrote: post #97
mirimir wrote: post #71

Might something like this be fscking unification in physics?

I doubt it. Physics doesn’t seem to have much which is clearly connected to proof systems. And, besides, from any (consistent) axiom system for which a given statement is undecidable, there is another axiom system which is the same except it adds that statement as an additional axiom, and the statement is therefore (trivially) provable in that system. And, it doesn’t seem like physics is constrained to use only some…

Thanks.

But doesn't that lead to systems with infinite numbers of axioms?

Re: List of Statements Independent of ZFC

#102

Post by OscarCunningham »

joe_the_user wrote: post #99
OscarCunningham wrote: post #86

Earlier quoted context omitted.

You're right that if ZFC is consistent then it has a model in which this polynomial has an integer root. The issue is that the "integers" in the model are different from the actual integers. So you can't take the root out of the model and use it to prove ZFC inconsistent.

That is certainly how I would understand the situation. However, it seems intuitively you construct the notion of finite, could engage in the ordinary computation and have a way of distinguishing these "weird roots" from regular roots. I suppose if you make the position of (something like) "every polynomial whose root cannot be found by finite calculation does not, in fact, have one" an axiom but then you would have…

You might be interested in https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theo....

Re: List of Statements Independent of ZFC

#103

Post by jfarmer »

jfarmer wrote: post #64
xigency wrote: post #58

Earlier quoted context omitted.

Well done. Has anyone named a set in between the rationals and the reals?

Whether a set of cardinality strictly between the rationals and reals exists is independent of ZFC. https://en.m.wikipedia.org/wiki/Continuum_hypothesis There are many sets which are strict supersets of the rationals and strictly sheets of the reals, of course.

"strictly subsets of the reals", thanks autocorrect.

For example, if ℚ is the rationals and ℝ is the reals then ℚ∪{√2} is a strict superset of the rationals but a strict subset of the reals. However, it still has the same cardinality as the rationals (cf. https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...)

Re: List of Statements Independent of ZFC

#104

Post by Dylan16807 »

dodobirdlord wrote: post #78
joe_the_user wrote: post #76

Earlier quoted context omitted.

OK, the way I'd figure it out is: for such a polynomial, you definitely can't find those integers. There isn't any concrete m1...m9 satisfying the condition. The stumbling block is you can't find a proof for this fact in ZFC. But this seems to go against the idea that for any proposition independent from an axiom system, there is a model of the axiom system where that proposition is true and another where it is false…

> OK, the way I'd figure it out is: for such a polynomial, you definitely can't find those integers. There isn't any concrete m1...m9 satisfying the condition. The stumbling block is you can't find a proof for this fact in ZFC. If this were formalizable it would be a proof. I encourage you to read through the excellent piece on the busy beaver function and computability. It's not entirely related, but it's fun! And i…

> If this were formalizable it would be a proof.

A proof, but not a proof that can be expressed inside ZFC.

Re: List of Statements Independent of ZFC

#105

Post by chrisweekly »

chrisweekly wrote: post #91

"The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC." - TFA

"ZFC" is an esoteric acronym; I posted the quote from the article to help others because the title was entirely opaque to me.

Re: List of Statements Independent of ZFC

#106

Post by dodobirdlord »

Dylan16807 wrote: post #104
dodobirdlord wrote: post #78

Earlier quoted context omitted.

> OK, the way I'd figure it out is: for such a polynomial, you definitely can't find those integers. There isn't any concrete m1...m9 satisfying the condition. The stumbling block is you can't find a proof for this fact in ZFC. If this were formalizable it would be a proof. I encourage you to read through the excellent piece on the busy beaver function and computability. It's not entirely related, but it's fun! And i…

> If this were formalizable it would be a proof. A proof, but not a proof that can be expressed inside ZFC.

Obviously, the original statement has been proved to be independent from ZFC.

Re: List of Statements Independent of ZFC

#107

Post by drdeca »

mirimir wrote: post #101
drdeca wrote: post #97

Earlier quoted context omitted.

I doubt it. Physics doesn’t seem to have much which is clearly connected to proof systems. And, besides, from any (consistent) axiom system for which a given statement is undecidable, there is another axiom system which is the same except it adds that statement as an additional axiom, and the statement is therefore (trivially) provable in that system. And, it doesn’t seem like physics is constrained to use only some…

Thanks. But doesn't that lead to systems with infinite numbers of axioms?

If you want to add axioms in order to be able to show each of an infinite number of statements which are all independent of the initial axiom system, and also none of them follow from the rest of them, that could involve an infinite set of axioms, yes,

but any statement we could make about stuff in physics, if it could be expressed in the language of the formal system, would, as a single statement, be something that could be added as a single axiom.

Re: List of Statements Independent of ZFC

#108

Post by daxfohl »

daxfohl wrote: post #60
xigency wrote: post #58

Earlier quoted context omitted.

Well done. Has anyone named a set in between the rationals and the reals?

The canonical example is https://en.wikipedia.org/wiki/Vitali_set You can kinda think of it as "the set of real numbers MOD the set of rationals". Kinda. And they're dorked up because the length is infinitesimal, but a countably infinite number of them add up to length 1. For a more precise explanation see here: https://math.stackexchange.com/a/137959/287133

Also note that the construction of the above set requires the axiom of choice. And, as we all know, the axiom of choice is equivalent to the continuum hypothesis in ZFC. So that's how it all fits.
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