Might something like this be fscking unification in physics?
List of Statements Independent of ZFC
71–80 of 108 posts
Re: List of Statements Independent of ZFC
#72Earlier quoted context omitted.
All of calculus.
It's possible to do quite a lot without infinities in calculus and trigonometry, whether it makes sense to bother is another question. https://en.m.wikipedia.org/wiki/Rational_trigonometry
> It's possible to do quite a lot without infinities in calculus and trigonometry
This is like Stacey King saying "I'll always remember this as the night that Michael Jordan and I combined for 70 points."
Pretty much the entire field of Analysis (of which calculus is a part) relies on 'infinities' of some kind. Even if you try to restrict to the rationals, you're still typically working with infinite series of them.
You can do some analysis over the rationals, but often this takes the form of Cauchy sequences of rationals which might be cheating.
[0] https://math.stackexchange.com/questions/387234/how-far-can-...
Re: List of Statements Independent of ZFC
#73What's a reasonable strategy of proving a statement like that is undecidable in ZFC? I think it must use some tools I'm unfamiliar with.
There are 2 popular ways.
1.Set-theoretic technique : forcing, e.g. Cohen forcing which was used in the proof of the independence of the continuum hypothesis.A good reference is Shelah's book on forcing.
2. Model-theoretic: construct a model for ZFC in which the negation of the statement is true and another model for which the statement is true.
There may be proof-theoretic techniques based on the non-existence of proofs involving finitely many steps, but I am not aware of these.
Re: List of Statements Independent of ZFC
#74What's a reasonable strategy of proving a statement like that is undecidable in ZFC? I think it must use some tools I'm unfamiliar with.
Show that it implies the consistency of ZFC, and that its negation also does so. Or show that it's equivalent to a statement that's already on that list.
Re: List of Statements Independent of ZFC
#75One can write down a concrete polynomial p ∈ Z[x1,...x9] such that the statement "there are integers m1,...,m9 with p(m1,...,m9)=0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent).
So you were to specify such a thing and the thing was small enough it's value could be determined by a command line program and you found integers m1.... m9 such that when you typed them at the command line, the value returned was 0, would "reality" have determined a "truth" that was not deducible?
Still, I can't see each of the steps wouldn't be easily determined by existing axioms.
Anyway, my head hurts.
Re: List of Statements Independent of ZFC
#76One can write down a concrete polynomial p ∈ Z[x1,...x9] such that the statement "there are integers m1,...,m9 with p(m1,...,m9)=0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). So you were to specify such a thing and the thing was small enough it's value could be determined by a command line program and you found integers m1.... m9 such that when you typed them at the command line, the val…
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.
Someone enlighten me (not sarcastic).
Re: List of Statements Independent of ZFC
#77Earlier quoted context omitted.
Try thinking about how you could probe that the integers have fewer subsets than the reals using ZFC.
That still feels intuitive. Like for any open interval there are uncountably many reals and a finite number of integers. It seems like nothing changes as you expand the interval.
The integers HAVE fewer subsets than the reals!
Cantor's theorem tells us that the power set of any set has more elements than said set. It's also well known that the reals have the same cardinality as the power set of the integers.
Therefore, |P(R)| > |R| = |P(N)| > |N|, so the power set of the reals has more elements than the power set of integers.
The continuous hypothesis is the belief that no cardinal exists between |N| and |P(N)|. My previous argument fails if we pick two sets A and B, such that |A| < |B| < |P(A)|.
Re: List of Statements Independent of ZFC
#78One can write down a concrete polynomial p ∈ Z[x1,...x9] such that the statement "there are integers m1,...,m9 with p(m1,...,m9)=0" can neither be proven nor disproven in ZFC (assuming ZFC is consistent). So you were to specify such a thing and the thing was small enough it's value could be determined by a command line program and you found integers m1.... m9 such that when you typed them at the command line, the val…
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 it touches on the same theme as your question here, that human intuition about mathematics is weak. The writeup describes and proves the non-computability of a particular function of positive integers with definite (but non-computable!) value.
Re: List of Statements Independent of ZFC
#79Earlier quoted context omitted.
Calculus would likely not be invented without infinity. Many theorems, identities, and techniques may be approximated but ultimately rely on proofs using infinities - I doubt we would have discovered them quickly or at all without infinity. After all, the very concept of a limit evokes the concept of an infinite sequence.
That is true historically but I doubt that it is true necessarily. I mean I suspect that starting from what we now know it should be possible to reconstruct calculus (at least for all practical purposes) without reference to infinities or infinitesimals. One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals. The later, at least, has been my ex…
>One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals.
Don't derivatives count? That's a pretty important and trivial calculation. Sure, you can approximate it when it's nicely behaved, but they aren't always. There's also lots of verrrrrrry slowly converging series that can't be easily computed numerically.
Re: List of Statements Independent of ZFC
#80Some of this was familiar, but I also saw this: > In 1973, Saharon Shelah showed that the Whitehead problem ("is every abelian group A with Ext^1(A, Z) = 0 a free abelian group?") is independent of ZFC. (This is the same Shelah who proved what is usually called "Sauer's Lemma" on set separability, upon which the "VC dimension" and the resulting VC learning theory are based.) Anyway, this was surprising because I didn…
Another famous example of that kind is the existence of outer automorphisms of the Calkin algebra, which is a simple question about a naturally occuring object which turned out to be independent of ZFC by work of Farah and Weaver.
Topology and set theoretic topology is full of those statements, are there Suslin lines? Are there S-spaces? Is the product of two ccc spaces also ccc? The list goes on
Another one from analysis is a very strong form of Fubini's theorem that was shown independent by Friedman.
There's surely more but those are the ones I could remember right now!