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List of Statements Independent of ZFC

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Re: List of Statements Independent of ZFC

#61

Post by reggieband »

empath75 wrote: post #57
reggieband wrote: post #56

Earlier quoted context omitted.

I think it feels easier to say that now that we are long past the point where the debates occurred. But this happened during a time when universities were still trying to figure out how to divide up sciences. Nowadays, the idea that math has a role to play in pretty much every science isn't really questioned at all. I mean, imagine I suggested that something other than math should be brought to bear on physics. I dou…

ZFC has very little to do with why math is used in universities or the sciences, and it would still be used even without it, because as you said, it works. It worked for 3000 years before we had ZFC after all. ZFC wasn’t even the end of the debate on mathematical foundations even in math. There are a lot of people trying to redo everything with types and category theory today.

ZFC follows along a path including (but not started by) Russell/Whiteheads Principia Mathematica, which famously (infamously?) takes several hundred pages to prove 1+1=2. I doubt very few have thought ZFC (or it's variants) would be the last word.

Almost no scientists cared about formalizing or proving the soundness of the mathematical tools they used. In the same way the majority of programmers do not care about proving the soundness of their programming languages. In general, people seem to be interested in the practical aspects of their work.

But the general idea that symbolic logic is the primary basis for understanding the world is something a bit different and something we rarely question now. I think people assume that this is some obvious thing but it is actually an idea that was coordinated and forwarded. It appears to me that the debate at the beginning of the 20th century around using set theory to establish the foundations of math by way of logic is when the scale seems to have heavily tipped towards that particular idea.

Re: List of Statements Independent of ZFC

#62

Post by CrazyStat »

deepsun wrote: post #51
ur-whale wrote: post #21

Earlier quoted context omitted.

So, does that mean that one can assume this to be true and build a perfectly consistent theory, or conversely assume it to be false (with - say - at least one counter-example) and build another perfectly consistent theory?

Well, it's not possible to prove the consistency, thanks to Godel. Maybe one of your new theories would contain a statement, inconsistent with the rest of ZFC.

This is incorrect. It absolutely is possible to prove consistency, what Gödel tells us is that in any consistent logic system there are true but unprovable (in that system) statements.

For this particular list, the statements have been proven to both be consistent with ZFC and for their negations to be consistent with ZFC.

Re: List of Statements Independent of ZFC

#63

Post by eximius »

tgflynn wrote: post #41
eximius wrote: post #33

Earlier quoted context omitted.

Tell all of the engineers using calculus that they don't need infinity. And you can't easily only partially include infinity.

Computers handle calculus quite nicely using numerical algorithms, which don't involve any infinite sets. > And you can't easily only partially include infinity. Sure you can. You just need to use a dx that's small enough for the particular functions you're working with and the degree of precision you need.

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.

Re: List of Statements Independent of ZFC

#64

Post by jfarmer »

xigency wrote: post #58
gjm11 wrote: post #46

Earlier quoted context omitted.

[The following isn't really "like you're five", but given how long it is already that's probably just as well.] Proofs and formal systems, and why we're kinda screwed Mathematicians like to prove things. What we would really like would be to be able to find, for every mathematical statement, either a proof that it's true or a proof that it's false. It wasn't until the early 20th century that mathematicians got a clea…

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.

Re: List of Statements Independent of ZFC

#65

Post by improv32 »

baddox wrote: post #43
new2628 wrote: post #30

Earlier quoted context omitted.

This is a fine and defensible (although not mainstream) viewpoint with several adherents among mathematicians and various levels of success in making it formal and precise. See e.g. https://en.wikipedia.org/wiki/Finitism Doron Zeilberger would like to pat you on the back.

How do finitists describe the decimal representation of the fraction 1/3?

I don't think finitists have a problem with convergent sequences, they just wouldn't talk about infinities, rather than extending the sequence gets you as arbitrarily to 1/3 as you wish to go

Re: List of Statements Independent of ZFC

#66

Post by tgflynn »

eximius wrote: post #63
tgflynn wrote: post #41

Earlier quoted context omitted.

Computers handle calculus quite nicely using numerical algorithms, which don't involve any infinite sets. > And you can't easily only partially include infinity. Sure you can. You just need to use a dx that's small enough for the particular functions you're working with and the degree of precision you need.

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 experience. I think I have a fairly decent practical intuition for calculus based on imagining dx becoming smaller and smaller until it's small enough, but I don't think my brain has any actual representation of "true infinitesimals" and my intuition breaks down completely if I try to imagine things like the relationship between the rational and irrational numbers. Maybe that's due to my intellectual limitations, but I wonder if it isn't because these concepts might be over-elaborate abstractions that don't really exist in our world.

Re: List of Statements Independent of ZFC

#67

Post by aetilley »

maest wrote: post #24

What'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.

To show that a sentence P is independent of a theory (that is, a set of sentences) T, generally one constructs or demonstrates the existence of a model of T + P and also produces a model of T + (not P).

https://en.wikipedia.org/wiki/Model_theory

Re: List of Statements Independent of ZFC

#70

Post by ithinkso »

CrazyStat wrote: post #62
deepsun wrote: post #51

Earlier quoted context omitted.

Well, it's not possible to prove the consistency, thanks to Godel. Maybe one of your new theories would contain a statement, inconsistent with the rest of ZFC.

This is incorrect. It absolutely is possible to prove consistency, what Gödel tells us is that in any consistent logic system there are true but unprovable (in that system) statements. For this particular list, the statements have been proven to both be consistent with ZFC and for their negations to be consistent with ZFC.

This is first incompleteness theorem. What deepsun was referring to is second incompleteness theorem - in a consistent system F the statement 'F is consistent' is in fact unprovable (in F).
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