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List of Statements Independent of ZFC

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Re: List of Statements Independent of ZFC

#51

Post by deepsun »

ur-whale wrote: post #21
triska wrote: post #3

A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding statements that are independent of ZFC? https://mathoverflow.net/questions/1924/what-are-some-reason... With the top voted result currently being: "If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." As is also mentioned i…

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.

Re: List of Statements Independent of ZFC

#53

Post by empath75 »

ArchReaper wrote: post #34

Is there any ELI5-type explanation for us non-mathers? Whenever I see stuff like this, I start trying to actually understand it, then fail miserably just trying to google terms I'm not familiar with. Is advanced knowledge of these math principles required to understand the significance of this page, or why it is interesting?

These are a list of theories that can’t be proven or disproven using the most common mathematical foundation.

Re: List of Statements Independent of ZFC

#54

Post by xelxebar »

The_rationalist wrote: post #26
correct_horse wrote: post #20

Earlier quoted context omitted.

It is easy to prove for finite sets (just count) but much harder for infinite ones. For example, which has more subsets, the integers or the positive integers? How about the integers and the reals? If you answered "the integers" to the first question, you aren't thinking about this right, as the integers and the positive integers have the same number of elements to start with. source: https://en.wikipedia.org/wiki/Al…

Another example of why mathematics are a wrong abstraction to be optimally useful. Mathematics should have bounds in the same way as our universe has bounds. Any theorem that has a different behavior if something is infinite doesn't matter at all and is a waste of time for real engineers who solve things in the real world. The niche of mathematics that describe things beyond what our universe has to offer should be a…

A formalist might say that mathematics has no infinities. Even infinite sets are just definitions of finite length, even if expanded all the way to ZFC axioms. The problem is your intuitive understanding.

A realist might say that mathematics exists within this universe, therefore it is equally subject to its constraints just like anything else. Problems arise via misapplication or misinterpretation of the math.

A model theorist might point out that even in models of math with only a countable collection of objects, it's still true that the reals are uncountable. The problem isn't with infinitie, it's that the finitary objects have subtle interactions.

A logician might object that ZFC already does start of with an intuitive notion of infinity, i.e. the counting numbers is a natural collection. The problem is that this inevitably has unintended consequences.

A historian might gently point out that this debate already occurred vigorously about 150 years ago, and the verdict was that we just gotta live with the weirdness of infinities. The problem is that we lose too much useful math by trying to throw them out.

Etc.

Re: List of Statements Independent of ZFC

#55

Post by tgflynn »

voxl wrote: post #44
The_rationalist wrote: post #26

Earlier quoted context omitted.

Another example of why mathematics are a wrong abstraction to be optimally useful. Mathematics should have bounds in the same way as our universe has bounds. Any theorem that has a different behavior if something is infinite doesn't matter at all and is a waste of time for real engineers who solve things in the real world. The niche of mathematics that describe things beyond what our universe has to offer should be a…

Since you feel so strongly about it, let me argue that you're simply wrong: Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers. In analysis, it's much easier to reason about infinitely differentiable, or smooth, surfaces then very rigid and complicated services. The fact that infinities ca…

> Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers.

I would argue that a few extra lines in a proof is a small price to pay to avoid the Godelian catastrophe.

Re: List of Statements Independent of ZFC

#56

Post by reggieband »

empath75 wrote: post #49
reggieband wrote: post #35

I recall once I read the short book "The Philosophy of Set Theory" [1] since I like philosophy and have an interest in Math. It contains much of the history that lead up to the decision to base significant portions of the soundness of mathematics on top of set theory (and by proxy: Cantor's work on infinities). My recollection is fuzzy since it was years ago but I recall it starts at Zeno's paradox and follows along…

I’m not sure that any side won. ZFC is merely a game with clearly defined rules that lots of people have agreed to play, but it is not the only game, by a long stretch.

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 doubt a single person in here would support such an approach on any level. I think that represents a clear win. Answering objections about the fundamentals of math using formalisms like ZFC was a component of that.

Re: List of Statements Independent of ZFC

#57

Post by empath75 »

reggieband wrote: post #56
empath75 wrote: post #49

Earlier quoted context omitted.

I’m not sure that any side won. ZFC is merely a game with clearly defined rules that lots of people have agreed to play, but it is not the only game, by a long stretch.

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.

Re: List of Statements Independent of ZFC

#58

Post by xigency »

gjm11 wrote: post #46
ArchReaper wrote: post #34

Is there any ELI5-type explanation for us non-mathers? Whenever I see stuff like this, I start trying to actually understand it, then fail miserably just trying to google terms I'm not familiar with. Is advanced knowledge of these math principles required to understand the significance of this page, or why it is interesting?

[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?

Re: List of Statements Independent of ZFC

#59

Post by Enginerrrd »

pvarangot wrote: post #40
egdod wrote: post #6

Earlier quoted context omitted.

That is a strange one.

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.

Re: List of Statements Independent of ZFC

#60

Post by daxfohl »

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?

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

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