[triska]

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 in the linked entry, this statement is independent of ZFC.

[ur-whale]

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?

[karatinversion]

Yes, this is the exact meaning of “independent from” here.

[deepsun]

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.

[ginnungagap]

When we say that a statement P is independent of ZFC what we're really saying is "if ZFC is consistent, then P is independent of ZFC (hence both ZFC+P and ZFC+not P are consistent)".

This is the only sensible way to interpret claims of independency, since if ZFC is inconsistent it just proves every statement so there's no independent ones

[CrazyStat]

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.

[ithinkso]

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).

[ginnungagap]

It gets worse than that, by Easton's theorem, except for two mild restrictions provable on ZFC, the function that sends a regular cardinal to the cardinality of its powerset can be anything

[egdod]

That is a strange one.

[yarg]

Infinities are strange.

[pvarangot]

Try thinking about how you could probe that the integers have fewer subsets than the reals using ZFC.

[Enginerrrd]

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.

[rodamaral]

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)|.

[The_rationalist]

WTF I'm pretty sure I can prove that

[correct_horse]

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/Aleph_number

[The_rationalist]

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 an optional extension to the mathematical framework instead of being the default, in order to not pollute discussions.

[new2628]

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.

[baddox]

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

[improv32]

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

[voxl]

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 cause some complications in the foundations itself is a drop in the bucket to the practical application and simplifications of day to day mathematics.

[tgflynn]

> 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.

[xelxebar]

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.

[tgflynn]

Yes, my current inclination would be to define truth (in a practical sense) as that which is computable. Everything else belongs to the realm of fiction or fantasy.

Note that I don't mean to imply that this is the entirety of my philosophy. Just that one should distinguish between what is practically relevant and what is not and I think the current state of mathematics and CS does a poor job of making this distinction.

[roywiggins]

The most prolific contemporary proponent of this view is probably Norman Wildberger, who takes a very unconventional approach and tries to build everything without real numbers, which he doesn't believe make physical or logical sense and therefore shouldn't be used as a basis for anything.

He's very nearly a minority of one, mind you.

[eximius]

Tell all of the engineers using calculus that they don't need infinity.

And you can't easily only partially include infinity.

[tgflynn]

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.

[eximius]

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.

[voxl]

Except for all that numerical analysis around floating point arithmetic...

[The_rationalist]

Could you give a use case?

[Filligree]

All of calculus.

[RossBencina]

Being "optimally useful" is not the goal of mathematics. See for example, https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

[0-_-0]

I know that feeling. It never lasts long.

[ur-whale]

>I'm pretty sure I can prove that

For infinite sets of arbitrarily large cardinality?

[The_rationalist]

No I can't. But see my answer to correct_horse