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A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding
by triska 7y ago
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-reasonable-sounding-statements-that-are-independent-of-zfc/6594 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.
- sansnomme 7y agoCantor bad.
- egdod 7y agoWhat does this have to do with Cantor?
- saagarjha 7y agoCantor did work on cardinalities, presumably.
- GraemeL 7y agoZFC was created to avoid the paradoxes inherent in Cantor's set theory work.
- JadeNB 7y agoI think it's fairer to say that ZFC was a response to paradoxes in naïve set theory of the sort discovered by Russell than in response to Cantor; see, e.g., > In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. (https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory). Cantor discovered a rigorous basis for the intuitively paradoxical situation of two sets where it seems that one is manifestly bigger, but they are actually the same size; but I think that this work is best viewed as a resolution of the paradox, not a paradox itself.
- dang 7y agoPlease don't post unsubstantive comments here.
- egdod 7y agoThat is a strange one.
- yarg 7y agoInfinities are strange.
- pvarangot 7y agoTry thinking about how you could probe that the integers have fewer subsets than the reals using ZFC.
- Enginerrrd 7y agoThat 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 7y agoThe 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)|.
- deleted 7y ago[deleted]
- The_rationalist 7y agoWTF I'm pretty sure I can prove that
- ur-whale 7y ago>I'm pretty sure I can prove that For infinite sets of arbitrarily large cardinality?
- The_rationalist 7y agoNo I can't. But see my answer to correct_horse
- correct_horse 7y agoIt 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 https://en.wikipedia.org/wiki/Aleph_number
- The_rationalist 7y agoAnother 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 7y agoThis 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 https://en.wikipedia.org/wiki/Finitism Doron Zeilberger would like to pat you on the back.
- ur-whale 7y agoSo, 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?
- deleted 7y ago[deleted]
- karatinversion 7y agoYes, this is the exact meaning of “independent from” here.
- deepsun 7y agoWell, 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.
- CrazyStat 7y agoThis 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 7y agoThis 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 7y agoWhen 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
- ginnungagap 7y agoIt 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