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Something I have puzzled about is the demonstrated reality continuum hypothesis can be neither proved nor disproved from Zermel-Fraenkel Set Theory. I'm not sure if my question makes sense, but I ask "what would be the consequences of adding the statement 'continuum hypothesis is false' to the axioms?"

Would that mean anything .... i.e. are there interpretable statements that could then be proved or disproved from the new axiom set? For example if there is an Aleph number greater $\aleph_0$ ($= \beth_0 =$ cardinality of natural numbers) but less than $\beth_1$ (= Cardinality of Reals) might there be infinitely many? If so might that be provable (or possibly not)?

David G. Stork
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jonathan birchley
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  • There are some related questions on MSE: https://math.stackexchange.com/questions/648550/, https://math.stackexchange.com/questions/2326252/, https://math.stackexchange.com/questions/2483306/. – Xander Henderson Sep 05 '19 at 17:21
  • Counter examples for the continuum hypothesis necessarily must be very abstract. For every "concrete" set (in particular for constructable sets), the continuum hypothesis is true. Nevertheless, models in which the continuum hypothesis is false seem to be "richer" than models for which it is true. – Peter Sep 05 '19 at 17:48
  • Of course, if we add the axiom CH, CH is provable. If we add "not CH", it is not assuming that ZFC is consistent. – Peter Sep 05 '19 at 17:51
  • Hey. Define (write out) CH, otherwise readers might mistake it for FT, DFT, RH, FLT, LT, ... – David G. Stork Sep 05 '19 at 17:58
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    @DavidG.Stork "CH" is the standard abbreviation; I think it's fine to use. – Noah Schweber Sep 05 '19 at 17:59
  • I've removed the "large cardinals" tag, since it's inappropriate (read the tag description, or see the relevant wiki page); it doesn't just mean "cardinals which are big." – Noah Schweber Sep 05 '19 at 18:00
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    It is rude and unhelpful to not take the time to write out acronyms... even standard ones. After all, not everyone knows every acronym, and this is especially true of non-native-English speakers. – David G. Stork Sep 05 '19 at 18:01
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    Re: the particular question you ask, see here. – Noah Schweber Sep 05 '19 at 18:05
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    @DavidG.Stork That's not really fair to the OP - or do you demand that people write out "ZFC" as a non-acronym, too? Not that writing them out is a bad thing, it certainly can't hurt in this case, but it's a far jump from that to an actual accusation of rudeness. If you feel strongly about it you can always edit the question of course. – Noah Schweber Sep 05 '19 at 18:12
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    Isn't the marked duplicate asking the opposite question? That post is asking what the consequences of the continuum hypothesis are; this post is asking what the consequences of the negation of the continuum hypothesis are. I'm voting to reopen. – Tanner Swett Sep 05 '19 at 18:52

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