> On 13 May 2019, at 08:55, Philip Thrift <[email protected]> wrote: > > > > On Sunday, May 12, 2019 at 9:40:12 PM UTC-5, Jason wrote: > > > > > Incompleteness disproves nominalism. Arithmetical truth was proven not only > to be not human defined, but to be not human definable. > > > > (This is something I posted a few days ago in another forum.) > > From Joel David Hamkins @JDHamkins - http://jdh.hamkins.org/ > <http://jdh.hamkins.org/> > > "Truths" in the set-theoretic multiverse (slides from a talk last week): > > http://jdh.hamkins.org/wp-content/uploads/Is-there-more-than-one-mathematical-universe.pdf > > <http://jdh.hamkins.org/wp-content/uploads/Is-there-more-than-one-mathematical-universe.pdf> > > > The final slides: > > ---- > > The Continuum Hypothesis is settled > > On the multiverse perspective, the CH question is settled. > It is incorrect to describe it as an open question. > > The answer consists of our detailed understanding of how the > CH both holds and fails throughout the multiverse, of how these > models are connected and how one may reach them from each > other while preserving or omitting certain features. > > Fascinating open questions about CH remain, of course, but the > most important essential facts are known. > > Ultimately, the question becomes: do we have just one > mathematical world or many > > ---- > > Mathematics is a language - with multiple dialects. > > Each dialect of mathematics has its own syntax (to some extent) and > semantics!
If it has a semantic, it is not just a language, there is a reality/model/semantic, and we have to distinguish languages and possible theories on that reality. It is obvious (for a mathematical logician) that there are many mathematical worlds, but like in physics, this does not interfere with realism, on the contrary. Now, I use only arithmetical realism, on which everybody agree. The standard arithmetical truth is definable with a bit of set theory, on which most people agree (as it is the intersection of all models of the theories RA or PA). That is as acceptable as any theorem in analysis. With Mechanism, Analysis, and physics, remains full of sense, but have became phenomenological. > > > > There is no settled "truth" in mathematics. > > For example (as Hamkins shows) the CH is true in one dialect (of set theory) > and false in another. That was shown by Cohen and Gödel. Interestingly, ZFC and ZF + CH does not prove more arithmetical propositions than ZF alone. The arithmetical truth is totally independent of the axiom of choice or the continuum hypotheses. Now, ZF proves much more theorems in arithmetic than PA, which proves much more than RA. Bruno > > @philipthrift > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/06ca3480-cdf1-426b-9f38-404bc2fa1550%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/06ca3480-cdf1-426b-9f38-404bc2fa1550%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/07CE2D6F-E36D-45E6-883E-E9A13C4812B3%40ulb.ac.be.
