> On 15 May 2019, at 19:50, Philip Thrift <cloudver...@gmail.com> wrote: > > > > On Wednesday, May 15, 2019 at 10:31:37 AM UTC-5, Bruno Marchal wrote: > > On 13 May 2019, at 08:55, Philip Thrift <cloud...@gmail.com <javascript:>> > wrote: > > 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 > > > > > The set-theoretic multiverse of Hamkins > > https://arxiv.org/pdf/1108.4223.pdf > <https://arxiv.org/pdf/1108.4223.pdf> > > > goes beyond the model-theoretic forcing methods of Cohen, with a framework > for a multiverse of dialects (my word) of set theory, each with their own > definition of "set". I'm not a set theorist, but can read the paper > approximately well, and it was enough to get him from City University New > York to Oxford. > > https://en.wikipedia.org/wiki/Joel_David_Hamkins#Biography > <https://en.wikipedia.org/wiki/Joel_David_Hamkins#Biography> > > In September 2018, Hamkins moved to the University of Oxford > <https://en.wikipedia.org/wiki/University_of_Oxford> to become Professor of > Logic in the Faculty of Philosophy and Sir Peter Strawson Fellow in > Philosophy in University College, Oxford > <https://en.wikipedia.org/wiki/University_College,_Oxford>. >
When assuming Mechanism, we cannot even add the induction axioms to RA. With a richer theory than RA for the ontology, the mathematical multiverse becomes so big that the white rabbits cardinality explodes. Hamkins theory is interesting, no doubt, and might be used in the phenomenology of numbers, but it should not be taken “ontologically”, unless we abandon Mechanism. 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 everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/992c5ede-0609-4ee7-bb8c-6da22283dded%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/992c5ede-0609-4ee7-bb8c-6da22283dded%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 everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/32AC7C89-3A20-4E86-BF47-EB7A5F31A673%40ulb.ac.be.
