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
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
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>.
@philipthrift
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