> On 10 Apr 2018, at 08:39, Brent Meeker wrote:
>
> I wonder if Bruno is familiar with this paper?
>
> https://arxiv.org/pdf/1312.0670.pdf
Not bad at all, interesting! I did not got the time to verify all details, but
it looks valid!
But all what they show is that , let me quote them:
<<
Thus, two models of set theory can agree on which natural numbers exist and
agree on all the details of the standard model of arithmetic, yet disagree on
which sentences are true in that model.
>>
The contrary would have been astonishing. That is part of why I insist
Mechanism is a theology, a risky invitation of an unknown at the table, which
looks strangely like yourself.
The Model of Set Theory have too much imagination. I am already happy that ZF
and ZFC captured the same arithmetical truth.
The paper does not illustrate that the notion of arithmetical truth is not
definite. It illustrates that machine with diverse beliefs in diverse
infinities will not find a common approximation oft that truth easily, and
might indeed interpret it differently, and disagrees on many, arithmetical
sentences.
The universal numbers are condemned to have difficulties to agree or disagree
already on their relations with the “other” universal numbers. Set theory is a
tool for understanding the numbers, but by incompleteness it cannot do that
completely, and the paper here illustrates that ZF + very different axioms can
get different part of the arithmetical truth, limited, and different from each
others. That would not be the case, with semis-computable sets, so this is
weakened, in the computationalist realm by the use of actual infinities.
You can relate this also to the fact that the Löbian universal machine
disagrees already with all complete theories you would claim about it/her.
Bruno
>
> Brent
>
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