On 9/25/2017 6:37 AM, Bruno Marchal wrote:
On 24 Sep 2017, at 21:02, smitra wrote:
On 23-09-2017 10:34, Bruno Marchal wrote:
On 22 Sep 2017, at 13:47, David Nyman wrote:
https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/
[1]
A rare progress on the continuum hypothesis (CH). Shelah is amazingly
smart. There is that story that he arrived one week to early at a
congress of logic, and decided to follow a congress on group theory
instead, and depressed everybody by solving most open problems of that
congress! His first question was "what is a group?", and people taught
he was retarted!
Now, this does not necessarily concern us. I think. Even ZF and
ZF+Choice proves the same theorems in arithmetic. That is probably not
the case for ZF and ZF + CH, but the comp ontology will not change.
For the phenomenology, that might change something though, making the
measure problem more easy or more difficult. We are not yet enough
advanced on this to decide, i think. model theory and set theory are
*quite* complex compared to arithmetic!
Bruno
Everything in physics suggests that infinities don't actually exists,
so perhaps more progress can be made if you use a finitistic logics
system.
That is the case for computationalism. It belongs to finitism. You can
interpret all the infinities which appears at the phenomenological
level as machine's inventions to study the finite realm. That is what
I do actually in the math treatment.
In fact, contrary to what I have thought some years ago, it even
admits an ultrafinist reading, although you need again some infinities
at the meta-level to prove this. Computationalism is consistent with
"there is a highest natural number". But no need of this to proceed,
unless we met a genuine ultra-finitist (that is very rare!).
How can that work? In a finite system Goedel's theorem doesn't hold.
Every proposition can be decided by exhaustive search.
Brent
Note that you cannot invoke a God or a Physical Universe to decide
what exists or not, or you beg the (metaphysical) question. Someone
could say that everything in physics suggest that there is no physical
reality existing per se, but only statistically interfering
computations "seen from inside". Look how quick people like Bohr and
Heisenberg were to abandon realism in physics. Fortunately Einstein
and Everett were not that quick, and computationalism go in that same
direction.
Bruno
Saibal
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[1]
https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/
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