On 26 Sep 2017, at 19:24, Brent Meeker wrote:
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.
Not at all. It is true that both PA and ZF cannot be finitely
axiomatized, but RA can, and RA is already essentially undecidable.
Gödel's theorem applies to RA. Some universal proposition (shape
(x)P(x)) will not been decided by exhaustive search.
RA is not Löbian, i.e. RA itself cannot prove its own incompleteness.
But is Gödelian.
Yet there are finitely axiomatizable system which are Löbian: the von-
Neumann-Bernays Gödel set theory (the most powerful theory knows today
in math, you can formalize Category theory in!) is Löbian, despite it
*can* be finitely axiomatized.
Bruno
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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