On 26 Aug 2015, at 22:22, meekerdb wrote:
On 8/26/2015 3:31 AM, Bruno Marchal wrote:
On 26 Aug 2015, at 12:06, Peter Sas wrote:
Personally my brain stack overflows at about 3 or 4 levels of
being aware that I am aware that ... I am aware. I think it would
require infinite memory to truly be aware of an infinite number of
steps in such a recursive relation.
Maybe the infinite hierarchy doesn't have to be thought/remembered
in full. Royce notes that the simple intention to be completely
self-aware is enough: the infinite hierarchy is logically implied
in the intention. Oskar Becker has a slightly different approach:
he says that as soon as one notices the endlessness of 1, 2, 3....
etc. one is already beyond the natural numbers and has reached the
first transfinite ordinal (small omega=N) which collects all the
natural numbers into one set. And then of course one can continue:
omega+1, omega+2...etc. Noting the endlessness of that series one
has grasped the next transfinite ordinal (omegaxomega). And so
on... Becker notes that Cantor's generation principles can be
recast as principles inherent in the structure of self-awareness.
I think that's a strong point.
That is very well known in recursion theory (alias theoretical
computer science).
We can both collapse the infinite regression, by using the second
recursion of Kleene, which I often explain with the diagonal Dx =
T(xx) ==> DD = T(DD) methode to get self-reference.
Then this can be used to define properly the constructive ordinal.
The first non constructive ordinal, which is also the set of all
constructive ordinal is omega_1^CK (for Church and Kleene). Then
reflexive machine can name ordinal beyond omega_ 1^CK, but using a
non recursive naming procedure. Machines and humans suffer from the
same limitation there (provably so if we assume computationalisme).
<iahibajj.png>
MATHEMATICIANS GONE WILD
Yes, it is one of the consequence of incompleteness.
A (first order logical) theory is consistent iff it has a model (a
semantic, a reality verifying the saying of the theory)
By incompleteness, to make a mathematical semantic of a theory (like
PA, ZF) you need a richer theory (like ZF, ZF + kappa). And to get a
mathematical semantic for ZF + kappa, you need much more and you go
can go wild, because the mathematical reality *is* wild. Now,
personally I don't really believe in sets, but I do believe in the
consistency of ZF and plausibly ZF + kappa. But sets are better seen
as epistemological constructs once in the frame of the
computationalist hypothesis.
When a Gödel-Löbian Post-Church-Turing-Markov little machine, like PA,
just tries to understand itself, it can get that if comp is true its
3p "Nous" are Pi_2 complete (for the first order modal extensions), or
(for the true Noùs) Pi_1 complete in the oracle of ... God
(arithmetical truth, the set of the Gödel number of the sentences
valid in the Model (N, 0, +, x).
But for the 1p truth it is even more inexhaustible, ineffable, etc.
Damascius wrote thousand of pages that you can sum up in one sentence
just saying that about *that*, even one sentence is already too much,
and as such, can only completely miss the point.
With computationalism, it is obvious, I think, that the study of the
high cardinals (mathematical logic) is part of the study of machine's
theology.
But set theory is extensional, and flats the intensional nuances, but
then we have the mathematical representation theorems which can be
used to handle in the 3p extensional way the 1p-intensional nuances.
Bruno
Brent
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