On 29 May 2012, at 19:27, meekerdb wrote:
On 5/29/2012 12:27 AM, Bruno Marchal wrote:
I doubt infinities.
I can doubt actual infinities. Not potential infinities, which
gives sense to any non stooping program notion.
Comp is ontologically finitist. As long as you don't claim that
there is a biggest prime number, there should be no problem with
the comp hyp. Infinities can be put in the epistemology, or at the
meta-level: they are mind tool, souls attractor etc.
Bruno
But diagonalization arguments assume realized infinities.
Set theoretical diagonalizations, à-la Cantor, assume realized
infinities (like analysis, by the way). I don't use them, if only to
explain diagonalization.
Computer science or "arithmetical" diagonalization does not assume
realized infinities, only potential. Kleene second theorem is
constructive. Gödel's diagonalization is constructive: for each
effective theory, it provides the undecidable sentences.
The intensional diagonalization, leading to reproduction, self-
generation and self-reference are all constructive concepts.
The theory of everything is really just logic and
Ax ~(0 = s(x)) (For all number x the successor of x is different from
zero).
AxAy ~(x = y) -> ~(s(x) = s(y)) (different numbers have different
successors)
Ax x + 0 = x
AxAy x + s(y) = s(x + y) ( meaning x + (y +1) = (x + y) +1) = laws
of addition
Ax x *0 = 0
AxAy x*s(y) = x*y + x laws of multiplication
The observer is the same + the induction axioms. To define it in the
theory above is of course a very long subtle and tedious exercise.
Bruno
http://iridia.ulb.ac.be/~marchal/
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