On 5/29/2012 11:39 AM, Bruno Marchal wrote:
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.
But they do depend on infinity (i.e. the axiom of succession).
The intensional diagonalization, leading to reproduction, self-generation and
self-reference are all constructive concepts.
Can you explain "intensional diagonalization"?
Brent
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/ <http://iridia.ulb.ac.be/%7Emarchal/>
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