On 31 May 2012, at 18:13, meekerdb wrote:
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 axiom of succession is not an axiom of infinity. It just says that
the numbers have each unique and different (from other's) successors.
All the standard numbers are finite.
In ZF there is an axiom of infinity, for you cannot prove infinity
from below. Unless you have more powerful axiom like the scheme of
reflexion.
The intensional diagonalization, leading to reproduction, self-
generation and self-reference are all constructive concepts.
Can you explain "intensional diagonalization"?
It is when you build an expression involving a formal diagonalization,
like writing a program which refer to itself.
For example like with the self-duplicating expression Dx = 'xx', so
that DD generates 'DD', i.e. its description. That is what Gödel did
to prove the existence of a sentence referring to itself, and notably
asserting that she is not provable. And that's what Kleene did to
prove the existence of a number e, such that phi_e (x) = T(e, x), or
generalization. In that case the program e compute T on itself with
parameter x.
Intensional diagonalization concerns codes.
Extensional diagonalization concerns set of functions, like the Cantor
one, showing that N^N is not enumerable, or Kleene one showing that
comp-N^N is not recursively enumerable. (comp-N^N = the partial
computable functions from N to N).
Bruno
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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