On 8/24/2012 12:02 PM, Bruno Marchal wrote:
As emulator (computing machine) Robinson Arithmetic can simulate
exactly Peano Arithmetic, even as a prover. So for example Robinson
arithmetic can prove that Peano arithmetic proves the consistency of
Robinson Arithmetic.
But you cannot conclude from that that Robinson Arithmetic can prove
its own consistency. That would contradict Gödel II. When PA uses the
induction axiom, RA might just say "huh", and apply it for the sake of
the emulation without any inner conviction.
With Church thesis computing is an absolute notion, and all universal
machine computes the same functions, and can compute them in the same
manner as all other machines so that the notion of emulation (of
processes) is also absolute.
But, proving, believing, knowing, defining, etc. Are not absolute, and
are all relative to the system actually doing the proof, or the
knowing. Once such notion are, even just approximated
semi-axiomatically, they define complex lattices or partial orders of
unequivalent classes of machines, having very often transfinite order
type, like proving for example, for which there is a branch of
mathematical logic, known as Ordinal Analysis, which measures the
strength of theories by a constructive ordinal. PA's strength is well
now as being the ordinal epsilon zero, that is omega [4] omega (=
omega^omega^omega^...) as discovered by Gentzen).
Dear Bruno,
What happens when we take the notion of a system to those that are
not constructable by finite means? What happens to the proving,
believing, knowing defining, interviewing, etc.?
--
Onward!
Stephen
http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html
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