Hmm ... I may come back on this difficult and interesting question (Is
Church thesis provable) when we will pursue the seventh step serie
thread.
My "feeling" is that if Church thesis was provable, we would become
capable of defining the natural number in a categorical first order
language theory, which is impossible.
Church thesis is a necessary informal truth. It can have many formal
counterpart (like some intuitionist Church thesis), but those are
different theses.
Yet CT is a scientific thesis, because it may be formally refute. It
is really at the intersection of science and philosophy/theology.
A weakening of Church thesis is: "It exists a universal digital
machine". From this you can prove already by one (double)
diagonalization the incompleteness of its many relative provability
powers, like you can prove, that IF ever that machine believes in the
induction axioms i.e. [{A(0) & for all n, A(n) -> A(n+1)} -> {for all
n, A(n)}], for all formula A expressible in its language, THEN the
machine becomes Löbian, and is able to prove its own incompleteness,
and is capable of "intuiting" the truth in its many modal (person
perspectives) ways.
The key of understanding Church thesis, is that it makes both the
notion of computability an absolute, and of provability a relative
notion. But all Löbian provabilities obeys to the same (double) self-
referential laws (G and G*).
But I am not sure everyone get the "8th" point (movie graph). If a
computer is active from t to t+dt, and does not use register 24 during
the computation, would the absence of that register (during t and t
+dt) break the supervenience of consciousness on the computation
implemented by that machine?
My opinion is that such a break would contradict either comp or
physical supervenience. But I am not sure how much
are convinced by this.
It is needed to understand that a universal machine cannot distinguish
physical, virtual, and arithmetical "realities", so that the first
person indeterminacy bears on the sigma_1 sentences and their
(possibly infinite) proofs (with oracle, ...).
Thanks for the link Mirek,
Bruno
On 13 May 2010, at 00:36, Miroslav Dobsicek wrote:
A link for those who like to wrap their brain around prospects of
proving Church-Turing thesis.
http://www.logicmatters.net/resources/pdfs/KreiselSqueezing.pdf
Mirek
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