Re: My investigations into Godels Incompleteness Theorem

John Jones wrote:
John Jones reply to R. Srinivasan :

I think I have got a handle on this, see last 2 paragraphs, but first I

explore an alternative.

In summary, if the mind neither asserts the cat is alive or dead, then
I need to know whence arises the assertion of it being in a superposed
state. For the SS, unlike being alive or dead, is not a familiar state
that I can naturally suggest as an alternative to being alive or dead.
I cannot understand this state other than as a linguistic construction
that I recognise when I come across it. But the 'state' of
superposition escapes my understanding.
So the SS cannot arise spontaneously in the human mind. I am forced to
conclude that it arises by another means. So if SS arises via QM or
newtonian axioms it is still independent of the human mind, which is in

conflict with the claim that truths are mind dependent.


You are right that SS is not consciously asserted by the human mind.
When I say that truths for formal propositions are mental axiomatic
assertions in NAFL, what I mean is the classical truths. The SS is a
non-classial truth which expresses that the human mind has not asserted
either of the classical possibilties -- namely, that "the cat is alive"
or "the cat is dead". So SS expresses the fact that the human mind is
blank with respect to the cat's state -- and hence the cat is not in
either of the classically possible states "alive" or "dead". Instead
the cat's state could be thought of as "neither alive nor dead" -- but
the human mind is not permitted to assert this explicitly, for that
would be equivalent in NAFL to asserting P&~P and adding it as an axiom
to QM. But this is not allowed in NAFL theories. The cat's state is
equivalent to P&~P in a non-classical model for QM, but since the human
mind has not explicitly asserted P&~P, we are saved from having to deal
with an inconsistent (or paraconsistent) *theory* QM+(P&~P) in NAF..
P&~P would only hold in a non-classical *model* of QM.


........On the other hand, if you say that being alive, dead, and
superposed are all abstractions independent of reality and that I can
make whatever truth statements and axioms I like, then by abandoning
the idea of the necessity of the real world and the familiar objects
that populate our physical and mental life, then it seems that NAFL is
pragmatism.


In other words, NAFL as a pragmatism, offers us the idea that being
alive, dead or superposed are linguistic tools to get things done, and
are not required to be made sensible to human understanding except in
so far as they can do the job for which they were created. Now it seems

to me that the job of superposition is to be able to synthesize or
present an alternative to 'is' and 'is not'. The PARTICULAR job SS must

perform is rationalising the empirical results of QM. But why would it
have to do that? Certainly not to make the results accessible to human
understanding, but only to make them accessible to use, for otherwise
SS merely replaces one mystery with another. I conclude that
superposition is a linguistic tool that enables certain empirical
results, from QM in particular, to be adequately (usefully) summarised
for use in theory.

Actually superposition is a logical necessitiy in NAFL. Consider the
following simplified argument. If T is a theory in which the
proposition P is undecidable, i.e., neither P nor its negation ~P is
provable in T, then T cannot prove the law of non-contradicition
~(P&~P), which is equivalent to the law of the excluded middle Pv~P in
NAFL.

The reason is as follows - let the human being have the theory T in
mind. In order to prove ~(P&~P) in T, the classical argument would be
equivalent to "If P is the case, then ~P cannot be the case; if ~P is
the case, then P cannot be the case". But in NAFL, "If P is the case"
and "If ~P is the case" are both axiomatic assertions by the human mind
of P or ~P; they are *not* to be viewed as Platonic truths independent
of the human mind. So the first argument is a refutation of P&~P in the
theory T+P and the second argument is a refutation of P&~P in the
theory T+~P. Hence there is *no* argument for refuting P&~P in the
theory T (one can also show that the intuitionistic argument for
~(P&~P) fails in NAFL).

Since P&~P cannot be refuted in the theory T, it follows that T must
"tolerate" P&~P, i.e., there must exist a non-classical model for T n
which P&~P is the case.So whenever a theory T tolerates both P and ~P
in separate classical models (as it must when P is undecidable in T) it
must also tolerate a non-classical model in which P&~P is the case. It
is this result of NAFL that leads to severe restrictions in classical
or intuitionistic infinitary reasoning, including non-existence of
infinite sets, failure of Godel's/Turing's incompleteness results as
well as classical/intuitionistic real analysis, non-existence of
non-standard models of arithmetic and inconsistency of non-Euclidean
geometries and the relativity theories, etc. At the same time the
requirement of the existence of the non-classical model demystifies
"weird" quantum phenomena like superposition and entanglement and also
provides a new way to do real analysis in which open intervals of reals
do not exist and dy/dx is just *defined* as 0/0 (where the numeratoir
and denominator are *real* zeroes, defined by Cauchy sequences) and so
on. See my paper <http://arxiv.org/abs/math.LO/0506575> for an outline
of these results; also <http://arxiv.org/abs/quant-ph/0504115> for
another potential application to quantum mechanics.

Regards, RS

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