Re: My investigations into Godels Incompleteness Theorem
- From: "John Jones" <jonescardiff@xxxxxxx>
- Date: 30 Sep 2006 12:21:01 -0700
Thankyou. My thoroughly depressing objections against the points made
in your labours, follows:
R. Srinivasan wrote:
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 --
Are we hiking above the peaks? From what perspective, if it is neither
from the human mind or the external world, could the SS command the
view of the human mind not asserting either of the classical
possibilities?
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"
There are four objections here. Not knowing whether the cat is alive or
dead does not mean that being alive or dead are not known to the mind,
even if either one or the other are not baptised in a particular case.
Second, there is no reason to accompany 'is alive' by 'is not alive' -
it may not be my concern that the cat is dead, merely that it is alive.
Third, is it possible to retain the 'state' of the cat when neither its
life or death states are in consideration - isn't NAFL's claim that the
SS is a 'state' founded on a grammatical mistake? Fourth, If the
status of the cat being alive or dead is not in consideration in the
human mind, how does NAFL construct an SS state in respect of it?
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.
It seems that this might be another hike above the peaks. While the
'alive' and 'not alive' states together are not allowable, yet it still
seems that I command a view that enables me to see the cat's 'state'.
Do I need to construct an NAFL alternative to fill the void of that
state? Would it not be easier just to abandon the idea of the 'state'
of the cat altogether? I am not sure what NAFL is trying to do here by
its creation of a Superposed state. It seems an unnecessary effort to
make.
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).
I am not sure what is being claimed here. I notice that the
undecideability of P is not a factor in T, even if it may be one of the
world in which T is placed. Also, I do not know what is meant by the
'negation of "P and not P"' (~(P&~P)). I have no guide as to what that
could be, nor what P&~P is or how a SS could solve it except by
invoking another mystery.
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.
That's our point of disagreement. P&~P should be written P, ~P. It
seems that Platonism might be very much alive in NAFL, except that it
has been removed from an external world (and with it possibly the
raison d'etre for the external world) and re-installed in the human
mind where it suggests P&~P 'both', instead of P, ~P.
In fact it seems that NAFL is employing classical familiar objects of
Platonism as the models for familiar objects of the mind.
So whenever a theory T tolerates both P and ~P
in separate classical models (as it must when P is undecidable in T)
Again, if it can be discerned from what I have written, I have
objections to this interpretation.
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.
Projects close to my heart, but I am not satisfied with NAFL's idea - I
raised some pertinent objections.
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
What is the existence of the non-classical model again, please? Is that
the SS? If so, I cannot see how it demystifies QM.
Ta
.
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