On Thu, 2 Jan 2003, Sarad AV wrote:
> An axiom is an improvable statement which is accepted
> as true.
An axiom is a statement which is -assumed to be universaly required-.
That is -not- equivalent to 'true' (eg "A point has only position" is not
'true' but a -definition- which is neither true or false, it just is). If
it's unprovable then it's 'truth' is irrelevant, derived statements can
be 'true' only if we accept the assumptions. Derived statements can
-never- be used to 'prove' the assumptions or else we have circular logic.
To talk of an axiom as being 'true' is a logic error (you've actually
switched into a meta-mathematics at this stage without recognizing it), it
can't be 'false' or everything falls apart (eg Godel's commentary about
PM being inconsistent means we can prove -any- statement 'true').
There are examples of such axiomatic statements turning out to be
problematic or not required; Euclids 5th and the PM assumption that
mathematics is complete (or as Godel says 'consistent'). There are other
definitions of 'parallel' that work just as well, but are -demonstrably-
(as compared to 'provably') different than Euclids original 5th. Euclids
5th isn't 'false', it's just different. With respect to PM, if we accept
the axiom that PM is consistent then we can't prove it, even though we
-must- accept it if we want to prove -any- (as compared to 'all')
statements 'true'.
> A Formula is a finite set of algebraic
> symbols expressing a mathematical rule. Proofs, from
> the formal standpoint, are a finite series of formulae
> (with certain specifiable characteristics).Hence any
> proof has a deterministic and well defined sequence of
> steps.
Godel says differently, what he says -via proof- is that there -are-
proofs that can't even be written because individual steps may be true but
are unprovably so. Hence, a proof that can't be written down can't be said
to have an end since it isn't complete. An algorithm for proving a
statement true when fed a unprovable statement -must not halt- or else it
is saying the statement is 'true or false', hence it is -not- required to
terminate or halt.
The primary result of Godel's work here is that 'true' and 'false' are
-not sufficient- to describe the behavior of PM. That -any- 'universal
algorithm' for proving statements 'true or false' can't exist since some
statements -in principle- (never mind practice) are -not provable-. Godel
in effect answers the 'Halting Problem' in the negative.
> This is true by the way I define a proof.
> You are right in ur context and I am right in my
> context.So both of us are right?yes,based on the
> *sense* of what we mean by a proof.
No, being 'right' isn't really the issue. I vote for Godel. If we accept
his proof then we have the unprovable assumption that PM is consistent
(which is ok for an axiom). This means that we have at least -an
implication- that it is so. Otherwise we are left with accepting it is
false, and hence PM is incomplete and -any statement can be proven
false-. How usefull would that be? I don't think very.
--
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