Re: What does Gödel's Incompleteness mean for the Working Mathematician?

On Jul 31, 6:37 pm, Bob Stewart wrote in sci.math:

BS>> ... one cannot, using the usual methods, be certain that the
axioms will not lead to contradictions ... As a working mathematician
(who is not an expert in logic), how worried should I be that the
statement that I am trying to prove is not one of these undecidable
statements? <<BS

Bob
===
Axiomatic languages do not lead to contradiction unless we have reason
to believe that either the axioms we have adopted, or the rules of
inference we have specified, harbour an inconsistency.

There is no reason to believe that either is the case in Peano
Arithmetic, which is, arguably, amongst the most intuitive of our
mathematical languages.

So any uncertainty regarding its consistency must reflect our
interpretation of Goedel's reasoning, rather than the substance of it.

Here is an alternative interpretation which suggests that concerns, of
the kind that you raised, could be unfounded:

PA is instantiationally complete, but algorithmically incomplete: An
alternative interpretation of Gödelian incompleteness under Church's
Thesis that links formal logic and computability

http://alixcomsi.com/PA_is_instantiationally_complete.htm

Intuitively, the thesis of the paper seems quite reasonable: If an
algorithm computes an arithmetical relation R, treated as a Boolean
function, as always true, then it must yield a proof sequence for [R]
in first order Peano Arithmetic.

The consequences are, however, quite drastic for the limitations on our
understanding of classical mathematics as suggested by standard
interpretations of classical theory.

On the other hand, the knowledge that every false arithmetic statement
is disprovable in some particular instance (even though this may take
an exhaustive search), or instantiationally provable in every instance
(even if only exhaustively, and not algorithmically), should actually
invigorate mathematics.

Although the search for neat, algorithmic, solutions may be
intellectually satisfying, the search for proofs that differ from case
to case is certainly more challenging - witness the solution of the
4-colour problem - and possibly more rewarding, since the most
significant of natural phenomena can rarely be seen to be amenable to
neat, algorithmic, expression and representation within mathematical
languages.

Regards,

Bhup

.

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