FOM: consistency equals existence
Walter Felscher
walter.felscher at uni-tuebingen.de
Thu Feb 24 04:18:04 EST 2000
When Hilbert stated that consistency means 'existence', he probably
meant only that there simply was no other way to secure things to
exist.
Mr. Simpson observed that Goedel's completeness theorem can be used
to provide a better explanation. Here I just want to comment that,
after initial hesitation, I find this a rather convincing argument.
If the completeness theorem is said to be
(1) if a set C of formulas semantically implies a formula w
then there is a deduction from C which produces w ,
then one may be skeptical: it has an indirect proof only, which
requires to conclude from
for every deduction from C : it does not produce w
upon
there is a deduction from C which produces w .
However, as Mr.Simpson made it clear, what he actually uses is
Goedel's original theorem
(2) if C is consistent then it is satisfiable ,
and no indirect argument is required. Moreover, the structure A , in
which C is satisfied, will (not use some arbitrary set, but) be
explicitly defined on the set of terms of the language, assumed as
given from the outset (and the satisfying valuation will be the
identity). Thus existence here simply is based on the existence of (a)
language.
If Boolean valued structures are admitted, then the values taken
by the relations of A simply are in the Boolean algebra B , formed by
the congruence classes of formulas modulo provability from C . As no
homomorphism from B to 2 is required, neither is Koenig's Lemma. Further,
if semantical consequence is considered with respect to Boolean valued
structures, then also (1) becomes accessible to a direct proof.
"Alles wendet sich zum Guten / in der besten aller Welten ... " ;-)
W.F.
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