Re: help with Godel's

In article <MPG.2054d528fefaa10e989d5e@xxxxxxxxxxxx>,
David Marcus <DavidMarcus@xxxxxxxxxxxxxx> wrote:

herbzet wrote:
[SNIP]

And, while we're on the subject, what are we to make of
a sentence like ~Con(PA), which, intuitively, says that
PA has no models, but, if PA does have models, is true in
some of them???

Our intuitive understanding of the sentence doesn't match its meaning in
such a model.

In particular, Con is defined in terms of Provable, Con(PA) being
equivalent to ~Provable(PA,"1=0"). Provable(axioms,theorem) is the case
if there is a sequence of proof-steps, each of which is either an
instance of one of the given axioms or an application of modus ponens to
2 previous steps in the sequence. Now, in the standard model of PA,
every number is finite, therefore every proof has a finite sequence of
steps. In this standard model, Con(PA) is true. So far, so good. But
in each of the non-standard models, there are numbers which are
infinite, and ~Con(PA) refers to a proof (of "0=1") that has an infinite
number of steps. But we (in standard-model-land) wouldn't consider that
to be a legitimate proof.

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