Re: incompleteness and inconsistency


lugita15@xxxxxxxxx wrote:
Nam Nguyen wrote:
Peter_Smith wrote:
We should be circumspect in distinguishing the technical facts from the
informal philosophical gloss that we put on them.

The second theorem tells us that we can't prove PA's consistency in a
theory which is strictly weaker than PA.


And you might well suppose
that it can't be terribly informative to prove PA's consistency in a
stronger theory which contains PA (which is therefore itself only
consistent if PA is).

Not a major issue but "only" is technically incorrect: PA's
consistency would be a requirement, but might not be the only one.

But as lugita points out, there can be a
consistency proof for PA in a theory which is stronger-in-some-respects
and weaker-in-others: Gentzen's proof is a case in point.

What lugita stated is "The most famous *relative* consistency proof of
PA is Gentzen's"; he didn't just say "a consistency proof for PA".


So far so familiar. Now, it isn't unusual to find people concluding
from a review of such technical facts (as laguta does) that "we don't
really know whether arithmetic is consistent or not". But why exactly
is that supposed to follow???

Because we don't know everything there is to know about infinity.

(Why, we might ask, doesn't Gentzen's
argument do the trick given that its premisses are surely compelling??
But set that question aside.)

I don't know much about his proof. My gathering though is that he used
PRA as a proving theory, but did he prove or assume that PRA is
consistent? If he proved that, on what ground did he do so? If not,
would it be wrong to say that "we don't really know if PRA is
consistent, or not"?
Yes, it would be wrong to say that. That is because David Hilbert gave
a direct, finitistic consistency proof for PRA. That means that PRA is
*certainly* consistent, as opposed to being consistent relative to some
other theory.

Well, that's only relative to some questionable philosophical
definition of "certainty". Edward Nelson is trying to prove that PRA is
inconsistent.


Technical facts in themselves rarely settle philosophical issues:
rather it is technical facts plus background philosophical assumptions
that entail further philosophical conclusions. So it is in this case.
To get a conclusion along the lines of "we don't really know whether
arithmetic is consistent or not" out of the technical facts here we
need some philosophical assumptions as input. What assumption would
justify laguta's "we don't really know" conclusion?

What technical facts would we have to state - beyond doubt - that
arithmetic is consistent? (And by arithmetic here I'm not referring
to PA, but rather to something like Shoenfield's 'N', a finitely
axiomatizable system).


I suspect the background assumption (along with some sort of dismissal
of the efficaciousness of Gentzen's argument as persuasive to someone
who seriously doubts PA's consistency) is a general thought along the
lines of: if we can't give a non-question-begging argument that
convince a sceptic about P that P is true, then we don't really know
that P. But this, of course, is a *highly* contentious general
epistemological principle which -- when we think through its
consequences elsewhere -- we can see we have little reason to believe.



--
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What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
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