Re: incompleteness and inconsistency

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). 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.

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

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?

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.

.

Relevant Pages

  • Re: incompleteness and inconsistency
    ... informal philosophical gloss that we put on them. ... consistency proof for PA in a theory which is stronger-in-some-respects ... PRA as a proving theory, but did he prove or assume that PRA is ... Technical facts in themselves rarely settle philosophical issues: ...
    (sci.logic)
  • Re: incompleteness and inconsistency
    ... informal philosophical gloss that we put on them. ... consistency proof for PA in a theory which is stronger-in-some-respects ... PRA as a proving theory, but did he prove or assume that PRA is ... Technical facts in themselves rarely settle philosophical issues: ...
    (sci.logic)
  • Re: incompleteness and inconsistency
    ... informal philosophical gloss that we put on them. ... Technical facts in themselves rarely settle philosophical issues: ... epistemological principle which -- when we think through its ... consequences elsewhere -- we can see we have little reason to believe. ...
    (sci.logic)
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