Re: incompleteness and inconsistency


Nam Nguyen wrote:

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.

Eh? Not a major issue but what I said was not "technically incorrect".
I said theory X is consistent only if PA is: "only if" doesn't mean
"if" :-)

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.

And why is *that* a reason? The fact that we don't know everything
about X doesn't imply that we know nothing about X. One thing we might
know is that PA is consistent!

(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?

He assumed PRA is consistent -- which is fair enough, because it is :)

What technical facts would we have to state - beyond doubt - that
arithmetic is consistent?

I'm not sure what that question means.

.

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