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Re: prove no largest prime
Posted:
Jul 5, 2012 8:31 AM
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quasi wrote:
>
> quasi wrote:
> >quasi wrote:
> >>
> >>To prove something doesn't exist, unless there's an axiom
> >(or prior theorem)
> >>that asserts such nonexistence, you pretty much have to
> >>start by assuming it _does_ exist, and then try to derive
> >>a contradiction.
> >
> >But in the absence of a nonexistence axiom, if it's based on
> >a prior theorem that asserts nonexistence, then at some lower
> >level, it's _based_ on a proof by contradiction.
>
> I think I claimed way too much.
>
> After all, if we accept DeMorgan's laws, then to prove
>
> "There does not exist x such that P(x)"
>
> one can simply prove the equivalent form
>
> "For all x, not P(x)"
>
> which does not have the form of a proof by contradiction.
It depends. One way of proving "For all x, not P(x)" is to assume P(a),
where a is any individual of the required type and nothing that follows
depends on which particular thing a is, deduce a contradiction from that
assumption, conclude not-P(a). But a was not special, so for all x,
not-P(x).
--
The animated figures stand
Adorning every public street
And seem to breathe in stone, or
Move their marble feet.
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