Date: Feb 26, 2013 6:48 AM
Author: Frederick Williams
Subject: Re: Matheology ? 222 Back to the roots

Nam Nguyen wrote:
>
> On 26/02/2013 1:16 AM, Virgil wrote:

> > In article <m%XWs.20125$mC2.392@newsfe29.iad>,
> > Nam Nguyen <namducnguyen@shaw.ca> wrote:
> >

> >> On 25/02/2013 10:25 PM, Virgil wrote:
> >>> In article <SDWWs.99982$Hq1.27823@newsfe23.iad>,
> >
> >>> Since I said "not always", any such situation shows I am right.
> >>
> >> I think you misunderstood my point:
> >>
> >> In the context of language structure truth verification,
> >> your original statement would _always_ fail: because for
> >> Ex[P(x)] to be true, P(x0) must be true for some _example_ x0.

> >
> > To know that something must be true for some x0, it need not be known
> > for which x0 it is true, only that it is true for SOME x0. Which was my
> > original point!

>
> Then, can you construct a _language structure_ that would illustrate
> your point?


Here is an example of a non-constructive existence proof.

Thm: There are solutions of x^y = z with x and y irrational and z
rational.

Prf: sqrt 2 is irrational and (sqrt 2)^{sqrt 2} is either rational or
irrational. Put x = sqrt 2, y = sqrt 2 so that x = (sqrt 2)^{sqrt 2},
which by hypothesis, is rational. If, on the other hand, (sqrt 2)^{sqrt
2} is irrational, put x = (sqrt 2)^{sqrt 2} and y = sqrt 2, so that z =
((sqrt 2)^{sqrt 2})^{sqrt 2} = (sqrt 2)^2 = 2, which is certainly
rational. Thus in ether case a solution exists.

That is classically valid, but intuitionistically not.

Dummett once attributed the example to Fred C Benenson, my philosophy
tutor at the time, who now seems to be a realtor[2] (is that the
word?). Now [1], he attributes it to Peter Rogosinski and Roger
Hindley.

[1] Dummett, Elements of Intuitionism, second edition, OUP, 2000. The
misattribution was in the first, 1977, edition.
[2] http://www.benensoncapital.com/

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting