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