```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 zrational.Prf: sqrt 2 is irrational and (sqrt 2)^{sqrt 2} is either rational orirrational.  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)^{sqrt2} 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 certainlyrational.  Thus in ether case a solution exists.That is classically valid, but intuitionistically not.Dummett once attributed the example to Fred C Benenson, my philosophytutor at the time, who now seems to be a realtor[2] (is that theword?).  Now [1], he attributes it to Peter Rogosinski and RogerHindley.[1] Dummett, Elements of Intuitionism, second edition, OUP, 2000.  Themisattribution 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
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