From: Bernie Cosell <bernie@fantasyfarm.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, November 13, 2011 12:27 PM Subject: Re: [math-fun] Request for example elementary non-constructive proofs with "witnesses"
On 13 Nov 2011 at 12:04, Marc LeBrun wrote:
Could anyone supply me with elementary examples that illustrate the idea of a non-constructive proof, for those with a "Martin Gardner reader" level of mathematical sophistication that also has a not-too-trivial but reasonably easily-verified case?
How about the existence of irrational numbers? The classic proof that sqrt(2) cannot be rational shows the existence of such numbers, but says essentially nothing about how to find more or how many of them there are, etc.
/Bernie\
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Another nice example is Cantor's diagonalization proof that there are uncountably many real numbers. Since it is easy to see that there are only countably many rational or algebraic numbers, it follows that there exist (uncountably many) irrationals and transcendentals, but Cantor's proof does not exhibit any example of such. An actual transcendental number was first discovered by Liouville in 1851, predating Cantor. -- Gene