```Date: Feb 22, 2013 8:27 AM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s

On 2/22/2013 5:05 AM, WM wrote:> On 22 Feb., 02:08, William Hughes <wpihug...@gmail.com> wrote:>> So>>>>       A) For every natural number n, P(n) is true.>>>>          implies>>>>       B)  For any n: There does not exist a natural number>>           between 1 and n such that P(n) is false>>>> However, we cannot conclude>>>>       B')  There does not exist a natural number>>            m such that P(m) is false>> In potential infinity there is no actually infinite set, i.e., there> is no completed infinity. That means there are only finite sequences.> However we cannot find a last element, although we can conclude that> it exists.>> Compare my example:>>       A              B> --> 1         -->{ }> --> 2,1      -->{ }> --> 2         -->1> --> 3, 2     -->1> --> 3         -->1, 2> --> 4, 3     -->1, 2> --> 4         -->1, 2, 3> ...> --> n         -->1, 2, 3, ..., n-1> --> n+1, n -->1, 2, 3, ..., n-1> --> n+1     -->1, 2, 3, ..., n-1, n> ...>> There is no last element in B. Every x from 1 to every n you desire> can be in B. Nevertheless there is never an empty A.>> That is all we can know about infinity. (Not my fault.)Yes.  But being rude and ignorant toward otherscertainly is.what you are describing here is the structure of thenatural numbers as a directed set.There is actual mathematics that describes this.There are logics in which your reasoning can beformalized.And the fact that this structure remains as absoluteinfinity relative to Cantor's transfinite arithmeticdoes not make that arithmetical calculus notmathematics.If you have an objection that may be called"matheology", it is precisely against individualssuch as yourself who have little appreciation forthe extent of the subject and create urban legendsby regurgitating facts about topics in which theyhave expended little effort to study.
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