```Date: Feb 16, 2013 5:01 PM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s

On 2/16/2013 3:37 PM, Virgil wrote:> In article> <e827583d-6246-4dd1-a860-bc7da80bfbcd@r3g2000yqd.googlegroups.com>,>   WM <mueckenh@rz.fh-augsburg.de> wrote:>>> On 15 Feb., 23:27, Virgil <vir...@ligriv.com> wrote:>>> In article>>> <8847efa6-6663-40e9-a61e-76bba7f34...@dp10g2000vbb.googlegroups.com>,>>>>>>>>>>>>>>>>>>   WM <mueck...@rz.fh-augsburg.de> wrote:>>>> On 15 Feb., 00:53, Virgil <vir...@ligriv.com> wrote:>>>>>>>>>> And just this criterion is satisfied for the system>>>>>>>>>> 1>>>>>>> 12>>>>>>> 123>>>>>>> ...>>>>>>>>>> For every n all FISs of d are identical with all FISs of line n.>>>>>>>> For every n there is an (n+1)st fison of d not identical to any  FIS of>>>>> line n.>>>>>>> That does not prove d is not in the list, but only that d is not in>>>> the first n lines of the list.>>>>>> For every n.>>>> For every n there are infinitely many lines following.>> You never can conclude having all n.>>>> Then WM must be willing to give up all induction arguments and proofs by> induction, as they are all have the same basis as Cantor diagonal> arguments: if something is true for the first natural, and whenever true> for a natural 'n' is also true for the natural 'n + 1',  then it is true> of ALL n.>> Thus true for the first natural and if true for n then true for n+1> DOES allow one to conclude having all n.>> At least outside of Wolkenmuekenheim.>> Does WM really want to give up the  proof by induction?>Yes.Once again, as I have watched these discussions,it is clear that what is involved here is thestructure of the natural numbers as a directedset.And, as I consider how WM never tires of the samemeager statements rather than substantive discussion,it occurs to me that his issue with the reversalof quantifiers in relation to directed set structureis the key.Could it be that the underlying sense of proof byinduction is the same as the construction of aforcing language?That is, it is Euclid's proof that there is nogreatest prime number which establishes thedirected set structure.  Euclid's proof involvesan application of a successor.In forcing, the order relation of a directedset is inverted.  From this, one constructs aforcing language through which a forcing modelis obtained.  What proves that the forcing modelis not the ground model?  It is the fact thatone can produce a name not in the ground model.The forcing model has names for every elementof the ground model plus at least one namedifferent from those names.This is not the diagonal proof.So, in relation to your question, is mathematicalinduction an arithmetical expression of whatthe algebraic methods associated with forcingdo.  Does mathematical induction rely on thereversal of order relative to the directed setstructure?If this is reasonable, then WM's assertionsconcerning the reversal of quantifiers do implythat WM rejects induction.Apparently, WM is a paragon ultrafinitist.But, then which set of prime numbers are *the**finite* set of prime numbers?This is where his problem of not understandingthe meaning of "singular term" reappears.
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