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 the structure of the natural numbers as a directed set.
And, as I consider how WM never tires of the same meager statements rather than substantive discussion, it occurs to me that his issue with the reversal of quantifiers in relation to directed set structure is the key.
Could it be that the underlying sense of proof by induction is the same as the construction of a forcing language?
That is, it is Euclid's proof that there is no greatest prime number which establishes the directed set structure. Euclid's proof involves an application of a successor.
In forcing, the order relation of a directed set is inverted. From this, one constructs a forcing language through which a forcing model is obtained. What proves that the forcing model is not the ground model? It is the fact that one can produce a name not in the ground model. The forcing model has names for every element of the ground model plus at least one name different from those names.
This is not the diagonal proof.
So, in relation to your question, is mathematical induction an arithmetical expression of what the algebraic methods associated with forcing do. Does mathematical induction rely on the reversal of order relative to the directed set structure?
If this is reasonable, then WM's assertions concerning the reversal of quantifiers do imply that 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 understanding the meaning of "singular term" reappears.
