On 3/22/2013 3:19 PM, Virgil wrote: > In article <d_OdnaqCGY6gO9HMnZ2dnUVZ_uidnZ2d@giganews.com>, > fom <fomJUNK@nyms.net> wrote: > >> On 3/22/2013 1:21 PM, WM wrote: >>> On 22 Mrz., 16:31, William Hughes <wpihug...@gmail.com> wrote: >>>> On Mar 22, 10:05 am, WM <mueck...@rz.fh-augsburg.de> wrote: >>>> >>>>>> If you want to >>>>>> remove all of the lines you have to remove the set of all >>>>>> lines that are indexed by a natural number. >>>> >>>>> But I don't want to remove a set. >>>> >>>> We have the set of lines. You do not want to leave >>>> any of the lines. >>> >>> I do not want this or that. >>> I simply prove that for every line l_n the following property is true: >>> Line l_n and all its predecessors do not in any way influence (neither >>> decrease nor increase) the union of all lines, namely |N. >>> >>> This is certainly a proof that does not force us to "remove a set". >>> But we can look at the set of lines that have this property. The >>> result is the complete set of all lines. >> >> So now you have two complete infinities. >> >> The infinity of objects comprising |N. >> >> The infinity of finite lines whose sequentially >> ordered elements are from |N. >> >> Which one is *the* infinity? >> >> Or, if there are many, how do they stand >> in relation to one another? > > One way out would be to use the Von Neumann zero-origin model for the > naturals, with the first natural, 0, being the empty set and each > subsequent natural being the set of all previous ones. > > With this model, each natural. except possibly 0. could also be its own > line, and we would no longer have to distinguish between naturals and > lines. > > WM only objects to vN naturals because they diminishe his abiity to > obscure his ubiquitous illogic. >
Actually, this is exactly what he uses.
Alan Smaill had some posts in one of these threads discussing the impredicativity implicit to his usage, and, about that time, I posted a few considerations of how that might be formulated with a functional notation.
In this regard, each of WM's natural numbers is a choice function choosing its own symbol from its defining initial segment of the completed infinity.
If I recall correctly, this is not unlike the definition of a limit point for subsets of the sequence of ordinals
If Y is a set of ordinals and x is an ordinal then x is a limit point of Y if
I hope I remembered that correctly.
But WM approaches this like a political activist and not a mathematician. So one day it is a "name" or a "symbol" that is "understood", and, the next day it is nonsense about removing elements from a list of lines without removing sets of lines.
So today we take the names and the symbols as understood and reject the possiblity of their interpretation as choice functions.
How does he compare his multiplicities of completed infinities?
He has problems with the use of singular terms. They are not consistently interpretable as referring uniquely.