On 4/11/2013 1:57 AM, WM wrote: > On 10 Apr., 23:24, Virgil <vir...@ligriv.com> wrote: >
>> >> WM is again ignoring the fact that, for a strictly increasing infinite >> sequence of any sort, the limit, of one exists, cannot be a member of >> the sequence. > > No, that is just my point.
Regardless of what is explained to WM, he does not make the distinction.
There are different logical types in relation to one another.
The arithmetical system governing the limits is distinct from the arithmetical system governing the grounding type from which the system of limits is formed.
It is easiest to see this in real analysis.
Hilbert reversed this relation with formalist axiomatization. However, that axiomatization corresponds with Cantor's call for treating the system of limits (equivalence classes of fundamental sequences) as an arithmetical system in its own right. Relative to that axiomatization, Dedekind cuts and Cantorian fundamental sequences become representable structures with respect to which defined topological properties are instantiated.
But, transfinite arithmetic arises in the same way.
Relative to the enumeration of successors, countably infinite collections are taken to be finished. The arithmetical system is ordered according to the premise that exactly one such collection contains an element without a successor and that no such collection has an element with more than one predecessor or less than one predecessor. The arithmetical system is axiomatized as a single system of ordinal numbers, among which are the transfinite.
Just as the real numbers partition into the rational numbers and the real numbers, the ordinal numbers partition into successor ordinals and limit ordinals (the empty set is vacuously a limit ordinal).
WM's statements have no relation whatsoever to the one he has claimed to be in agreement with.