On 7/4/2013 11:42 PM, apoorv wrote: > > > But ordinals form a chain in their natural order,
One of WM's finitist supporters (I believe he posts as Albrecht) wrote the following:
"Numbers count themselves"
I have thought about that statement in several contexts. Since you bring up the subject of "names" through mention of nameability, consider my restatement:
"Numbers name their counts"
"Numbers count their names"
In my opinion this properly reflects the relation between ordinality and cardinality.
That is, I find myself unable to differentiate the ordinal relation of numbers from the semiotics of naming.
Do not misunderstand me. I realize how the philosophical investigation of 'names' is far removed from the nature of number. However, the idea of a logical foundation for mathematics mandates consideration on how that philosophy manifests itself in the foundations of mathematics.
If forced to consider "intuitive representations of number" corresponding with the Kantian schema of a sequence of homogeneous representations placed side by side, then I transform Albrecht's statement to my own. This is because modern mathematics differentiates ordinal numbers from cardinal numbers.
It is only with respect to consideration of Cantor's completed infinities that these concepts are clearly different, however.
But, to have that theory, limit ordinals must be assumed to exist (in the abstract). They cannot be constructed. But, they can extend the chain which is their natural configuration.
This had been implicit in your original post (assuming the logic did not have some fatal error with which I have no particular concern).