On 6/17/2013 7:38 AM, email@example.com wrote: > On Sunday, 16 June 2013 21:38:26 UTC+2, Virgil wrote: > >>> Necessarily! endless = without end = not ended. > >> Infinite might be interpreted to imply endless but but endless does not imply infinite! Circles are endless but not infinite. > > The way on a circle is infinite. > >> All sorts of cyclic processes are endless but not infinite. > > These processes are infinite. >
In Hilbert's metamathematics, "concrete, intuitive" representations of numbers correspond with your "numerals-as-strokes", that is,
1 11 111 ...
His argument to get from these well-ordered sequences of marks, representing numbers according to an ordinal construction, to a notion of cardinality is based upon the symmetric permutation groups.
Now, one may use modern reasoning to claim that the combinatorial group theory one might associate with universal algebra is logically more sound than the group theory that arose from geometric considerations, but that is historically inaccurate and fundamentally dishonest.
So, if one wishes to view the situation through group theory, one has cyclic groups and the symmetric groups associated with them.
Now, I am too stupid to know what you mean by finite.
That is, you have just declared cyclic processes to be equivalent to infinite processes.
In support of your remarks, Albrecht has written that numbers count themselves. But, they do not really. Numbers can be distinguished from one another by a sequence of representations that are inscriptionally distinct. But to correspond with a count -- that is, to have a cardinality -- requires something more.
You never really get around to explaining that much.