On 6/28/2013 3:53 AM, email@example.com wrote: > On Friday, 28 June 2013 03:39:04 UTC+2, fom wrote: >> by the way, a Cantorian "theory of ones" is not a "strokes-as-numerals" representation of number > > It is precisely the same! For all cardinals. (In Germany the 1 differs a bit from a simple stroke, not in England, by the way.) Zermelo criticized Cantor's attitude. >
Cantor insisted that numbers were sets.
Consequently, the identity of numbers is related to the axiom of extensionality.
Zermelo's system -- prior to revision -- defines identity on its domain with respect to denotations. This suggests that Zermelo had been taking Frege's arguments into account. Frege argued against a "theory of ones". Thus, Zermelo's criticisms may be surmised, although I have no English translations to validate that speculation.
Frege later retracted his logicist views. That does not mean that anyone knows which of his logicist arguments fall.
I do not know of how the revisions to Zermelo's system had been introduced. The issue associated with identity in the domain would have been related to the presuppositions of Hilbert's formalism.
Upon turning from formalist axiomatics to proof theory, Hilbert invokes a 'strokes-as-numerals' representation for a concrete, intuitive basis (It would seem that he misunderstands Kant in so far as 'strokes-as-numerals' is a schema rather than an intuition.) Hilbert then goes on to explain the difference between 'strokes-as-numerals' and cardinals by invoking the action of symmetric groups on systems of objects.
This account of cardinality reflects the Cantorian view that cardinal numbers are given through the obfuscation of any given order with respect to a system of objects. It also corresponds with the idea that ordinality is irrelevant to the role of sets as domains for logical systems. All that seems to be relevant is the cardinality of a system, and, Tarski is one of the authors who seem to have pursued arguments to this effect in order to ground opinions concerning the generality of logic.
My posts may be long, but your statements are simpy inaccurate.