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Topic: Matheology § 295
Replies: 24   Last Post: Jun 30, 2013 3:32 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Matheology § 295
Posted: Jun 28, 2013 5:48 AM
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On 6/28/2013 3:53 AM, mueckenh@rz.fh-augsburg.de 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.
>


Wrong again.

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.




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