Re: Cantor's circular "proof" that evens = integers

"R. Srinivasan" <sradhakr@xxxxxxxxxx> wrote in message
news:1178802244.270897.225970@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On May 10, 4:44 pm, G. Frege <nomail@invalid> wrote:
This is what Solomon Feferman has to say regarding Cantorian set
theory in his article "Why a little bit goes a long way: Logical
foundations of scientifically applicable mathemtaics":

\begin{quote}

3. What's wrong with set-theoretical foundations?

Philosophically, set theory - even in its "moderate" form given by
Zermelo's axioms - requires for its justification a strong form of
Platonic realism. This is not without its defenders, most notably
Gödel (1944 and 1947/1964) (cf. also (Maddy 1990)). For its critics,
however, the following are highly problematic features of this
philosophy:

(i) abstract entities are assumed to exist independently of any means
of human definition or construction;

(ii) classical reasoning (leading to non-constructive existence
results) is admitted, since the statements of set theory are supposed
to be about such an independently existing reality and thus have a
determinate truth value (true or false);

(iii) completed infinite totalities and, in particular, the totality
of all subsets of any infinite set are assumed to exist;

(iv) in consequence of (iii) and the Axiom of Separation,
impredicative definitions of sets are routinely admitted;

(v) the Axiom of Choice is assumed in order to carry through the
Cantorian theory of transfinite cardinals.

\end{quote}

Can anyone give up an entity that cannot be defined/constructed by any
means?

karl m


.

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