[FOM] 3 postings from Alexander Zenkin

Robert M. Solovay solovay at math.berkeley.edu
Sun Feb 2 18:30:12 EST 2003

Can you give any citations [preferabbly URLs] for the work you talk about?

--Bob Solovay

On Sun, 2 Feb 2003, Peter Apostoli wrote:

> A. Zenkin :
> But it does not mean that today, 'in the early 21st century', anybody
> knows how to solve the "Liar", the "Barber", the Russell's, Richard's,
> Berry's, Grelling's etc. paradoxes.
> I mean just to solve, but not to evade the paradoxes problem as it was
> done by Russell, Hilbert, Brouwer, etc.
> Professor Zenkin's comment offers a fair assessment of the state of the
> foundational arts pre-1995. However, already in the 1960's Dana Scott and
> the developers of Domain Theory pointed Frege's way out of paradox. Recall
> that whats at stake is finding a solution to the antinomies that (unlike
> type theory) Russell himself would find adequate to his paradox. Building
> upon this domain theoretic foundation in 1995, a Canadian Akira Kanda
> discovered the *canonical* solution to the antinomies of set theory,
> including Russell's. The solution involved the discovery of the reflexive
> domain of hyper-continuous functions. These characterize Cantor's
> "consistent multiplicities" as precisely the concepts which are "smooth", a
> variety of continuity familiar from Synthetic Differential Geometry extended
> to the domain of abstract sets. It turns out (I thank Professors Mike Dunn,
> Bob Myers and Danny Daniels for pointing this out to me in 1997) that second
> order logicians such as Nino Cochiarella had predicted that an acceptabe
> solution to Russell's paradox would take the form of a rejection of the
> principle of the identity of indiernibles. The discovery of the domain of
> hyper-continous functions verifies this remarkable prediction.
> "Discovery", rather than "invention" may apply here, since the variety of
> continuity under consideration seems to capture some essential aspects of
> the physical continuum. R's paradox ends up looking suspiciously like Zeno's
> on the analysis considered. The root cause of the paradox for which Russell
> searched in vain seems to involve the mathematical idealization of
> "dimensionless point" and the cognate physical idealization of "point mass".
> The former ignites the contradictions of set theory in much the same manner
> as the latter ignites the more horrible infinities of quantum field theory.
> Peter Apostoli
> Toronto
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