[FOM] 3 postings from Alexander Zenkin

Peter Apostoli apostoli at cs.toronto.edu
Sun Feb 2 18:57:35 EST 2003

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

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