[FOM] Zenkin's argument

[Dean Buckner]

Can't comment on Zenkin's argument (except that it rambles a bit).  However,
it does seem very similar to one made by a bona fide mathematician, which
I'll quote in full (adapted to plaintext).

"... by a kind of metaphysical accident, but, in any case, as a contingent
matter of fact, we humans are capable of directly apprehending only
totalities which are finite, and only such configurations as are finitely
articulated, where "finite" here must be understood in the traditional,
pre-Cantorian sense ...  That is why countable, first order languages
capture the formal aspects of our reasoning so well.

"But if we could directly apprehend totalities and configurations [which are
transfinite], then we might actually incorporate uncountably many names, cx,
for the elements, x, of an uncountable set A in our language, and employ
generalised inference rules

    From the (uncountable) set {phi(cx): x in A} of premises, infer
    (x) in A [ phi(x) ] as conclusion

which would enable us to decide questions like the continuum hypothesis.
The impossibility of our doing this is not, on the [Cantorian] view a
_logical_ impossibilty .. but [rests] rather on contingent facts about our
nature and the nature of the world in which we live.

"... [this constitutes] an embarrassment to the Cantorean standpoint.  For
the new [transfinite] species ... that the Cantorean view posits are not
directly manifested in the physical world.  The Cantorean transfinite is
"there", so to speak, but like the luminiferous aether, is not detectable by
any means available to us.

Mayberry, The Foundations of Mathematics in the Theory of Sets, p. 267-8

To RS (bcc): is similar the point you were making?






Dean Buckner
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