FOM: Re: infallibility and inexpressibility
neilt at mercutio.cohums.ohio-state.edu
Thu Jan 21 14:54:36 EST 1999
> > Theorem: If the concept "x is an infallible cardinal" is expressible
> > in the language of set theory, then there is no infallible cardinal.
> This follows most simply from the undefinability of truth.
Sure!---but only *once you have established the undefinability of truth*!
If, however, you insist that all the reasoning behind the result be
taken into consideration for the judgment of simplicity, then, by contrast,
it would seem to follow much more simply from the proof of the Theorem
that I gave. The theorem on the undefinability of truth requires much
more machinery. In particular, it requires coding of syntax so that one
can establish something like a fixed-point theorem that can then be
applied to the negation of the formula supposedly defining truth.
Supposes only that one has some formula T(x), and some way of
forming a term x* denoting x, such that for all sentences S,
T(S*) is interdeducible, in the theory, with S. To get a contradiction
from this assumption via Liar-type reasoning, we need to find some
sentence L (a Liar) such that L is interdeducible with ~T(L*).
Then, given also that L is interdeducible with T(L*), we get the
Unfortunately, without coding and diagonalization to exploit, such
an L would not seem to be forthcoming.
By contrast, the reasoning that I gave for the theorem above would be
accessible to anyone who had been acquainted with the iterative conception
of set, and with the concept of an ordinal number. They wouldn't need to
know about G"odel-numbering, numeralwise representability, and the
construction of fixed points.
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