[FOM] Paradox on Ordinals and Human Mind
praatika@mappi.helsinki.fi
praatika at mappi.helsinki.fi
Sun Dec 19 05:50:46 EST 2004
Lainaus "A.P. Hazen" <a.hazen at philosophy.unimelb.edu.au>:
> A) Resolution (1) doesn't look good. If you don't believe
> in infinite SETS, you probably shouldn't be happy with idealizations
> according to which humans can formulate infinitely many
> "definitions," and the same basic logic comes back to you with
> Berry's Paradox (the one about "the least integer not nameable in
> fewer than nineteen syllables").
Exactly!
> C) My own sympathies are with something like (2), but
> "meaningless" is too strong. One can think of "define" or "possible
> language" or "idealized human-like mind" as MEANINGFUL notions, but
> ones with an ineliminable vagueness which makes it inappropriate to
> reason CLASSICALLY about them. One can have a consistent theory
> quantifying over, say, possible definitions (see SKETCH below) if
> you use a formally intuitionistical logic: the inference from "Not
> all ordinals are defined" to "THERE IS a least indefinable one" is
> blocked.
This brings to my mind some memories... Some years ago, when I was a
graduate student, I wrote a paper (never published it) on Berry's paradox
in formalized arithmetic. First I noted that if one uses the standard
notion of (arithmetical) definabilility, one can just prove that the
definability relation is not arithmetical. Next, I used the notion
of "naming", borrowed from Boolos: A formula F(x) names a number n in a
theory T, if T proves: (For all x)[F(x) <-> x = n]. The solution of the
paradox in this case is the fact that the paradoxical formula only defines,
not names, a number.
Best
Panu
Panu Raatikainen
Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki
Finland
Tel: +358-(0)9-191 23437
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Email: panu.raatikainen at helsinki.fi
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