# [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

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