From: holmes@garnet.idbsu.edu (Randall Holmes)
Subject: Re: Belief and Logic (was re: Belief and science (Einstein))
Organization: Boise State University
Date: Wed, 18 Nov 1992 17:40:38 GMT

In article <1992Nov17.153512.17541@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>In article <1992Nov15.221156.22036@guinness.idbsu.edu>
>holmes@garnet.idbsu.edu (Randall Holmes) writes:
>
>>In article <Bxo0IB.H7y@mentor.cc.purdue.edu>
>>hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>
>>>In article <1992Nov12.174400.4716@guinness.idbsu.edu>
>>>holmes@garnet.idbsu.edu (Randall Holmes) writes:
>
>RH:
>>>>It is mathematically provable that the class of all true sentences in
>>>>a reasonable language is impossible to define in that language
>>>>(Tarski's Paradox).  In order to avoid this phenomenon, one has to
>>>>make the expressive capabilities of one's language _weaker_ (making
>>>>the language more expressive does not help!).  In particular, it can
>>>>be applied to ideas in the mind of God, as follows:
>
>HR:
>>>This is the case if the language has only proofs of finite length.  Since
>>>we do not know how to do infinitary operations, although we can handle 
>>>some of them, we cannot be sure of what God can or cannot do.
>
>RH:
>>The argument works just as well with infinitary syntax, as long as the
>>lanaguage used is powerful enough to describe its own syntax (which
>>certainly seems likely in the Divine case).
>
>Herman is absolutely right in the literal Cantorian sense.  Genuine
>infinitary syntax, as distinguished from merely transfinite Tarskian
>languages, is _a priori_ ineffable.

I was referring to the latter.  If God uses Absolute Infinite syntax,
then even He cannot understand the overall structure of His own
thought.


  (Would Randall agree that, at
>least in the present theological context, Cantorian set theory is
>inherently superior to the godless NF variants?)

NFU is better for theology; it allows one to distinguish between God
(the Absolute Infinite, still present in NFU, but a little harder to
find) and the created Universe, the oneness of which many theologians
have also maintained.  This distinction is impossible to draw in
Cantorian set theory.


  Note that there
>exists a very respectable diagonal argument to the effect that the
>syntax of natural languages falls in the former category.  (See _The
>Vastness of Natural Languages_ by Langendoen and Postal.)  Perhaps
>that is the reason why the notion of natural language proof has
>managed to resist all analytic efforts for two and a half millenia.

I do not find this argument convincing; natural languages are finitary.

>
>>>-- 
>>>Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
>>>Phone: (317)494-6054
>>>hrubin@snap.stat.purdue.edu (Internet, bitnet)  
>>>{purdue,pur-ee}!snap.stat!hrubin(UUCP)
>
>>-- 
>>The opinions expressed		|     --Sincerely,
>>above are not the "official"	|     M. Randall Holmes
>>opinions of any person		|     Math. Dept., Boise State Univ.
>>or institution.			|     holmes@opal.idbsu.edu
>
>cordially,
>mikhail zeleny@husc.harvard.edu
>"Le cul des femmes est monotone comme l'esprit des hommes."


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu