From: emc@doe.carleton.ca (Eli Chiprout)
Subject: Re: Belief and science (Einstein)
Organization: Dept. of Electronics, Carleton University
Date: Fri, 13 Nov 1992 01:11:52 GMT

holmes@garnet.idbsu.edu (Randall Holmes) writes:

>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:

>Let a "concept" in the mind of God be an idea of the form "x has
>property P" (where property P may be utterly inconceivable to us -- it
>might be infinitely complex).  Certainly God is aware of all of His
>concepts.  
Infinite and self-referential sets cannot be dealt with in the same 
manner as all other sets.  In fact God, as you have defined, is equivalent 
to the definition for the set of of all sets, which is not considered
a set, since no set can be a member of itself.  It is rather called
the class V.  The further problem is representing "utterly
inconceivable" properties with the conceivable statement "property P".
Giving it that association no longer makes it inconveivable.

>If a is an object and "x has P" is a concept, we say that a
>satisfies "x has P" exactly if a has P.  Examples of concepts: "x
>loves God" is satisfies by Amir if Amir loves God.  "x loves x" is
>satisfied by Amir if Amir loves Amir (note that the variable x can
>appear in more than one place in the definition of a concept).  Now
>consider the following concept in the mind of God: "x is a concept in
>the mind of God and it is not true that x satisfies x".  Call this
>concept C.  Does C satisfy C?  This asks whether the statement which
>results from substituting C for x in the definition of C is true: "C
>is a concept (true!) and it is not true that C satisfies C".  In other
>words, the question of whether C satisfies C is precisely equivalent
>to the question of whether C does _not_ satisfy C.  This is a
>contradiction.  But it certainly seems reasonable that God is familiar
>with concepts in this sense and that God can conceive C.

>	The solution to the paradox in its normal form (which does not
>mention God!) is as follows: an adequate language will be able to
>express the idea of a concept (by talking about sentences in the
>language and substituting names in the language for the variable x)
>and thus the only ingredient of the argument which is open to
>challenge is the idea that we can define truth for sentences of a
>given language in that language (which blocks the definition of C).
>This solution seems unsatisfactory for the theological form of the
>paradox.  God must know the truth or falsehood of any proposition.

A paradox is by nature not resolvable, or it would not be a paradox!
You have assumed the conclusion "God must know the truth or 
falsehood of any proposition" and have shown it false in the context
of your definitions.  Mathematical logic states that you must
then reject your tentative conclusion.  If I define "omnipotent"
as being able to do anything, and I submit that God is omnipotent,
then God should be able to create a free moral being that will always
follow God.  But even God cannot create such a contradiction!
The man will have either have free moral will with the possibility
that he may choose to reject God, or be an automaton doing God's will.
I can only conclude that God cannot be omnipotent according to my
definition.

To illustrate this paradox, there is the story of the statue that
is called the liar's statue.  It is said that when one puts a hand in 
its mouth, and tells a lie, God makes the hand unremovable.  A clever man
decided to test the story by putting his hand in its mouth and saying:
"I will not be able to pull my hand back out!"
Now if you assumed that God can decide the truth or falsehood of all
propositions, how would He decide on the truth of this one?

We either reject our assumed conclusion or face a completely 
contradictory set of propositions.  Many Jews that have concepts
of God, have not thought those concepts through to their ultimate
implications. 


>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
--
Eli Chiprout                 
Dept. of Electronics, 
Carleton University, Canada
emc@doe.carleton.ca                                     
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