From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rpi!usc!wupost!uunet!psinntp!scylla!daryl Thu Jan 16 17:19:55 EST 1992
Article 2665 of comp.ai.philosophy:
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>From: daryl@oracorp.com
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan13.175936.2755@oracorp.com>
Organization: ORA Corporation
Date: Mon, 13 Jan 1992 17:59:36 GMT

Mikhail Zeleny writes:

> Our understanding of the results of first-order ZFC is related to our grasp
> of its standard model.  I invite you to meditate on the difference between
> the latter and its countable submodels before you emit any further
> exclamations.

Okay. Jai Guru Deva Om. I'm finished meditating.

I repeat: there is no standard mathematical result that is not in fact
a theorem of ZFC. There is therefore no evidence that our
understanding goes beyond what is captured in ZFC. Nonstandard models
don't change this empirical fact.

> While you are at it, feel free to characterize "sufficiently
> well" the predicate "... is finite" in a first-order language of your
> choice. I eagerly await the results.

A set is finite if there is no function mapping it onto a proper
subset of itself. That is definable in ZFC, if you interpret "function"
as a set of ordered pairs.

>> There is no evidence that humans can go beyond formalized reasoning, so
>> it is not clear what you are demanding be explained.

> On the contrary, any semantic consideration goes beyond formalistic, purely
> syntactical symbol manipulation.

I didn't ask whether you believed that our understanding goes beyond
formal reasoning, I asked for evidence that our understanding goes
beyond formal reasoning. Please don't take offense at the fact that
I don't consider your beliefs to be evidence.

Daryl McCullough
ORA Corp.
Ithaca, NY


