Date: Nov 1, 2013 12:11 PM
Author: Dan Christensen
Subject: Re: Formal proof of the ambiguity of 0^0

On Friday, November 1, 2013 11:31:11 AM UTC-4, fom wrote:
> On 11/1/2013 9:37 AM, Dan Christensen wrote:
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> > You can't just pull functions out the air.
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> Interesting remark.
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> Have you ever heard of Alonzo Church and
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> the lambda calculus?
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> The trick is to understand how to
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> use the sign of equality.
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> http://mathforum.org/kb/plaintext.jspa?messageID=7933608
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> Since you will not understand, let me observe
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> that Feferman once made a nice comment about
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> the fact that paradoxes seem to arise when you
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> combine negation with self-similarity.
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> The "functions" pulled out of thin air are
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> precisely those functions which express the
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> compositionality of truth functional relations
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> without unary negation.
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> In summary, of course you can simply pull
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> functions out of thin air.


Then, how about pulling out of thin air the "function" f: {0} --> {0,1} such that f(0)=0 and f(0)=1? What, I can't do that? Something about that tricky equal sign?

You don't need any lambda calculus to construct an exponent-like function with 0^0=1. Just ordinary logic and set theory will do. The problem is, simple convenience notwithstanding, there doesn't seem any logically compelling reason to choose it as The Definition of exponentiation on N.

Dan
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