```Date: Oct 23, 2013 5:36 PM
Author: fom
Subject: Re: Formal proof of the ambiguity of 0^0

On 10/23/2013 11:55 AM, Dan Christensen wrote:> As some have suggested here, context may have a bearing on this matter, but care should be taken so as not confuse the subtly different concepts.>> We could have two different functions: what might be called multiplicative exponentiation (x^y, the usual kind), and what might be called combinatorial exponentiation (f where f(x,y) = 1 if x=y=0, = x^y otherwise). Perhaps a different notation should be used for each where it matters, e.g. in formal, foundational proofs.>>Let me thank you for remindingme of that with these threads.By the way, you should have takenthe challenge.  Quasi found a fatalerror.  It seems easily fixed, though.The new sentence will beAxAy( x mdiv y <-> ( Az( y pdiv z -> x mdiv z ) /\ Az( z pdiv x -> Aw( z mdiv w ) ) ) )The expressionAz( y pdiv z -> x mdiv z )says that if y properly divides any z,then 1 and all of its prime divisors alsodivide zThe expressionAz( z pdiv x -> Aw( z mdiv w ) ) ) )says that if any z can properly divide xthen it must be a monadic divisor for everyw.  But, a monadic divisor is either 1 ora prime.  So, it has to be 1.As for context here in contrast with othercontexts, Skolem introduced primitive recursivearithmetic precisely because he wished toavoid the use of quantifiers.  Where the arithmeticof 0 has no place in a construction like this, itis extremely useful in the recursive definitionsoften used in first-order Peano arithmetic.My thanks is sincere.  I had been wondering how Imight extend certain ideas of my own into an arithmeticaltheory.  Your threads motivated me for better or worse.
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