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Topic: An Infinity of Exponent-like Functions on N
Replies: 31   Last Post: Oct 23, 2013 10:48 AM

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 Dan Christensen Posts: 6,575 Registered: 7/9/08
Re: An Infinity of Exponent-like Functions on N
Posted: Oct 5, 2013 1:40 AM

Just trying to use some of these ideas in the writing of a proof just now and came across the weird notion that you might be able to use 0^0 in a proof even if it is "undefined!" You assume ^ is function on N -- you just don't know what value 0^0 has. So, you could prove 0^0 * 0 = 0 since x * 0 = 0 for ANY x in N. But you can't prove 0^0 = x for any x in N. This is getting really interesting!

As I said in my original posting, it can be shown (in 1038 lines in the DC Proof format) that there exists infinitely many binary functions ^ on N that have

(1) x^0 = 1 for x=/=0
(2) x^(y+1) = x^y * x

We can't determine the value of 0^0 from these equations alone, but it seems we can assume it is a natural number!

Comments? Have I missed a key point?

Dan

Date Subject Author
10/4/13 Dan Christensen
10/4/13 Graham Cooper
10/4/13 fom
10/4/13 Graham Cooper
10/5/13 Dan Christensen
10/5/13 Peter Percival
10/5/13 fom
10/5/13 Dan Christensen
10/5/13 Graham Cooper
10/5/13 fom
10/5/13 Dan Christensen
10/5/13 fom
10/5/13 Dan Christensen
10/5/13 Graham Cooper
10/5/13 Dan Christensen
10/5/13 Peter Percival
10/5/13 Graham Cooper
10/6/13 Peter Percival
10/5/13 Peter Percival
10/5/13 Dan Christensen
10/10/13 Shmuel (Seymour J.) Metz
10/10/13 fom
10/10/13 Peter Percival
10/23/13 Shmuel (Seymour J.) Metz
10/23/13 Michael F. Stemper
10/10/13 Michael F. Stemper
10/5/13 Dan Christensen
10/6/13 Dan Christensen
10/6/13 Brian Q. Hutchings
10/6/13 Dan Christensen
10/5/13 William Elliot
10/5/13 Peter Percival