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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: 8,219 Registered: 7/9/08
An Infinity of Exponent-like Functions on N
Posted: Oct 4, 2013 11:27 AM

It can be formally shown that for every x0 in N, there exists a unique binary function pow on N such that:

(1) pow(0,0) = x0 (0^0 = x0)
(2) pow(x,0) = 1 for x =/= 0 (x^0 = 1 for x =/= 0)
(3) pow(x,y+1) = pow(x,y) * x (x^(y+1) = x^y * x)

If we also have a binary function pow' on N such that:

(1) pow'(0,0) = x1 (the only difference)
(2) pow'(x,0) = 1 for x =/= 0
(3) pow'(x,y+1) = pow'(x,y) * x

then it can also be shown that pow(x,y)=pow'(x,y) except for the case of x=y=0.

Notwithstanding combinatorial analogies and arguments of convenience, there does not appear to be any logically compelling reason to choose any particular value for 0^0.

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