fom
Posts:
1,968
Registered:
12/4/12


Re: The ambiguity of 0^0 on N
Posted:
Sep 19, 2013 11:07 PM


On 9/19/2013 9:19 PM, Dan Christensen wrote: > On Thursday, September 19, 2013 7:36:04 PM UTC4, fom wrote: >> On 9/19/2013 1:26 PM, Dan Christensen wrote: >> >>>> >> >>>> Why apart from? Why are you leaving it out? >> >>> >> >>> We can't divide by 0. Unless you want to assign a value to 0/0 as well. >> >>> >> >> >> >> And, what about division is "number theoretic"? >> > > As you can see in my proof, I am actually using the rightcancelability property of natural number multiplication: > > x*y = z*y & y=/=0 => x=z > > "Dividing" both sides by y, to cancel off the factor of y, OK? >
Not really.
Cancellability is a feature of group operations or, more generally magmas.
http://en.wikipedia.org/wiki/Cancellation_property
So, again, you are invoking a property that is not strictly "number theoretic".
The axiom of Peano's number theory that addresses equality between natural numbers (other than transitivity, symmetry, reflexiveness, and closure) is given by
x=y <> ( x+1 ) = ( y+1 )
Wikipedia lists this as a conditional,
( x+1 ) = ( y+1 ) > x=y
It is axiom 8,
http://en.wikipedia.org/wiki/Peano_axioms#The_axioms
Now, since you continue to invoke "number theory", perhaps you could provide the formal theory which you seem to think that everyone knows and loves. It is clearly not Peano arithmetic.

