Rotwang
Posts:
1,684
From:
Swansea
Registered:
7/26/06


Re: The ambiguity of 0^0 on N
Posted:
Sep 18, 2013 8:33 PM


On 19/09/2013 00:27, federation2005@netzero.com wrote: > On Wednesday, September 18, 2013 6:11:53 PM UTC5, Rotwang wrote: >> Definitions cannot give rise to contradictions. > > A definition is generally based on premises (called "preconditions") and those premises can be false, in which case the definition gives rise to contradiction. For instance, the very statement of a definition may make reference to an object in the singular that itself happens (in fact) to not be uniquely defined, such as a "function" that is not a function.
Well, that depends what one means by "definition"; I would rather say that if e.g. I attempt to define a f to be the unique function that has some property when no such function or more than one such function exists then any contradiction at which I arrive was the result of my false assumption that led to the definition, rather than the definition itself. Empty semantic games, perhaps, but this is irrelevant to my admittedly badlyexpressed point, which is that Dan could define ^ to be any one of the functions in N^{NxN} and that would not, by itself, cause any trouble; a contradiction would only arise if he also assumed that the definition satisfied some property that it doesn't.
The problem is that Dan has things backwards; rather than picking a definition and determining what properties it satisfies, he's picked a few properties satisfied by the usual definition, noticed that those properties aren't sufficient to derive the usual definition, and is now insisting that we should therefore abandon the usual definition unless someone can show that the other possible definition that satisfies those properties gives rise to a contradiction. This makes no sense.

