Re: Axioms v. definitions.


|-|erc wrote:
> "Jim Spriggs" <jim.sprigs@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in
> : |-|erc wrote:
> : >
> : > "Jim Spriggs" <jim.sprigs@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in
> : > : In something like:
> : > :
> : > : A group is a set G with a binary operation * on it such that
> : > :
> : > : blah, blah, blah.
> : > :
> : > : Are the blahs axioms? The whole thing is clearly a definition and I am
> : > : tempted to call the blahs "defining axioms" rather than just "clauses in
> : > : a definition". I suspect that it is only a matter of taste what one
> : > : calls them, but it may be that the pros have a definite preference about
> : > : using the word "axiom" or avoiding it in a case like this.
> : >
> : > does each 'blah' stand true on its own?
> :
> : Each is true in models of a particular kind, the models in question
> : being the groups. I don't think that talk of the _truth_ of axioms (or
> : definitions) is helpful, certainly not in this case.
> :
>
> Defn square
> - square has 4 sides
> - each side has equal length
>
> the second clause has no global scope, and its not identifiable, no axiom number.

I see what you're getting at, but I couldn't resist this:
>
> Prove a right angle triangle has 3 60 degree angles in it.

Disproof:
Definition?/Axiom? Every triangle has only 3 angles.
Definition?/Axiom? Every right angle triangle is a triangle.
Definition?/Axiom? Every right angle triangle has one 90 degree angle.

>>From that, it can be proved that no right angle triangles have 3 60
degree angles in them.


> from axiom (square second bit), each side has equal length,
> therefore all 3 angles are the same, 180/3 = 60.
> therefore, by the axiom in 'defn square', a right angle triangle has 3 60 degree angles.
>
> ------------------------------------------------------
>
> Defn square
> s1 - square has 4 sides
> s2 - each side of a square has equal length
>
> Now I can refer to the assertion made in s2, so I would consider that an axiom but not
> the clause in the top example.
>
> Herc
> --
> yes, uh huh.... oh I see

.