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[Phys-L] Re: arithmetic



Unfortunately the game is FUN, which led to a waste of 30 minutes in the
office yesterday.

| > |
| > | The proof that (S0 + 0) is equal to (0 + S0) is easy.
| > |
| >
| > Am I allowed to use associativity for "+",
| >
| > i.e. may I use (a+b)+c = a+(b+c) for any, (a,b,c)
|
| Well, if you're going to "play the game", then you
| have to prove associativity; it is not axiomatic.
|
| > And may I use
| >
| > a+c=b+c => a=b
|
| You can prove that, too.


I suspected the above, but was unable to quickly prove (come up with
proper combinations) that above commutivity of the identity element with
its successor from the first two axioms.
|
| ================
|
| On the other hand, there is a lot of stuff you do get
| to use that hasn't heretofore been mentioned in this thread,
| including the axiomatic properties of the "=" operator
| (reflexive, symmetric, transitive).
|

I was assuming the above properties of "="; BTW

Reflexive A=B => B=A is the symmetric property isn't it? Or are you
referring to something else?


I read the book a good 25 years ago, I'll have to dig it out.

| When I said "it's easy" I didn't mean to suggest that
| re-inventing all of number theory from scratch would be easy.
| What I meant was that after you've seen 20 pages of
| explanations and examples, then little puzzles like (S0 + 0)
| = (0 + S0) are easy. They're also kinda fun.

I'm glad you said that. :-)

Joel R.