random

[Alex Martelli]

"Lulu of the Lotus-Eaters" <mertz at gnosis.cx> wrote in message
news:mailman.991514764.4040.python-list at python.org...
    ...
> But if you are not happy with mere physical/empirical impossibility, it
> works equally well for mathematical impossibles:
>
>   <2+2=5> IMPLIES <2+3=6>
>
> works fine by application of Peano arithmetic.  Using the successor
> operator ' and some permitted reasoning:
>
>   Premise: 2+2 = 5
>       (1): 2+2'= 5'   # successor of equals maintains equality
>       (2): 2+3 = 6    # rewriting terms in conventional manner
>
> The reasoning is valid, so the conditional holds.  The antecedent is in
> extremely bad shape--mathematically impossible--on its own.  But so
> what?

So you can prove "_anything_ else" just as well -- any WFF in
the system is now a theorem.  You took a simple case, but
it should be clear how with a little more work you can prove
any other equation for example -- as given x=y you may
legitimately add N*x to one side and N*y to the other, &c.

Thus, if "I can predict the next bit" was a WFF in the system,
    <2+2=5> implies <I can predict the next bit>
and thus the negation of this implication doesn't hold.


Alex