Re: Why to accept that the present set of arithmetical axioms are sufficient?


Arturo Magidin wrote:
> In article <1134167035.820307.293950@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> Proginoskes <CCHeckman@xxxxxxxxx> wrote:
> >
> >Arturo Magidin wrote:
> >> In article <1134146078.970473.193370@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> >> <luiroto@xxxxxxxxx> wrote:
> >> >Why to accept that the orthodox set of arithmetic axioms
> >> >are sufficient?
> >>
> >> Do you have someone saying they are "sufficient"? Sufficient for what?
> >>
> >> >This is not in contradiction to Goedel's results?
> >> >If Pascal in 1654 introduced the Complete Induction
> >> >axiom, and in modern times someone invented
> >> >the Axiom of Choice, why we cannot introduce new ones?
> >>
> >> Who says we cannot?
> >>
> >> >Which are the conditions that must fulfill a new axiom?
> >>
> >> A new axiom should be reasonable, interesting, and useful.
> >
> >And not contradictatory.
>
> I would say that an axiom which is contradictory (i.e., that leads to
> a contradiction) would not qualify as "interesting", since axiomatic
> theories in which one can prove both a proposition and its negation
> are highly uninteresting. (-:

And highly non-useful as well. 8-)

--- Christopher Heckman

.

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