Re: what makes it true?


Keith Ramsay wrote:
> mensanator@xxxxxxxxxxx wrote:
> |But the incompleteness theorem says there are P that hold that
> |cannot be proven. Does Euclidean geometry not meet the requirement of
> |"strong enough"? If so, shouldn't you mention that important issue?
>
> There is a language of "elementary" Euclidean geometry
> weak enough for there to be a decision procedure for it,
> an algorithm for determining for each sentence whether it's
> true or false. Take points, lines, and circles to be the
> fundamental entities, incidence and between relations as
> the relations, and then use only first-order logic, i.e.
> include the connectives "and", "or", "not" and "implies",
> and the quantifiers "for all" and "there exists" as applied
> only to the fundamental entities.
>
> That language can be translated, using the method of
> coordinates, into the analogous "elementary" language of
> the real line with +, *, <. Tarski provided an algorithm
> for deciding that language.
>
> One key thing here is that we don't allow in these languages
> references to general sets of points. Usually people would
> consider statements referring to sets of points to belong to
> Euclidean geometry as well, but they are excluded from this
> elementary language. For instance, the puzzle, "how many
> colors are required to color all the points in the unit
> square so that no two points of the same color are a
> distance of 1 from each other" isn't solved by this algorithm,
> since the puzzle refers implicitly to sets of points (having
> the same color).
>
> If we add to the elementary language of the reals the
> predicate "is an integer", then the resulting system does
> satisfy Goedel's requirement for strength of the system.
> The fact that one can't define "is an integer" in terms
> of the other terms in the language is why it's so weak
> to begin with.

Thanks for explaining that. Much more enlightening
than simply shouting "Wrong!".

>
> Keith Ramsay

.

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