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Hal Finney wrote

*> In other words, just as the falsity of the GC would seem to imply
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*> that there must be a specific number which ISN'T the sum of two
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*> primes, so the falsity of the PP would seem to imply that there
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*> must be a specific distance to a point which is the farthest
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*> possible point of intersection. For either proposition to be
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*> unprovable would seem to imply that there is no such number or
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*> point, which would mean that they are true. In the case of the
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*> Parallel Postulate, we can measure the distance to the
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*> intersection point as the lines approach parallelism.
*

I think that I see the flaw here, but I had to write a little

story before I could, and so, even if it is redundant, I might

as well post the story since someone besides me may need it too:

Okay, so we are standing in the lobby in the Restaurant at the

End of the World, and overhearing an interesting conversation

between a well-known great 20th century mathematician and Euclid.

I mean to say that with the help of some Tiplerian technology,

they've both been resurrected.

In the middle of their discussion, a student rushes up to

the mathematician and says, "I've discovered that Goldbach's

Conjecture is unprovable!"

"Nonsense," says the mathematician. "That is impossible.

If you had, then you would have proved it true, because

you would have prevented any possibility of me exhibiting

an even number with the (easily verifiable) property that

it is not the sum of two primes. Go away, charlatan!"

Euclid takes this all in, and is very impressed. But

just then, another student rushes up to Euclid, thinking

to have some fun. "By the gods, Mr. Euclid, I can show

that your parallel postulate can be neither proved nor

disproved from your first four axioms!"

Euclid starts talking almost without thinking: "That

is impossible. Suppose that you had shown such a

thing. Then that would mean that for any point on

the (lower) line, the upper line couldn't possibly

cross there, and so you would have shown that the PP

is true---not undecidable! Go away, charlatan!"

"But sir," retorts the student, "I've shown that you

can take it (the PP) and not reach a contradiction,

or that you can leave it and not reach a contradiction.

Yes, in some models it's quite true, but in others,

actually false."

Being a genius, Euclid somehow sees the point. But the

first student was still listening in on all this, and

noting Euclid's ashen countenance, decides he can pull

the same thing on the 20th century mathematician:

"Listen," he says, "you can have a model where every

even number is the sum of two primes", or, "you can

have a model where some even number IS NOT the sum

of two primes".

"And what becomes of that number in the first model?",

angrily roars the 20th century mathematician. "You see,

we are talking about THE SAME NUMBER, idiot! A particular

integer is a REAL THING by the grace of GOD! (screaming)

SEVENTEEN IS PRIME NOT BECAUSE OF ANY STUPID AXIOMS BUT

BECAUSE IT **IS** PRIME!!!". At this point, the 20th

century platonist mathematician becomes apoplectic and

has to be taken outside for some fresh air.

Says Euclid to the Riemannian geometry student, "So if

you are right, then a HalFinney point, i.e. a point that

is at a maximum "distance" from where a transversal cuts

two parallel lines, in your model---is that the right

word?---of course wouldn't have that property in a model

where parallel lines never meet. Hmm. I guess that's

another indication that geometry is richer than

arithmetic," he says with a smile.

So I'll guess from my little story that all models

of arithmetic that have enough axioms to support the

concept of "even number" either all have a the same

even number that isn't the sum of two primes, or all

have no such even number.

We reify "seventeen" and give it a particular identity,

and rightfully so, because you can prove many things

about seventeen, whereas "a point" is an undefined

concept in geometry.

*> In other words, just as the falsity of the GC would
*

*> seem to imply that there must be a specific number
*

*> which ISN'T the sum of two primes,
*

evidently in any model,

*> so the falsity of the PP would seem to imply that
*

*> there must be a specific distance to a point which
*

*> is the farthest possible point of intersection.
*

but only in some particular model, that "distance"

depending on which.

Lee Corbin

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