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I want to go back to this very old thread, which I have continued to

think about, and add a couple of points.

Earlier I wrote a response to Lee's puzzle about whether it makes sense

to say that Goldbach's conjecture is unprovable:

*> Goldbach's Conjecture is that every even number is the sum of two primes.
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*> If it is false, there must be an even number which is not the sum of
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*> two primes. If such a number exists, then by showing it we could show
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*> that GC is false and GC would not be unprovable (that is, it would be
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*> provably false). For GC to be unprovable, therefore, would mean that
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*> no such number exists. But this is simply the statement of GC itself,
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*> so to show it to be unprovable (undecidable) is to show it to be true.
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*> Hence you can't show GC to be unprovable.
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Other people gave a similar response.

I wrote earlier about Euclidean geometry minus the parallel postulate

as a specific historical example of an issue where provability was very

much in question. Most of Euclid's axioms seemed completely self-evident:

two points determine a line, and so on. But the parallel postulate, that

through every point not on a line there is exactly one line parallel to

the given line, never seemed as obvious. (This is especially true because

Euclid's actual parallel postulate was worded in a more complicated way

[1], although it was mathematically equivalent.) It didn't fit well

with the other postulates.

So there were a number of attempts over the years to eliminate the PP by

showing that it could be derived from the other axioms, but none of them

succeeded. In the end it was shown that the PP was independent of the

other axioms, and in fact you can get non-contradictory and interesting

versions of geometry by assuming either that there are multiple parallel

lines through a point or no parallel lines.

Now, here is how the PP is like GC. Consider a line and a point above

the line. Let there be a 2nd line through the point, not parallel but

almost. Say it makes a 5 degree angle with the base line. Then the

intersection point is well off to the side.

Rotate the top line so that it moves through the parallel angle.

The intersection point moves off, faster and faster. As we pass through

the angle of parallelism, the intersection point zooms off to infinity.

At the same instant, a new intersection point appears at infinity on

the opposite side, and zooms towards us, slowing down progressively.

Let's suppose we want to ask whether a weaker version of the PP is true,

which is whether or not there is at least one parallel line, versus

whether there are no parallel lines. We know in fact that this question

is unprovable. But here is an argument for why it "can't" be unprovable:

The PP states that there is an angle at which there are no points of

intersection as we do the rotation above. If it is false, as we rotate,

the 2nd intersection point exists before the first one disappears.

Therefore there must be some maximum distance away that an intersection

point can reach; beyond that distance the 2nd intersection point will be

closer. The PP is therefore a question of whether such a point of maximum

distance exists, or whether that point is in effect infinitely far away.

This point is like the number which is the counter-example to the GC

in the argument above. Now I am going to follow sentence-by-sentence

paralleling the GC argument above.

"If such a point exists, then by showing it we could show that the PP is

false, so the PP would not be unprovable. For the PP to be unprovable,

therefore, would mean that no such point exists. But this is simply

the statement of the (weak) PP itself, so to show it to be unproveable

(undecideable) is to show it to be true. Hence you can't show the PP

to be unprovable."

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, 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.

For either proposition to be unprovable would seem to imply that there

is no such number or point, which would mean that they are true.

In the case of the PP, we can measure the distance to the intersection

point as the lines approach parallelism. We can see it getting farther

and farther away, while confirming that no intersection appears within

the same distance in the other direction. This is like experimentally

verifying the GC by testing it with larger and larger numbers.

The question is, will the point get to infinity? And can we keep

finding even numbers that don't violate the GC, all the way to infinity?

It's really the same kind of question.

We know that the reasoning is false in the case of the PP; there exist

consistent geometries where there are no parallel lines. We might

even live in a universe where that is the case; it's possible that if

we actually tried the rotating line experiment, we would find that the

intersection point never gets farther than X billion light years away.

Given that this seemingly valid reasoning is actually mistaken with

regard to the parallel postulate, I think we need to be very cautious

about accepting it with regard to Goldbach's Conjecture. The two cases

are more alike than might seem at first.

Hal

[1] Euclid's actual Parallel Postulate: If A and D are points on the same

side of segment(BC) such that measure(angle(ABC)) + measure(angle(BCD)) <

pi, then ray(BA) intersects ray(CD).

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