Re: infinitely many nn's = infinite nn's?

G. Frege wrote:

On Mon, 12 Mar 2007 16:31:30 -0400, David Marcus
<DavidMarcus@xxxxxxxxxxxxxx> wrote:


...why don't you try to simply answer the question? Is it in fact possible for infinitely many reals to exist on the segment [0,1], when ALL of them are a finite distance apart?


I suppose you mean "positive" not "finite".


C'mon... ;-)

I'll guess what he has in "mind" (haha) here is a subset S c [0,1] for
which the following holds:

there is a number d e IR, d > 0, such that for all x,y e S with x =/= y: |x - y| >= d.

Of course, for any d e IR, d > 0, the number of elements in S is bound by
1/d + 1, hence is finite.

On the other hand,

[we] can prove that in a complete ordered field, [and hence in IR] there
are infinitely many numbers in [0,1] and that if x,y in [0,1] satisfy x =/= y, then |x - y| > 0.


Yeah.

Quantifier dyslexia strikes again. Seems to be epidemic among cranks.

Phil seems to be confused about those two (different) things:

Well I doubt that I am confused about the fact that there are two proofs, one which shows that there is a FINITE distance (or difference, if you must) between any two reals on the line segment (which is why I can use the word "distance," duh) [0,1], and another which shows that there are infinitely many reals -- in fact, infinitely MORE than there are natural numbers (aleph-C versus aleph-0) -- on that same segment. The question that most of you seem unable or unwilling to address, even though you know that it is not a GIVEN that all proofs use compatible premises, is whether the an infinite sum of FINITE distances (or differences) is can result in a FINITE distance (or sum) of one, equivalent to the line segment [0,1].

Now, DISHONEST people (I'm sure we have none here) would simply refer to the two proofs in question, and CLAIM that SINCE these two proofs exist -- thereby implying that they are too stupid to remember that proofs that use incompatible premises can produce contradictory results -- that there must therefore be no contradiction at all, and that an infinite sum of finite values can indeed produce a finite distance.

Tell me G., since you at least seem to be more honest, and less of a ***, than David, do you think there is some FINITE distance from the door such that we can double that distance an INFINITE number of times and wind up in the middle of the room? We could simply ADD that distance to itself an infinite number of times and get the same result. Are you going to ACTUALLY CLAIM here, for all to see from now on, that you believe that the "existence" of infinitely many reals, all a finite distance/difference apart, does not even HINT at a contradiction? Sure, this apparent contradiction may be okay, meaning that the two proofs use completely compatible premises, but don't you HONESTLY think that it at least hints that there be an incompatible premise or two, that we should at least THINK about it?

I am not asking you to say that these "incompatible premises" actually exist, mind you, I am just asking whether you want everyone to know, from now on, that when you combine the results from two proofs, and that combination states that (1) there is a FINITE distance between ANY two points (real numbers) on the segment [0,1], and (2) there are INFINITELY many points (real numbers) on that segment, that you see NO HINT OF A CONTRADICTION AT ALL??? Are you going to say that you see NO REASON AT ALL to investigate a bit further, even though you think it is probably just an apparent contradiction, and not a real contradiction? Please remember, the fact that these two proofs exist does NOT prove that they can both be applied to the same object without producing a contradiction, since we have NOT yet proved that the premises, hidden as well as stated, are the same in both proofs.

for all x,y e [0,1] with x =/= y: there is a number d e IR, d > 0, such that |x - y| >= d.
and
there is a number d e IR, d > 0, such that for all x,y e [0,1] with x =/= y: |x - y| >= d.

I'm not sure what you mean here. This first is obvious, there exist a real non-zero number d which is smaller than the difference between any two real non-equal numbers, but I don't understand the significance of stating it "in reverse." Yes, I understand that we often need to do that, but I don't see what that accomplishes here.

Phil

F.


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