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September 11th, 2012, 08:12 PM   #11
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Re: Dichotomy paradox reverse engineering

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Originally Posted by Maschke
As I indicated in my first post, you would have to tell us. Because your process is not defined at t = 1.
If set theory tells us, that the union of [0, 0.5], [0.5, 0.75], [0.75, 0.875], ... is [0, 1), isn't this the same as defining the limit case?
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October 2nd, 2012, 12:31 PM   #12
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Re: Dichotomy paradox reverse engineering

If we allow only finitely many copies of a line segment s in [0, 1], where the size of s is an element of (0, 0.5], how can we allow infinitely many line segments { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], where the size of each line segment is an element of (0, 0.5]? We wouldn’t allow an infinite number of copies of any element of { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], right?
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October 2nd, 2012, 04:07 PM   #13
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Re: Dichotomy paradox reverse engineering

Huh?
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October 2nd, 2012, 10:44 PM   #14
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Re: Dichotomy paradox reverse engineering

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Originally Posted by netzweltler
If we allow only finitely many copies of a line segment s in [0, 1], where the size of s is an element of (0, 0.5], how can we allow infinitely many line segments { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], where the size of each line segment is an element of (0, 0.5]? We wouldn’t allow an infinite number of copies of any element of { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], right?
I guess number of copies is ambiguous. For example, we allow four copies of the segment [0.5, 0.75] in [0, 1], which are [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1]. So, for each element of the set { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } is true, that the number of copies in [0, 1] is finite, right?
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October 3rd, 2012, 11:33 AM   #15
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Re: Dichotomy paradox reverse engineering

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Originally Posted by netzweltler
I guess number of copies is ambiguous. For example, we allow four copies of the segment [0.5, 0.75] in [0, 1], which are [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1]. So, for each element of the set { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } is true, that the number of copies in [0, 1] is finite, right?
So when you say that "we allow n copies of A in B" you seem to mean that n|A| <= |B| < (n+1)|A|, where |[a, b]| = b - a. In that case you're asking if each element of {2, 4, 8, 16, ...} is finite (it is).
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October 5th, 2012, 01:08 PM   #16
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Re: Dichotomy paradox reverse engineering

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Quote:
Originally Posted by netzweltler
I guess number of copies is ambiguous. For example, we allow four copies of the segment [0.5, 0.75] in [0, 1], which are [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1]. So, for each element of the set { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } is true, that the number of copies in [0, 1] is finite, right?
So when you say that "we allow n copies of A in B" you seem to mean that n|A| <= |B| < (n+1)|A|, where |[a, b]| = b - a. In that case you're asking if each element of {2, 4, 8, 16, ...} is finite (it is).
Does it make some kind of sense, that there is enough space for infinitely many segments { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1] even if there is no space for infinitely many copies of one of these segments of this set?
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October 5th, 2012, 01:30 PM   #17
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Re: Dichotomy paradox reverse engineering

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Originally Posted by netzweltler
Does it make some kind of sense, that there is enough space for infinitely many segments { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1] even if there is no space for infinitely many copies of one of these segments of this set?


Yes it makes perfect sense. The infinitely many intervals keep getting smaller and smaller. The sum of their lengths converges.

You're asking if it makes sense that

1/2 + 1/4 + 1/8 + /16 + ... = 1

but

1 + 1 + 1 + 1 + ... is infinite.

Well yes, it makes perfect sense. Doesn't it?

In fact it's a necessary condition of a convergent infinite series that its terms go to zero. All you've done is illustrate that fact, which is proved in freshman calculus.
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