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

herbzet wrote:

Phil wrote:

I'll correct my sloppy writing first ...

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 an infinite sum of FINITE distances (or
differences) can result in a FINITE distance (or sum) of one,
equivalent to the line segment [0,1].


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

--
hz
Yes, but again, this is an example where you have one proof that states that all of these values are finite, and another that states that there are infinitely many of them. Once again, I am NOT asking you whether two such proofs exist (or two pairs, or four pairs ...), which is what you have done here. Since this is (eventually) equivalent to the example above using the finite distance between every real number on the line segment [0,1], where the sum of INFINITELY many FINITE distances adds up to a FINITE distance, I am asking you whether this seems to you to even vaguely indicate a potential problem (given that normally, infinity times a finite value ALWAYS results in infinity). Yes, many such pairs of proofs exist, but that was NOT my question. My question is whether you think the resulting paradoxes even possibly indicate that the two proofs in each pair are using different sets of premises.

In fact, your example is one half of the "infinitesimal paradox" that I posted both recently and several years ago. Since there are infinitely many elements in the sequence, we can match these elements with a set of infinitely many infinitesimals g -- if you prefer, from non-standard analysis, so we do not even appear to be violating the natural numbers -- in the following manner:

F = {1/2, 1/4, 1/8, 1/16, ...,}
| | | |
I = { g, g, g, g, ...,}

Where g is defined as having an infinitesimal value such that the infinite sequence adds up to 1. Therefore, we have two infinite sums, both with equally many elements, in which the first series consists of nothing but finite elements, the second series consists of nothing but infinitesimal elements, and yet the two sums are equal. Furthermore, since the first 3 finite elements sum to a value quite a bit larger than the first 3 infinitesimal elements, the remaining finite elements sum to a value that is LESS than the sum of an equal number of infinitesimal elements. Even though EACH AND EVERY member of F is infinitely greater than ANY member of I, infinitely many F elements sum to a value that is equal to or even less than an equal number of I elements.

Now, does that give you just the slightest pause, the slightest feeling that the proof showing that there are infinitely many natural numbers, and the proof that all the natural numbers are finite, just MIGHT rest on different sets of premises? That IS one of the three possible explanations for the apparently contradictory results, although the other possible explanations are that (2) infinity is very weird or (3) there is a hidden error in my paradox, meaning that it is NATURAL for a series of infinitesimals to add up to a value equal to or greater than an equal number of finite values.

However, do you think, YOURSELF, that infinitely many FINITE values REALLY CAN sum to a finite value? Remember, the fact that {1/2, 1/4, 1/8, 1/16, ...,} sums to 1 after INFINITELY many elements could simply be a result of the fact that only finitely many (potential rather than actual infinity) of these elements are finite, and that infinitely many of them are infinitesimals. And all it would take for that to be true would be to use ACTUAL rather than potential infinity when determining not only the number of natural numbers, but also the maximum value that natural numbers can have. As I said elsewhere, if you use either potential infinity OR actual infinity for BOTH the number of numbers and the maximum value of a number, this particular paradox instantly vanishes. Does that even slightly suggest to you that potential and actual infinity might NOT be the same thing?

Do you think, in situations like these, that SOMEONE -- preferably a good mathematician who really knows what he's doing, but someone, in any case -- should at least look into the possibility that the proofs rest on incompatible premises? Or do you believe that, despite all the errors made concerning infinity for 2500 years, that starting about 150 years ago, we got everything right, everything perfect and error-free, and there's no need to even verify that no problem exists, no need to even investigate any paradoxes? It is as much the implicit claim that mathematicians obtained godlike perfection toward the end of the 19th century, as much as anything, that does amazes me -- or disgusts me -- about this group.

Phil
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