Q&A: Infinite Questions
Infinite Questions
Question
1. On a number line segment between 0 and 1 there are infinitely many points. But there are also infinitely many points between 0 and 10. Can we say that the second infinity is bigger than the first? On the other hand, that’s a bit strange, because we’re talking about infinity, and both are supposed to be the same.
2. A point has no length, but a line, which is a collection of infinitely many points, does have length.
A. On a line of length 3 cm, are there more infinitely many points than on a line of length 1 cm? (I’m not sure whether this is exactly the same question as the previous one; I’m taking my first steps in philosophy… thanks to you / because of you…)
B. We see that a collection of infinitely many particulars can have a property that does not exist in each and every individual part.
I was thinking about the cosmological argument: it’s true that every material particular in the universe has a cause, but the universe is infinite; it is a collection of infinitely many such particulars. So maybe that collection of particulars has no cause?
I thought this was a bit different from the example of a point and a line, because the line (the infinite whole) *receives* the property that the point (the individual part) does not have, whereas here the universe (the infinite whole) is *deprived of* the property that the individual part has. But I can’t quite see why that is what makes the difference… (if it makes any difference at all).
3. Is there an end to the numbers? Does the Holy One, blessed be He, know the end of the numbers, or is that one of the impossibilities? And then too it wouldn’t be disrespectful to say that He does not know it.
Thank you
Answer
My response was deleted. Here it is:
If you’re interested in questions like these, then you didn’t choose the right track. This is mathematics, not philosophy. Philosophers say quite a lot of nonsense in the area of infinity. 1. You are mistaken here in two ways: it is not true that there are no different infinities. There definitely are, as Georg Cantor showed. And the second mistake is that here there really is no “more.” It is exactly the same infinity. Cantor showed that there is a one-to-one correspondence between the two sets. 2. A line is not merely a collection of infinitely many points. It has the cardinality of the continuum and certain additional properties. A. Not true. See above. B. See the second and third conversations in the first volume of The Foundational Reality. 3. There is nothing to know, because they have no end. M.
And to this Zenon asked:
It seems that everything depends on section 1. Have you written about this somewhere?
No. Search online for Cantor’s infinities.
Discussion on Answer
I had a technical glitch and your comment was accidentally deleted. Let me just complete that you asked where I wrote about this and where one can read about it.
There isn’t a subfield. Here’s a video: https://davidson.weizmann.ac.il/online/mathcircle/clips/%D7%9E%D7%94%D7%95-%D7%92%D7%95%D7%93%D7%9C%D7%95-%D7%A9%D7%9C-%D7%94%D7%90%D7%99%D7%A0%D7%A1%D7%95%D7%A3
You can also read here:
https://he.wikipedia.org/wiki/%D7%A2%D7%95%D7%A6%D7%9E%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)
And plenty more.
Thanks,
I think you wrote the answer to me and not to him in the original question.
If you want, put it back…
What subfield is this called there?