Monday, July 03, 2006

Infinity II

The Chovos Helevovos says that because something has a PART of it which is
finite, the whole thing must also be finite, since all infinities are the
same size and there cannot be an infinity which has a finite part and an
infinite part. There seems to be a Kasha with this. The set of numbers (0,1,2...) is
infinite. A part of the set is (5,6,7), that set is finite, and the other part
which is left over, (0,1,2,3,4,8,9,...) is infinite, and it is a smaller
infinity than the original infinite set. Certainly in mathematics they talk
about infinities of different sizes, the infinite number of points on a line is
smaller than the infinite number of points on a plane, for example. So how does
his proof work, that there were a finite number of people from Noach to Moshe so
therefore the total number of all people ever was finite? Why is that true?

You can have a theoretical, imaginary infinity, such as the set of all numbers - which represents the idea that you can always add more numbers to whatever amount of them you already have. But you can never reach that amount called "infinity" (since you can always add more), and therefore, we conclude that any quantity of measurable things that you have already reached, in real life, cannot be infinity. Because you can always add more to whatever quantity you already have. In other words, you cannot have, in real life, an already accumulated infinite quantity of any measurable, finite, things. That includes grains of sand, or moments in the time stream.
The set of numbers is infinite because it represents a progression that can never be completed. But if a progression of numbers WAS completed, it is not infinite.

You can say that the set of numbers never ends, meaning, no matter how many numbers you’re going to add to that set, you can always add some more. It is the process of adding to the set that never ends. But that means that if you finished adding numbers to the set, then you do NOT have infinity, since infinity here means that you can never finish adding.

The set of infinite numbers represents infinity, but in real life, you can never have that quantity of anything by simply adding more and more until you reach that point that we call infinity - that point is never reached.

Therefore, infinity can exist in the sense that a certain concept or progression WILL never end, but if the progression ended, then it cannot be infinity.

So if you take grains of sand, for instance, you can never have an infinite number of them in real life, because no matter how many you pile up, there can always be more added.

You could, however, make a set of {an infinite amount of sand grains} because that set represents a theoretical number. But in real life, if you have a pile of sand, no matter how big it is, it does NOT contain an infinite amount of particles.

Therefore, you can never have in real life, an infinite quantity of anything. Since any quantity of finite things means a completion of the group of things in that quantity, it cannot be infinite.
Same thing with time. You can say, in theory, that time will never end, meaning, no matter how much time has progressed in the past, you can always add more, all the way into the future with no end in sight.

But you can never finish an infinite amount of time, and that means that you cannot have had an infinite amount of time in the past, since that would mean, as of yesterday or today, an infinite amount of time has ALREADY passed. That can't be. It would mean that an infinite amount of time has already been compiled. And that’s impossible since at you can always add more time to whatever has already occurred. At what point are you going to stop the compilation of time and say "Okay - we have an infinite quantity of moments here."?

In any case, when we talk about an infinite amount of "points" we means points that do not actually take up any measurable space in terms of inches or fractions thereof. And that’s exactly what the Chovos Halevovos means: If you have an infinite amount of things already here, then you cannot measure by any yardstick each of those things individually. Like the points between 2 spaces. And when I say they cannot be measured, I mean they too are infinite - i.e. they can be infinitely divided over and over again. Because they do not take up physical space. Or time.

In short: You cannot have already accumulated an infinite amount of finite things.

So there’s 2 types of infinity - (a) infinity "on paper", where you deal with the "fact" that things can theoretically be endlessly added (such as the set of numbers) or endlessly divided (like points between spaces), and (b) real life infinity - where an infinite progression has HAPPENED already, or an infinite amount of divisions has ALREADY taken place.

If the past was infinite it would have to be type (b) infinity; the future can be type (a). Type B cannot exist in real life. Since infinity/X=infinity, each divided part of the infinite set must also be infinite - and that's not going to happen.

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