EMUNAH & BITACHON-----proofs/ infinity

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Moshe00 Posted - 09 January 2003 7:09
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I have a question about Chovos HaLevavos, Shaar HaYichud, Perek 5, in the proof that the world is finite "V'Im Na'aleh B'Machashavteinu"..."B'Reishisan Ka'asher Hikdamu". I have not been able to find a satisfactory answer to this question:

It seems to be that the Chovos HaLevavos is saying that because something has a PART which is finite, the whole is also finite, since all infinities are the same size and there cannot be an infinity which has a finite part and an infinite part.

I don't see how this makes sense. 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?


MODERATOR Posted - 09 January 2003 12:28
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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. But that means that if you finished contributing 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 an 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."?

And even though there are different "amounts" of infinity, that is also theoretical. Five inches theoretically can be divided into an infinite amount of points, and so can 10 inches, yet 10 inches is longer than five. That’s a sticky mathematical and philosophical issue, that has led some philosophers who did not have the benefit of a Torah education (in particular, a guy named Zeno) to conclude that motion is an illusion. If there is an infinite amount of points in any given distance, then how can you ever traverse so many points in a finite amount of time?

The answer to Mr. Zeno's question will be something in the direction that, even though there is an infinite amount of points between any two spaces, those points are also infinitely small - the more you divide them up the smaller they get, so an infinite division would demand infinite smallness of each piece - which would mean you should not need any time to traverse any distance, since all the spaces are infinitely small. So the infinite quantity of the total cancels out with the infinite smallness of each part.

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.


grend123 Posted - 09 January 2003 13:51
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A little math might help here...

"it is a smaller infinity than the original infinite set"

Incorrect. It is an equal infinity because there is a one to one correspondence. The set of numbers LESS than zero is the SAME SIZE as the set of all real numbers, even though that seems to make no sense, because infinity - ANYTHING = infinity

You are correct that the number of points in the plane is a higher order infinity, called continuum, or aleph-1. Mathematically, this is infinity RAISED TO THE POWER OF infinity. After that, we go to lines in the plane, called aleph-2 (continuum raised to continuum) and curves on the plane, or aleph-3, (aleph-2 raised to aleph-2). Yes, aleph as in the Hebrew letter - this proof was written by a Jew. As of yet, aleph-3 is the highest "infinity" with useful properties that we know of.

Therefore, the chovos halevovos is saying as follows:

If you have a set divided into TWO RATIONAL PARTITIONS (i.e., the size difference can be expressed as one part's size being a certain fraction of the other), then if one part is finite, the other part must be finite, because otherwise the ratio between the parts is undefined, and we already claimed that it WAS defined.

What you were talking about when you wrote:

<"(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">

is something else entirely - a set with no definable ratio (the limit of the ratio is 0 or infinity, depending which direction you take it). That's something else entirely.

Hope this helped,

--Grend


Braunshlo Posted - 09 January 2003 14:05
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Mod, whets the difference between what you wrote and what grend wrote?


MODERATOR Posted - 09 January 2003 14:08
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Not much. Grend just used more loshon hakodesh words ;-)
It’s the same idea, explained 2 diff ways. Bottom line's pretty much the same.


Moshe00 Posted - 09 January 2003 16:04
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Grend, how does the Chovos Halevavos prove that the set of time from Noach to Moshe, and the set of all other time are rational partitions of time? How can we prove that it is not true (c"v) that: time is infinite, the time from Noach to Moshe is finite, and the ratio of time from Noach to Moshe to all other time is 0? Like my example with the numbers? How do we assume that they are rational partitions?

And also, I don't think I understand the other part of what the Moderator said:
"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 cant be. It would mean that an infinite amount of time has already been compiled. And thats impossible since at you can always add more time to whatever has already occured."

You can't add more things to infinity? {...-2,-1,0,1} is infinite. {...-2,-1,0,1,2} is also infinite. They are (as Grend corrected me) the same size infinity (both aleph-0, right?), but does that matter?


MODERATOR Posted - 09 January 2003 16:09
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Doesn’t matter if they’re the same size --- once you END the progression of time then you stopped before infinity. In infinity of progression of time, or numbers, that you can always add on more. If that’s so, you cant pass the point of infinity.

Or in other words, if there was an infinite amount of days that had to take place in the past before today, we never would have reached today, since you can never reach the number "infinity" in a real-life progression of finite moments.


Moshe00 Posted - 10 January 2003 10:44
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I'm still missing something. "if there was an infinite amount of days that had to take place in the past before today, we never would have reached today, since you can never reach the number "infinity" in a real-life progression of finite moments"

Why is that so, if you started the real-life progression of finite moments an infinite amount of time ago? Of course it's true that a FINITE number of finite points in time won't be an infinite amount of time, but if we "assume" (c"v) that the past could be an infinite amount of time, then just because we "stop" at any arbitrary point in time, when we look at the past, we would see an infinite amount of time. I don't see why that is a contradiction. Like this: (...-2,-1,0,1). I ended the progression of integrals with the number one. But I didn't stop before infinity.


MODERATOR Posted - 10 January 2003 11:02
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The past cannot be an infinite amount of time because the past is finished, and an infinite amount of time never finishes.

What you have to remember is, infinity is not a very large number which is you keep counting you will eventually reach. Even if you count forever, you will not reach "infinity". When we say that there are an infinite amount of finite natural numbers, we mean that no matter how long you were to count, you will never get to the end of those finite numbers.

So therefore, if you did count and reached a certain number, that number cannot be infinity.

Thus, if the past would consist of an infinite amount of time, it would never be over. AT no point would you be able to say "we have reached infinity", since that point is unreachable. The past, however, is over. Therefore, the amount of time that has already transpired in the past could never have reached infinity.

As a syllogism:

If the amount of moments in the past is infinity, those moments would never be finished.
But the past has finished.

Therefore, the amount of moments in the past is not infinity.

This is the answer to Zeno's paradox. Zeno is the one who said that everything in the world is infinitely large (and infinitely small but never mind that for now), since even an inch can be divided up an infinite amount of times. Which means an inch and a mile - which also is divisible an infinite amount of times - are really the same length.

But he's wrong, obviously, and the reason is, because you can never divide up an inch, or a mile, an infinite amount of times. No matter how many times you divide up the distance, the resultant amount of parts will always be a finite number. So you will never, ever have an infinite amount of parts in any given line.

Infinity cannot be reached in real life, ever. You can never count until infinity. You can never have an infinite amount of anything that has magnitude. Therefore, if we already had a certain amount of moments in time, since each moment does take up time, the total amount of moments cannot be infinity.


Moshe00 Posted - 19 January 2003 16:11
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I think that I may have understood. I think that this is a paradigm of what this argument expresses.

"Assuming"(C"V) time goes infinitely back in the past, and there is a person who has been counting integrals forever (he never started counting, he has always been counting), so that he has counted aleph-0 integrals. So he has finished counting all the integrals, since there are aleph-0 elements in the set of integrals. But he has also just finished with the negative numbers, since there are aleph-0 numbers in that set, also.

In fact, he could have reached any number and still have completed aleph-0. So the number that he is currently counting is undefined, which doesn't make physical sense, so it is impossible. (But I'm not quite sure that what I just wrote shows that there is a proof. Although it doesn't make physical sense, it does make theoretical sense, doesn't it?

...since each of these sets and the set of moments in the past all have aleph-0 elements it DOES make sense that there will be a one-to-one correspondence between them.)

What the Moderator said before and what Grend said, about rational partitions, I do understand (is it perhaps the same thing stated differently?), and that seems like what the Chovos Levavos means when he says that the time from Noach to Moshe is a "chelek" of all time: that it is a way of rationally partitioning time. My only question is, how do we know that the time from Noach to Moshe must be a RATIONAL partition of time? Why do we assume that?


MODERATOR Posted - 19 January 2003 16:46
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If any partition of time would be infinite it would never end. Since the time-partition was finite it therefore can not be comprised of an infinite amount of parts.
In theory and in reality, infinity cannot be reached. The time period from the beginning of time until today HAS been reached. Therefore it cannot be infinite.