Logic and Infinity

[Ibn Taymiyyah]

Hello everyone, I hope that you accept me as a new member of this wonderful forum. I also hope that you accept my broken English because it is not my mother tongue.

A few days ago I came across a logical argument which attempted to disprove infinity. It goes like this:

1) We cannot imagine two objects that are infinite in size to be anything but equal in size.

2) If a relatively small object increases in size at a constant daily rate (10 units of size per day), and,

3) A larger object increases in size at the same constant rate.

4) If both objects begin increasing in size at the same time, we can’t imagine the smaller object ever catching up with the larger object. There will never come a time when they become equal.

5) Therefore they will never become infinite in size because they will never become equal (point (1)).

Deep inside I know that there is something wrong with this argument but I can’t put my finger on it.

What do you think?

[bingley]

If they are starting from a finite size presumably your two objects would need an infinite number of days to become infinite in size. If it is possible for either object to become bigger the next day that objct is not yet infinite in size. Infinite doesn't just mean very big. It means there is no end to the object. I think there is a qualitive difference between a finite object and an infinite one, it's not just that one is bigger than the other.

[Kalailan]

the first assumption is wrong. if there is anything infinite it must be just one. if there are two they have to have an end, don't they?

[annis]

Deep inside I know that there is something wrong with this argument but I can’t put my finger on it.

There is an entire branch of set theory dealing with different sorts of infinities. I don't really understand it very well, but this link seems useful:


http://www.mathacademy.com/pr/minitext/infinity/

Some googling on the words "infinity" and "aleph-null" will get you close to relevant material.

[threewood14]

you gotta remember this about infinity. look at a ruler. how much space is between the one unit mark and the 2 unit mark. if you are using metric, probably about 1cm. now look at it this way. how many point in space time are between those two lines. well first of all, a point in space time is infinitly small. therefore, there are an infinite amount of points in space time between those two objects. we always look at our surroundings at human intervals. now take time. how many point in time are in one second. infinite right? one second is just a measurement of time at the rate we humans experience it.

__________
threewood14

[Ibn Taymiyyah]

* Qualitative not quantitative … makes sense … however, most scientists approach the concept of infinity by using quantitative methods, including the one referred to by annis (thanks annis).

* Only one infinitely large object can exist … also makes sense ... then lets assume two (2) straight threads, one shorter than the other increasing in length at the same rate, will they ever become equal?

* There are infinite numbers between the 1 and the 2 and between the 2 and the 3 … this proves that more than one (linear) infinity exist at the same time.

[Kalailan]

* There are infinite numbers between the 1 and the 2 and between the 2 and the 3 … this proves that more than one (linear) infinity exist at the same time.

it does not really. it is the same infinity all along the way, and the numbers are just like the numbers on a ruler - they ruler doesn't pass between the numbers, the numbers are set on the ruler.

i think it's also the difference between the greek theory that the world is constructed from the math or logic that humans found and the idea that these are actually subjective; a creature that is water-drop like would think like this:
"hmm, 1+1=<1".

>whoops... kalailan has driven off the road...<

[xn]

Ibn Taymiyyah: Your English is flawless! To my thinking (which is colored by my mathematics background), the problem in the argument is point #1, because it is imaginable that different infinite sets of objects can have differing number of members (or different “cardinalities”).

Mathematically, a set is considered infinite if it can be placed in a one-to-one correspondence with a proper subset of itself. For example, the set of natural numbers [called “N”, comprising the integers 0, 1, 2, 3, …] is infinite because it can be placed in a one-to-one correspondence with the subset of N that contains only the members that are evenly divisible by two. Because for every member n of N, there exists a member n×2 within the subset (e.g. 0 to 0, 1 to 2, 2 to 4, 3 to 6, &c.), N is deemed to be infinite. Since there is a one-to-one correspondence between N and the subset, both N and the subset have the same number of members — the same cardinality. The cardinality of N is known as “aleph null” — that is, the Hebrew letter aleph followed [left-to-right] by a subscript zero. This represents the smallest infinity.

The mathematician Georg Cantor showed that the set of real numbers [called “R”, comprising all rational numbers and all irrational numbers] has more members (a greater cardinality) than N, which demonstrates two differently sized infinities. Look for “Cantor’s diagonal” in your preferred search engine for the reductio ad absurdum argument.

xn

[Evito]

We múst remember that we are part of infinity as something we aren't part of cannot be infinitely big. Also, we cannot imagine "everything" not being infinite as something non-infinite has boundaries.

Of course in mathematics we can speak about infinity as if it were possible to have some infinite outside of ourselfs. Here, though, we only use it to solve problems. Thus, it is not important if it were really possible to create such a situation. In fact, creating infinity and working with it is not possible, as there is nothing outside the infinite itself to either create or work.

[threewood14]

Here is something that is mind boggling. If something moves an infinite small distance, it would take an infinite amount of time for it to get from point A to point B whereever they may be. Let us say that we give it 100 years to move one inch. Since the distance it is moving is infinitly small, it would obviously not get there by then. But, it would have not moved at all. Because an infinitely small distance plus an infinitely small distance equals and infinitly small distance. If it did move a distance, then it would take a finite amount of time to get from a to b. But since it doesn't, an infinitly small distance can be thought of as 0!!

[xn]

threewood14: Are you restating Zeno’s Dichotomy here? One must remember that there is an infinite set of finite distances, each member of the set being longer than that infinitely small distance …

xn

[threewood14]

Who is Zeno?

[Raya]

Who is Zeno?

He is an ancient Greek philosopher who discussed ideas similar to yours about infinite distances. He illustrated these ideas through the example of a footrace between Achilles ('fleetest of foot of all mortals') and a tortoise.

If I remember correctly (and someone correct me if I've got it wrong, because it's been a while since I read this...)
Achilles and the Tortoise decided to run a race of an infinite distance. The tortoise was allowed a headstart over Achilles, and this allowed him to win the race because - the distance being infinite - Achilles could never catch up to the tortoise no matter how fast he ran!

[Kalailan]

He also "proved" that movement is impossible.

if you want to pass a distance, you first have to pass half that distance. but to pass half that distance you have to pass half that distance, a quarter of the original distance. you have to move the smallest distance that isn't 0 in order to move anywhere; but what is that number? the "smallest" number doesn't exist. there is always a smaller one...

[Ibn Taymiyyah]

The smaller the distance the less is the time needed to pass it. The smallest distance that isn't 0 will be passed in the smallest time that isn’t 0. So there is no problem as I humbly see it.

[Kalailan]

the problem is that between zero and the smallest number there must be another number; therfore the first number isn't the smallest. if there isn't such a thing as the smallest distance, how could you possibly pass it?

[threewood14]

I guess just about all things that have a velocity are moving an infinitely small distance. But that is not the actual distance they are moving. If I move one inch, I have also moved an infinitly small distance. Its like if I move 1 foot, I have also moved 1 inch> But of course that has nothing to do with my last post. LOL

[threewood14]

infinity is just a concept of no limits

[aot]

The problem is that point 5 is a non sequitor. It doesn't matter if we can 'imagine' them catching up in size. infinity is not something you can add numbers to get to.

[threewood14]

It really is hard to do anything with infinity. But I had a thought.

1) We cannot imagine two objects that are infinite in size to be anything but equal in size.

2) If a relatively small object increases in size at a constant daily rate (10 units of size per day), and,

3) A larger object increases in size at the same constant rate.

4) If both objects begin increasing in size at the same time, we can’t imagine the smaller object ever catching up with the larger object. There will never come a time when they become equal.

5) Therefore they will never become infinite in size because they will never become equal (point (1)).

Here is why either will never become infinite in size. They have a finite amount of time to do so! If they were to become an infinite size, they would need an infinite time to do so.

It doesn't matter if we can 'imagine' them catching up in size. infinity is not something you can add numbers to get to.

Yes of course because we would need an infinite amount of time to do so.

[Democritus]

5) Therefore they will never become infinite in size because they will never become equal (point (1)).

Deep inside I know that there is something wrong with this argument but I can’t put my finger on it.

Well, it didn't disprove infinity, but it did demonstrate that you have to be careful when thinking about in infinity. Infinity is not a number. You cannot ever "reach" infinity.

For any number x you can imagine x+1. So you cannot imagine a number x such that x=infinity. There is no such number. But you can imagine incrementing x forever, so that x increases towards infinity.

1) We cannot imagine two objects that are infinite in size to be anything but equal in size.

If an object is "infinite in size," it means that we can't evaluate its size. Whatever number x we propose as being the size, there is always another number x+1, larger than the one we propose. Ordinarily, we cannot say that two objects of "infinite size" are equal in size, because we have no sizes to compare.

But check out the link annis posted about Georg Cantor and transfinite numbers.

Here is one of Canor's observations: Imagine the set N of all natural numbers (1,2,3,4,5,6.....). Now eliminate all the even numbers, so we are left with set O=(1,3,5,7,9...). Common sense tells you that set O is half as big as set N, right? But both O and N are still "infinite in size," even though we removed every other item from set N to get set O. In fact, for every item in N, we can find another item in O to pair it with, as if they were equal in size. Weird, huh?

[threewood14]

Here is one of Canor's observations: Imagine the set N of all natural numbers (1,2,3,4,5,6.....). Now eliminate all the even numbers, so we are left with set O=(1,3,5,7,9...). Common sense tells you that set O is half as big as set N, right? But both O and N are still "infinite in size," even though we removed every other item from set N to get set O. In fact, for every item in N, we can find another item in O to pair it with, as if they were equal in size. Weird, huh?

I think infinity is not a number. Its really hard to use it anywhere without running into a lot of trouble. I totally agree with you. That is weird

[Emma_85]

Infinity can't be a number, but it's a useful thing in mathematics. But we should always remember that we made the rules of mathematics. Infinity is not something we can imagine, it's something we came up with too. I'd like to know who came up with the idea of infinity? Was it a philosopher, mathematician, or is this a concept we don't know the origins of because they are so far back?
Mathematics has come up with some 'strange' things, zero, negative numbers and such like, but those concepts have always existed, mathematicians just formulated them. What about infinity? Anyone know?

[threewood14]

I had a thought that kind of has to do with infinity.

Everything that has a begining has an end. Its from the Matrix actually, but I think its a good concept.

If you think about it, everything that has had a begining has had an end. Look at the Romans, the Greeks, the Nazis, and of course many other powers. It doesn't exclude modern day countries either. Look at people. People are born, and they die. They are not infinite.

I guess the only stand one could take would to use a geometric ray. But if you think about it, where does a ray exist other that in our minds? It really is a mathematical concept in our heads. I don't think that there are any objects that are rays in the universe. You can still specify a ray, but its really just a bunch of points in space time put together with a starting point. It's geometry.

Other than that, what would you say?

[mingshey]

Infinity is a helplessness for a fish.

[mercutio]

you gotta remember this about infinity. look at a ruler. how much space is between the one unit mark and the 2 unit mark. if you are using metric, probably about 1cm. now look at it this way. how many point in space time are between those two lines. well first of all, a point in space time is infinitly small. therefore, there are an infinite amount of points in space time between those two objects. we always look at our surroundings at human intervals. now take time. how many point in time are in one second. infinite right? one second is just a measurement of time at the rate we humans experience it.

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threewood14

Reminds me of that drunken clown Zeno & his paradoxes, the arrow one this time. To go from a to b it must past through c, to go from a to c it must pass through d, to go from a to d it must go through e, blah blah blah ad infinitum et absurdum. Therefore the arrow can never arrive at b because there are an infinite amount of points it must pass through before it gets there. Are you right, Mr Zeno, are you right? No, you are wrong. Why? Because if you were alive I would shoot an arrow at you & it would not pass through an infinite amount of points & never hit you, it would hit you & you would die. Again. RIP.

[Michaelyus]

Remember: the arrow has a distance (a length) too.

[messalina]

you gotta remember this about infinity. look at a ruler. how much space is between the one unit mark and the 2 unit mark. if you are using metric, probably about 1cm. now look at it this way. how many point in space time are between those two lines. well first of all, a point in space time is infinitly small. therefore, there are an infinite amount of points in space time between those two objects. we always look at our surroundings at human intervals. now take time. how many point in time are in one second. infinite right? one second is just a measurement of time at the rate we humans experience it.

__________
threewood14

Reminds me of that drunken clown Zeno & his paradoxes, the arrow one this time. To go from a to b it must past through c, to go from a to c it must pass through d, to go from a to d it must go through e, blah blah blah ad infinitum et absurdum. Therefore the arrow can never arrive at b because there are an infinite amount of points it must pass through before it gets there. Are you right, Mr Zeno, are you right? No, you are wrong. Why? Because if you were alive I would shoot an arrow at you & it would not pass through an infinite amount of points & never hit you, it would hit you & you would die. Again. RIP.

It's the mathematicians vs. the engineers here -
so the joke goes, a room full of mathematicians and engineers are asked to line up against a wall, each across from a large stack of money. they are told that if they are able to cross the room to the money they can keep it, but the catch is that each step they take must be half the distance of the previous step. All the mathematicians step down, knowing that this is impossible and that they will never get to the money. the engineers, on the other hand, all stride across the room confidently, pick up the money, and shrug their shoulders saying, "eh, close enough!"

note: i don't mean to insult the practical knowledge of mathematicians, or the theoretical knowledge of engineers here, i just think it's a funny joke.

[messalina]

Here is something that is mind boggling. If something moves an infinite small distance, it would take an infinite amount of time for it to get from point A to point B whereever they may be. Let us say that we give it 100 years to move one inch. Since the distance it is moving is infinitly small, it would obviously not get there by then. But, it would have not moved at all. Because an infinitely small distance plus an infinitely small distance equals and infinitly small distance. If it did move a distance, then it would take a finite amount of time to get from a to b. But since it doesn't, an infinitly small distance can be thought of as 0!!

It can practically be thought of as 0, but it is still not quite 0, as long as it has distance, and thus a size. if we think of the distance as the length 1/n (for n=1,2,3,...etc.) and then let n approach infinity, then we know that this length will never actually be 0 (since 1/anything is nonzero), but 0 can be considered as a limit for the distance we travel, since the length of the distance will never be less than 0 either.

rereading this, i'm not sure if this clarifies or confuses anything further, but i'll throw it out and see if anyone has anything to say. It's been a long time since i've read anything Zeno wrote, so maybe he dealt with this after all and i have forgotten/not read it..

[Michaelyus]

Zeno, Zeno, Zeno. Motion is relative; did you know that your ink bottle there is travelling at over 100km/h? Spacetime is continuous and it cannot be grouped into class intervals. It may be possible to split the distance up, but there is a continuous flow throughout the points. Things don't happen at various points; they happen during/through it.

You cannot ever "reach" infinity.

So true. Infinity is not specified, although it is the quotient of anything and zero. This is why dividing by zero is illegal in mathematics; it is a concept including everything, anything and nothing (but not something); it cannot be split (a true [face=SPIonic]atom[/face] then).

[nefercheprure]

Bernard Bolzano was a Bohemian (i.e. from Bohemia, not Gypsy [this remark is for the sake of clarity only, for the meaning of Bohemian can be easily confounded by an English-speaking reader]) mathematician, philosopher and theologist. He addressed the problem of existence of infinite sets. First he has shown that there indeed exists an infinite ammount of something. Then from the existence of God he has shown the actual existence of a infinite set.

The proof goes as follows: (beware: matemathical induction ahead)

Theorem 1. There exists at least one valid theorem.
proof: by contradiction. Let there be no valid theorems. Then the theorem
``There is no valid theorem'' is valid. Q.E.D.

Theorem 2. There exists infinitely many valid theorems.
Proof: follows from Lemma 3.

Definition 1. Let Tk denote a theorem ``There are at least k valid theorems,'' for integral k > 1, furthermore let T1 denote ,,There is at least one valid theorem''

Lemma 3. For every integral k > 0 the theorem Tk is valid.
Proof: by induction. As it has been shown, there is at least one valid theorem, namely ``There is at least one valid theorem.''
Therefore T1 is valid.

Let n be integer and Tk be valid for all integral k > 0, n > k.
We do ask, whether Tn is valid. Suppose it is not. Then there is a valid theorem stating ``There are no more than n-1 valid theorems,'' which is absurd [face=SPIonic]o(/per a)/topon[/face].

Now some useful mathematical properties of the God. (postulated as axioms [note: this axioms are postulated for this proof only, your axiom set and/or belief concerning entity and/or entities commonly referred to by similar names may be different]
The God will be referred to as ``the God'', and ``G'' hereinafter.

Axiom 1. There is the God.
Axiom 2. The God knows everything.
Axiom 3. The God can think of everything.

Theorem 4. There actually exists an infinite set.
Proof: The God can think of all the Tk at once (from A3). He does so (A2)
((as G knows all Theorems that are true, G knows all Tk as well, therefore they are in G's mind, therefore G thinks of them))

Therefore there actually exists a set of Tk in G's mind, if G exists and has the properties A2 and A3.
Therefore (A1) there exist an infinite set, for we have shown that the number of Tk is not finite.

More information on Bolzano can be found eg. here: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Bolzano.html.