Thread: infinite sets

Hi, I wonder if any of you can help me understand something that is considered 'basic', but I just can't get my head around it.

I was talking with a Canadian friend, who is a mathematician, and at some point we got to talk about chemical molecules (I am a chemist).
To cut a long story short, he was puzzled by me saying that the number of conceivable molecules is finite, so he made up a simplified example where all the molecules you can imagine are made of n repeating units M, joined together by a chemical bond. So your generic molecule would be . This is what we chemists would call a polymer.

Here's the point that I didn't get. He told me that you can have an infinite set of such molecules without the need for n to go to infinity. (Of course you can't have a molecule made of infinite units, because each unit has finite size, so this molecule would be infinite in size, which is absurd.)

So I kept telling him, how is it possible that n is finite but the sum of all the molecules in the set of is infinite? And he kept telling me that this is a basic concept of set theory, and he couldn't see why it was so difficult for me to understand.

Then we didn't have time to talk about it any further, and I'm still left with the doubt.
I read something about sets, and if I got it right, a set as infinite if you can establish a correlation with the set of natural numbers. But there you go, back to my original objection: if we take n as the natural number, then if n is finite, the sum of all molecules from 1 to n is a finite number, not infinite. Only if you allow n to go to infinity you have a complete map of your set to the natural numbers, don't you?

Does anyone know where I'm going wrong with my reasoning?
(I would only add that the number of conceivable chemical molecules is indeed finite - OK, it's a very big number, but it is finite; so this discussion is more about maths than the original problem itself).

He's saying this: Instead of talking about molecules, let's talk about something we probably both know about--chain links. Let M_n be a chain with n links. Then there are an infinite number of possible chains, because for any chain M_n with n links, we can always theoretically create a new chain M_{n+1} by adding one more link.

Now, in reality, the number of genuine chains we can form is finite because there's only a finite amount of matter in the universe. But theoretically, there is no upper limit.

Originally Posted by lavoisier

Hi, I wonder if any of you can help me understand something that is considered 'basic', but I just can't get my head around it.

I was talking with a Canadian friend, who is a mathematician, and at some point we got to talk about chemical molecules (I am a chemist).
To cut a long story short, he was puzzled by me saying that the number of conceivable molecules is finite, so he made up a simplified example where all the molecules you can imagine are made of n repeating units M, joined together by a chemical bond. So your generic molecule would be . This is what we chemists would call a polymer.

Here's the point that I didn't get. He told me that you can have an infinite set of such molecules without the need for n to go to infinity. (Of course you can't have a molecule made of infinite units, because each unit has finite size, so this molecule would be infinite in size, which is absurd.)

So I kept telling him, how is it possible that n is finite but the sum of all the molecules in the set of is infinite? And he kept telling me that this is a basic concept of set theory, and he couldn't see why it was so difficult for me to understand.

Then we didn't have time to talk about it any further, and I'm still left with the doubt.
I read something about sets, and if I got it right, a set as infinite if you can establish a correlation with the set of natural numbers. But there you go, back to my original objection: if we take n as the natural number, then if n is finite, the sum of all molecules from 1 to n is a finite number, not infinite. Only if you allow n to go to infinity you have a complete map of your set to the natural numbers, don't you?

Does anyone know where I'm going wrong with my reasoning?
(I would only add that the number of conceivable chemical molecules is indeed finite - OK, it's a very big number, but it is finite; so this discussion is more about maths than the original problem itself).

Perhaps what salsaonline has told you will answer your question. What he said is correct.

But I think you are having your problem with a different question.

I think this analogy might help. The the natural numbers are the numbers 1,2,3,4,5,6,...... is an infinite set. But each of the natural numbers in that infinite set is a finite number.

Now consider your polymer molecules with a repeating unit M. In principle you can make a molecule with any finite number n of those repeating units. So you can label each such molecute with a natural number n, say . And then you can list the molecules that you build in this way
It is now easy to see that we have a one-to-one correspondence between the molecules constructed in this way and the natural numbers . From that it is also easy to see that the set of all such molecular structures is infinite.

In fact you can prove it is infinite, in the same way that you can prove that is infinite. Suppose that there only finitely many such molecular structures. Then there is one with the largest number of repeating groups, call it . But then you could always add one more group to construct , resulting in a contradiction, since supposedly was the largest such molecule. Since the logic is correct, the original assumption that there are only finitely many such structures must be wrong. Hence there are an infinite number of such structures.

Thanks DrRocket,
this is exactly the same argument my Canadian friend was using.

Then I see the real problem here is that I have a concept of infinite which is not the same as the mathematician's one.

I appreciate the argument that for any molecule of size k you can always build a molecule of size k+1, but that's what I'd call 'illimited' or 'unbounded', i.e. there is no finite upper 'limit'.
On the other hand, infinite as I understand it is an abstraction of something that doesn't exist in the physical universe. E.g. no matter how large an integer k, k is always smaller than infinite.

That's why I don't really see how a collection of k objects can be infinite in size, unless k goes to infinite. But that's because I think about it in terms of real objects, not abstract mathematical entities.

As far as my original problem is concerned, the number of conceivable molecules is finite because of practical limitations, certainly not in purely theoretical terms.

So thanks again, salsaonline too, this was a very interesting discussion.

Originally Posted by lavoisier

Thanks DrRocket,
this is exactly the same argument my Canadian friend was using.

Then I see the real problem here is that I have a concept of infinite which is not the same as the mathematician's one.

,,,

That's why I don't really see how a collection of k objects can be infinite in size, unless k goes to infinite. But that's because I think about it in terms of real objects, not abstract mathematical entities.

A set of k objects (k some natural number) cannot be infinite in size. In fact a set that has k objects for some natural number k is, by definition, finite.

The point of what I was trying to convey is that, in theory, the set of molecules formed from your repeating groupd is NOT a set of k objects. In fact we showed that it cannot be, because whatever k you miight propose as the descriptor, there is a set of such molecules that is larger, We constructed a 1-1 correspondence between a set of such molecules and the natural numbers.

k doesn't "go to infinity". It doesn't "go" anywhere. What we showed is that no finite k gets the job done, so the number of molecules in the set that we constructed must be infinite.

Hi DrRocket,
I confirm that this concept of infinity escapes me. I have to accept it as a 'rule', and apply it to be able to do calculus etc., but I don't really grasp it in concrete terms. It's a personal limitation that I have no problem to admit. It didn't stop me from passing all my maths and physics exams, luckily.

I think of molecules as real objects with their own physical properties and behaviour, and that means I'm sure there must be some k (or a distribution around a maximum value of k) for which you can't actually add any more repeating units. And this disrupts this mechanism where you decide that for any k there can be a k+1. This can be true in some abstract mathematical space, but it is not true in the chemical space.

I'm sure my Canadian friend would still be puzzled by this, but abstraction sometimes takes you too far from reality, and often you can't really use this sort of theoretical approach to deal with real systems.[/i]

Originally Posted by lavoisier

Hi DrRocket,
I confirm that this concept of infinity escapes me. I have to accept it as a 'rule', and apply it to be able to do calculus etc., but I don't really grasp it in concrete terms. It's a personal limitation that I have no problem to admit. It didn't stop me from passing all my maths and physics exams, luckily.

I think of molecules as real objects with their own physical properties and behaviour, and that means I'm sure there must be some k (or a distribution around a maximum value of k) for which you can't actually add any more repeating units. And this disrupts this mechanism where you decide that for any k there can be a k+1. This can be true in some abstract mathematical space, but it is not true in the chemical space.

I'm sure my Canadian friend would still be puzzled by this, but abstraction sometimes takes you too far from reality, and often you can't really use this sort of theoretical approach to deal with real systems.[/i]

Of course what I have presented is a mathematical abstraction, but mathematical abstractions are useful.

In reality, so far as we know there are only a finite number of elementary particles in the universe -- a stependous number but a finite one. Given that, it follows that you cannot actually construct a molecule of arbitrarily long chain length. I am sure that other physical constraints kick in long before the finiteness of matter in the universe takes effect, but in any case I am quite sure that there are limits to the length of a polymer chain.

But that doesn't change the mathematics either, and it is useful to understand both the mathematics and the physics/chemistry. Understanding a theory includes understanding its limits. But don't start to think that you can't use a solid theoretical approach to deal with real systems. You in fact need a strong theoretical approach to understand them deeply. Just include a knowledge of the limits of a theory in your bag of tricks.

No, by all means, I do appreciate the importance of mathematics, and I was not trying to say abstraction isn't useful.

Actually, I quite like applying mathematics to chemistry. E.g. once I tried to solve a problem (chromatographic gradient elution) that involves partial differential equations, and I got a partial result; I got stuck when I obtained an equation that apparently can't be solved analytically (it's a mixed algebraic-transcendent expression). Maybe one day I might post the problem here and see if someone can come up with a solution.

This is to say that, in the finite :wink: 'set' of organic chemists, I am quite a mathematics-oriented one. Most of my colleagues would dismiss this whole discussion from the very beginning as 'a lot of faff', to put it politely.
Then of course you have physical chemists who are, dare I say, half mathematicians. But this is a different debate.

Concerning the original problem, you're right about the physical limitations of the system. In fact if you're interested, there is some interesting literature on how the actual chemical space was defined as being made of approximately 10^60 compounds:

http://en.wikipedia.org/wiki/Chemical_space

Well, don't confuse yourself into thinking that the subject of this thread is of any importance to mathematicians either. I mean, the reason we're talking about it is because you brought it up. So if the subject is fluff, the fluff is originating from your corner.

Originally Posted by lavoisier

This is to say that, in the finite :wink: 'set' of organic chemists, I am quite a mathematics-oriented one. Most of my colleagues would dismiss this whole discussion from the very beginning as 'a lot of faff', to put it politely.
Then of course you have physical chemists who are, dare I say, half mathematicians. But this is a different debate.

Don't confuse physical chemists with mathematicians. Physical chemists are an entirely different breed of cat. A useful breed, but a different breed.

A friend with whom I shared an apartment in grad school was a chemist at the time, an organic physical chemist no less. His description of the mathematics of typical organic chemists was that they "take one data point and extrapolate to infinity". He was also best man at my wedding.

Believe it or not, decades later, he is in graduate school and will receive a Ph.D. in mathematics, algebraic topology, in August. If he is not the oldest U.S. Ph.D. graduate in math, at least this year, I would be very surprised. But it does show that even chemists can learn advanced mathematics if they want to , although some are rather late to the party.

Originally Posted by salsaonline

Well, don't confuse yourself into thinking that the subject of this thread is of any importance to mathematicians either. I mean, the reason we're talking about it is because you brought it up. So if the subject is fluff, the fluff is originating from your corner.

Wow! This haughty and resentful reply was totally unnecessary.
First because I didn't say this was faff to me, but to some people I know. Otherwise, why would I have asked the question in the first place?. Second because I stated pretty clearly that I was grateful for your input and for taking the time to answer me. So no need to put me back in my place; I was already there.

Besides, if the subject of my post was such a boring and unimportant thing, it's a bit contradictory that you took the time to reply in the first place. But then perhaps I shouldn't try and analyze contradictions using my impaired, moth-like logic.

And don't worry, I know I am intellectually inferior to any of you because I can't do advanced maths, but I'm OK with it, and I don't feel any need to humiliate other people.
I know what I do is useful because there are people who are paying me to do it, and possibly patients who will ultimately benefit from my work.
That's good enough for me.

I answered the question out of a desire to help clear up a confusion. That doesn't mean I find the topic inherently interesting.

Imagine someone coming up to you on the street asking "how do I get to union station?" By answering their question, does that somehow imply that you're deeply fascinated by the subject? Clearly not, and normally that would go without saying.

However, in your post, you said something to the effect that other chemists would find the topic of our conversation rather airy. The implication was that mathematicians find this stuff important by comparison. And I was saying that, no, they don't. But of course any mathematician would happily take up the conversation--not because it is of mathematical interest, but to help clarify confusion regarding the concept of infinity.

A lot of mathematicians relish the chance to educate people about their subject. But it's a bit annoying having someone ask a question in earnest only to point out that most people they know would think the whole conversation was poofle.

OK, I see your point now.

I thought I had made pretty clear that I did value very highly your and DrRocket's input on my problem. My comment about other chemists' over-pragmatic view on maths (based on my direct experience) was in reply to DrRocket, who seemed to think I was disparaging the importance of abstraction and maths in general. I tried to point out that I am among those who like maths, and that's why this problem fascinated me, but I wouldn't find my same interest in it, and most probably not enough competence to clarify it, among fellow chemists.

Unfortunately, I have to conclude that my command of the English language isn't good enough to convey what I mean without involuntarily annoying/offending entire categories of people.

So please do not waste any more of your time for me. The original problem is pretty much solved now. I caught my train. Thanks.

Originally Posted by lavoisier

OK, I see your point now.

I thought I had made pretty clear that I did value very highly your and DrRocket's input on my problem. My comment about other chemists' over-pragmatic view on maths (based on my direct experience) was in reply to DrRocket, who seemed to think I was disparaging the importance of abstraction and maths in general. I tried to point out that I am among those who like maths, and that's why this problem fascinated me, but I wouldn't find my same interest in it, and most probably not enough competence to clarify it, among fellow chemists.

Unfortunately, I have to conclude that my command of the English language isn't good enough to convey what I mean without involuntarily annoying/offending entire categories of people.

So please do not waste any more of your time for me. The original problem is pretty much solved now. I caught my train. Thanks.

My impression is that your English is just fine. I am under the impression that you are British -- and as Patton observed, the U.S. and Britain are two peoples separated by a common language. I think you are quite clear.

I am not the least bit offended, and never was (at least by you). Your view of mathematics seems pretty reasonable and in keeping with the view of most chemists of my acquaintence (I know a lot of chemists). No problem. When I need to know about chemistry, I talk to a chemist.

I am not quite sure where this particular discussion went south, but I see no need for it. If you have any further questions regarding mathematics I hope you will feel free to bring them here.