What topic in mathematics deals with the functions of infinite variables (if any at all)? Does this have any significance? Does an infinite-dimensional geometry exist?

There is a course here called Functional Analysis which deals with infinite dimensional spaces. In particular, spaces of infinite sequences and spaces of functions. For instance, you can have a function that takes another function as it's variable, and this is infinite dimensional. For instance, the "functional" I[y]=Integral from 0 to 1 of y(x) dx is a function from the space of real continuous functions on the interval [0,1] to the real numbers. Therefore, if y(x)=x² , then I[y]=1/3.

As far as an infinite dimensional geometry, lots of physicists would have you believe we are in fact living in one! I think that infinite dimensional Hilbert spaces are used in String theory and Quantum theory, although someone more knowledgeable will probably correct me here ;).

The cardinality of an infinite dimensional space will depend on the space itself, it could even be countable. For instance, the space of infinite sequences on the field Z mod 5 (integers modulo 5) is countable* , but infinite dimensional. The space of continuous functions of the interval [0,1] is definitely NOT countable.

[* See the post at the end of this thread. - The Editor]

Perhaps Pooya means:

Suppose you have a point which is free to move in a normal Euclidean space, with Cartesian co-ordinates (x,y,z,t,p,q,...) where the no. of dimensions is infinite but countable. In this case, the points will have the same cardinality as the irrational numbers.

Imagine the co-ordinates of the point is:

(0.4635251..., 0.2452323..., 0.2625353...,......)

We need to store the infinity of irrational numbers in a single irrational number.

To do write down the first digit of the first number. Then second digit of the first number. Then the first digit of the second number.

Then the 3rd of the 1st no.
Then the 2nd of the 2nd no.
Then the 1st of the 3rd no.

Then the 4th of the 1st no.
Then the 3rd of the 2nd no.
Then the 2nd of the 3rd no.
Then the 1st of the 4th no.

Then the 5th of the 1st no.
Then the 4th of the 2nd no.
Then the 3rd of the 3rd no.
Then the 2nd of the 4th no.
Then the 1st of the 5th no.

I hope you can see what's happening now. (It's easier to see if you draw a grid.) We can store the infinity of irrational numbers in a single irrational number, so the cardinality of a point in a Euclidean infinite space, is the infinity of the irrationals.

Yours,

Michael

Thanks guys.
I really enjoy using this site!

Actually my question is this: Is there a set which has a cardinality larger than c or aleph-1?

There are certainly sets of cardinality greater than c (we'll suppose c = aleph-1 for simplicity). One such set is the set of all functions from R to R (R being the set of real numbers), which has cardinality aleph-2, I think - anyway it certainly has cardinality strictly greater than c.

In fact, there are sets with arbitrarily large cardinality. If X is a set with cardinality A, then P(X), the power set of X (i.e. the set of all subsets of X) has cardinality strictly greater than A, so by repeatedly constructing X_i = P(X_i-1 ) you can make a sequence of sets with strictly increasing cardinalities.

This is relatively easy to prove, but I can't remember how to do so off the top of my head and it's quite late now, but if you're interested then I look up the proof for you when I have a minute (or maybe someone else who knows it better than me will).

James.

Thanks James.

You have certainly thought about this:

Let i®¥.

What is the cardinality of X_¥?

Isn't this true: X_¥=P(X_¥)? What is the cardinality of the set of all sets?

By the way, in which year in university do you do cardinality/foundation of mathematics/Godel's theorem etc?

I studied most of this stuff before coming to university, but didn't do any of the proofs until getting to university. The proof I gave you above about P(X) and X is straight from the notes for a course in my first term in my first year at university. I haven't yet done any courses on the foundations of maths or Godel's theorem, but I'll probably do them next year, I'm in my second year at the moment. However, I've just ordered a copy of Godel's proof from amazon.co.uk, but I don't know if I'll be able to follow it yet.

Just a quick followup about 'the set of all sets'. One way that Mathematicians resolve the paradoxes which arise from having a 'set of all sets' is by introducing things called 'classes'. A class is like a set, but more general. So while there isn't a 'set of all sets', there is a 'class of all sets'. However, you can't talk about the cardinality of a class.

This is only one way of going about things though - there are other approaches as well (or at least, I've heard of one other approach, but I don't know anything about it).

James.

Here is a different way in which functions of an infinite number of variables come up:

The r^th elementary symmetric function in variables x₁ ,...,x_n are given by:

s₁ = x₁ + ... + x_n (the sum of the elements)
s₂ = x₁ x₂ + x₁ x₃ + ... + x_n-1 x_n (the sum of products of distinct pairs of elements)
s₃ = the sum of products of distinct triples of elements.
etc...

Now a polynomial in the variables x₁ ,...,x_n is called symmetric if it is unchanged by permuting the variables.

Eg. Suppose our variables are x,y,z then the following are symmetric:

x+y+z; x² + y² + z² ; xyz+x+y+z; (x-1)(y-1)(z-1) ...

The following are not symmetric:

x; x+y; x+y+z² ; xyz+x+y; (x-1)(y-2)(z-1).

Now the big theorem is that any symmetric polynomial can be expressed in terms of the elementary ones:

So in the above case we have

s₁ = x+y+z
s₂ = xy+yz+zx
s₃ = xyz

x+y+z = s₁
x² + y² + z² = s₁ ² - 2s₂
xyz+x+y+z = s₃ +s₁
(x-1)(y-1)(z-1) = s₃ - s₂ + s₁ - 1

Now if I had used 4 variables instead of 3 and written down the symmetric function which correspond to the above ones:

z+y+z+t; x² +y² +z² +t² ; xyzt+x+y+z+t; (x-1)(y-1)(z-1)(t-1)

then (amazingly) all the formulae in terms of the symmetric functions would have been identical.

So all the formulae in terms of the elementary symmetric functions do not depend on how many variables I'm using (for example the sum of the squares of n variables is ALWAYS s₁ ² - 2s₂ ) and so these can legitimately be considered as formulae in an infinitely large numebr of variables.

This is a simple example of a mathematical method called 'taking inverse limits' of doing problems in an arbitrarily large number of variables in one go.

This is all useful because theorems proved in the case of the elementary symmetric functions then give lots of theorems depending on how many variables we restrict them to. So it gives a unified method of doing many (obviously) different problems in the same framework.

AlexB.

Michael Dore posted a message on June 5.
In one of his first sentences he said :
...where the number of dimensions is infinite but countable.

Isn't it contradictory to say that if something is infinite it can be countable?

In case someone else than Michael can give me the answer, please do it.

The word "countable" in maths is used to describe a set that is only as big as the positive integers. The reason is that the positive integers are sometimes called the counting numbers, or something like that.

A few clarifications:

1. The set of infinite sequences whose elements are integers mod 5 (Dan's second reply) is not countable. (It's the same "size" as the set of real numbers, or the set of subsets of the natural numbers.)
   
   
2. Here's something countable that you could reasonably describe as being "infinite-dimensional". Take Dan's set of sequences, and throw away all the sequences that have infinitely many non-0 elements. What's left is countable.
   
   
3. The set of functions from R to R has cardinality 2^c . That needn't be the same thing as aleph-2, even if it happens that c = aleph-1.
   
   
4. Perhaps I should explain this aleph business.
   
   aleph-0 is the size of the set of integers (or the set of natural numbers, or the set of rational numbers; they all have "the same number of elements").
   
   aleph-{n+1} is the smallest "size" that's larger than aleph-n.
   
   If you have any set of size aleph-n, then its power set (i.e., the set of all its subsets) has size *at least* aleph-{n+1}, but it might be larger than that. (Or perhaps it's better to say: aleph-{n+1} might be smaller than that.) Let's write "2^{aleph-n}" for the size of the power set.
   
   2^aleph0 is the size of the set of real numbers, or the set of all sets of integers. It's sometimes called "c", which is short for "continuum" because the real numbers form a "continuum".
   
   The first person to think really clearly about all this business of infinite sets was Georg Cantor. He conjectured that c = aleph-1; this is generally called the "continuum hypothesis", or "CH". It's natural to generalize this and say that 2^aleph-n = aleph-{n+1} for all n; this is (more or less) what's called the "generalized continuum hypothesis" or "GCH".
   
   In 1940, Kurt Goedel (as in the "Incompleteness Theorem") proved that CH and GCH *might* be true: more precisely, he proved that the usual axioms for set theory are not capable of disproving either CH or GCH.
   
   In 1963, Paul Cohen proved that CH and GCH *might* be false: more precisely, he proved that the uxual axioms for set theory are not capable of proving either CH or GCH.
   
   So, the axioms of set theory tell us nothing at all about whether CH or GCH is true. Gosh. (Phenomena like this lead some mathematicians to say that it's a mistake to ask questions like "Is GCH true?"; that's not a meaningful question, and we should content ourselves with asking "Does GCH follow from such-and-such axioms?".)
   
   
5. There is actually more than one way of doing set theory. Some of them are *genuinely* different (i.e., it's not just a matter of using a different set of axioms but getting all the same theorems). One fairly important one, called "NF", *does* have a "set of all sets". So it's not really fair to say that "the set of all sets isn't a meaningful thing to talk about". In the usual variety of set theory, there is no set of all sets, though.