h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

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sign me up please, im really interested in maths

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I'm not entirely sure what to put in this box, I can't find the instructions, unless of course there are none, which would explain why I wasn't finding them...off down the rabbit hole... Pure maths, not statistics, perish the thought. 0+infinity=0, cyclic. Geometry is good fun.

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Sorry for the delay.


GTB

Not done any serious maths since 21th December 2001 (the day I retired), I didn't do much on that day either! It was a Triple_Friday. A Single_Friday is just an ordinary friday when you're due to come back to work the next monday. A Double_Friday is a friday when you're going away on leave and not back to work on the next monday. A Triple_Friday is when you're leaving work and you're never coming back!. Oh there's also a Zeroal-Friday thats when its friday you're at work and you'll have to come in over the weekend.

Hi! I can't believe it took me this long to discover this page, but now I have, and here I am! I hereby dedicate my own membership to all members who have never dedicated anything to themselves.

Random favorite math fact: I just learned in Set Theory class how to mathematically prove that there are (infinitely many) different sizes of infinity.

Other favorite math fact: If you take every term of the Fibonacci sequence and divide it by the previous term, you get a new sequence that converges on the Golden Ratio.

Welcome MuseSusan.

Ah set theory and its infinities... Just wait til you try to work out the ordinal number of the set of all ordinals!

GTBacchus

People may be interested in the entry I did on this subject:

http://www.bbc.co.uk/dna/h2g2/brunel/A593552

That's great! I'm just starting to learn about set theory, and I find it very exciting. Though I wish I had seen your entry a few weeks ago; then I would have gotten the last problem on my final completely right with time to spare.

Is anyone else on this thread, like me, rather unhappy with the power set version of the size of infinities. I rather like the notions that there are more positive integers than positive even integers, and more positive even integers than squares of integers, and so on. These infinities have different sizes to me, but are all the same from the set theoretic point of view (I think).

The idea that log(n)/n goes to zero as n increases is rather useful, and more useful than the idea that if one infinity is a, the next one up is 2^a, especially when the set theory concludes some infinities are just too big.

Any comments?

"I rather like the notions that there are more positive integers than positive even integers, and more positive even integers than squares of integers, and so on. These infinities have different sizes to me, but are all the same from the set theoretic point of view (I think)."

Well, not quite, you have the one set (even integers say) strictly contained within the other (all integers), and so there are more integers than even integers. The trouble is when you start asking questions about whether there are more or less elements in sets which overlap but neither is contained within the other. I think it's more useful to think about "partially ordered sets" (posets, amusingly). A partial ordering is one where you can sometimes say that A is bigger than B, or B is bigger than A, but sometimes you just can't compare them. (Compare with a total ordering, where you can always say AB or A=B.) Another approach is measure theory. Unfortunately there's no obvious measure on the set of integers, but it does work quite nicely on the reals.

One of the big ways of determining the sizes of infinities is the concept of countable versus uncountable sets. A countable set is one whose elements can be put into a sequence such that you are absolutely guaranteed of covering all of them. For example, the set N of natural numbers is countable. 1, 2, 3, 4, 5, 6, … The set of even integers is also countable: 2, 4, 6, 8, 10, … Now, there is a theorem that states that if you can find a bijection (a function that is both one-to-one and onto; I believe these terms are defined in the link above) between two different sets, then those sets must have the same number of elements. Thus, if you can find a bijection between N and E (even integers) then they must be the same size. And in fact, we CAN, because we can assign to each even number the natural number that denotes its position in the sequence. Thus, the set of even integers IS the same size as the set of natural numbers, despite the fact that only half the natural numbers are even.

In fact, any countable set can be shown to be the same size as the set of natural numbers in this way, by making a sequence out of it and then matching up each term in the sequence with the natural number that tells its position.

The set of integers {…-2, -1, 0, 1, 2, …} seems at first to be uncountable, because a sequence has to start somewhere, and where do you start it to ensure that you get all the negatives AND all the positives? But the answer is, we rearrange the numbers. If we start with zero and go 0, 1, -1, 2, -2, 3, -3, … then we are sure to get all the integers, and so it is countable. And thus, the set of integers is the same size as the set of natural numbers, even though there are twice as many integers as natural numbers.

Then there's the set of rational numbers (numbers that can be written as the ratio of two integers). Is it possible to put that set into a sequence? Well, if we start with, say, 0, and then take the next rational number, what do we get? Well, if I said 1/2, I'd be wrong, because there are lots of rational numbers between 0 and 1/2. What about 1/4? Nope, there are still rationals between 0 and 1/4. In fact, between any two rational numbers, there is another rational number, the average. So it seems impossible to produce a sequence that is guaranteed to cover all the rationals. However, we CAN produce an "infinite matrix", which has infinite rows and infinite columns, that covers all the rationals. Consider the matrix in which, for each element, its numerator equals the row number, and its denominator equals the column number.

1 1/2 1/3 1/4 1/5 1/6 …
2 2/2 2/3 2/4 2/5 2/6 …
3 3/2 3/3 3/4 3/5 3/6 …
…

Then, we start with 1 and then list all the elements in each diagonal: 1, 1/2, 2, 1/3, 2/2, 3, 1/4, 2/3, 3/2, 4, … It's a pretty messy sequence to have to look at, but it covers all the rational numbers and proves that the set of rationals is countable.


So it seems like maybe we could come up with other fancy tricks to prove that any infinite set is countable, and that therefore all infinite sets are the same size. NOT TRUE. Take a look at Cantor's Diagonalization Process (http://www.bbc.co.uk/h2g2/guide/A479180), which proves that any function mapping the set of natural numbers to the set of real numbers is NOT onto, and therefore is bigger than N, and uncountable. From here we can show that the set of irrationals is uncountable, and we can also define the set of algebraic numbers (any number that can be the root of a polynomial with integer coefficients; this now includes square roots, cubic roots, and so on) and show that it is countable, and lots of other fun things.

Ouch. Sorry, folks. This is what happens when we let Susan talk about math late at night: she starts writing essays and trying to prove the continuum hypothesis and bombarding poor, defenseless H2G2 researchers with set theory.

Susan, you can even show that the set of "computable" numbers is countable. That is, you can say that the set of numbers for which you can (in principle) write a computer program to calculate are countable. This includes all the algebraic numbers, as well as a few transcendentals as well, like pi and e.

I've only just taken a class called "Intro to Logic and Set Theory", so I've only seen the tip of the iceberg, but it's definitely a subject I would like to learn more about.

Yes, I was familiar with most of the points you have made. But if you insist that the number of positive even integers is the same as the number of positive integers, by your one-one and onto mapping, just how do you define the probability of a fair coin leading heads (even)?

Or do you, in that case, return to my idea that only half the integers are even? How do you choose the point of view? (This is but one of several arguments against the set based definitions of sizes of infinity, which lead to the doubt expressed earlier.)

Of even greater practical importance, how do you make the calculus work? It seems to be absolutely neccesary to have infinities of minutely variable size, rather than just A or 2^A (where A represents aleph nought).

Old Hairy, there is a measure of sets which tells you that there are half as many evens as odds, and extends to other sets. For a subset of the natural numbers, call it A, you define

density(A)=lim as n tends to infinity of [ |A intersect {1,...,n}|/n ]

That is, for each n calculate the percentage of the integers from 1 to n which are in A, and take the limit as n tends to infinity. This quite often gives you a sensible answer, but there are sets for which the limit doesn't converge (e.g. it could oscillate between 0 and 1 depending on how large n is).

Regarding infinities of minutely variable size and calculus, you should look into the "hyper-real" numbers and "nonstandard analysis" (Robinson), or if you want even more of a challenge the "surreal" numbers (Conway).

Ooh, sounds interesting. I'll have to look that up. Unfortunately, now that classes have started again, I have limited time to hang out and study set theory (or rather, I only have time to study set theory as it pertains to abstract algebra, which is the class I'm taking this term.)