matt grime

Only if you think that your dealing with angles and can't mix units, but since this isnt' physics that has no bearing on anything.

what do you think sin(pi) is? Or am I missing some joke/pun here?

Apostol's is the correct formal definitoin of a probability space (the possible outcomes) and measure (the probability of some event) and is an axiomatization of the informal idea you're used to. The strong law of large numbers means that the two ideas are equivalent. Take the THEORETICAL ideal of rolling a perfectly fair die once. The sample space is {1,2,3,4,5,6} and we assign the measure of P(die reads x)=1/6 for all x in the set {1,2,3,4,5,6} I don't actually need a fair die to hand to talk about this. This is the point of abstract mathematics: we make abstract models of things in real life then we can forget real life entirely and think purely of the model. If you simulate the sampling of a large number of rolls then the proportion of times x occurs will approach 1/6. So we idealize and talk about theoretical random variables. There is no such thing as a perfectly fair dice for instance.

I googled the phrase recursively enumerable, and got an answer in the very first hit, try doing the same.

It's called implicit differentiation; google is your friend.

That is more than just a mere idea of leibnitz, we commonly refer to a circle of infinite radius as being a straight line, especially in the plane (ie in C), it is the limiting case as the radius tends to infinity of a circle. Eg consider a circle of radius y and centre (y,0), then letting y tend to infinity the resulting shapes tend to the x axis. I would want world peace and a million pounds, not necessarily in that order.

Do you still need an answer? For anyone else wondering you simply draw a picture and think about it for a bit. To be honest the *hardest* to understand is the first one, the one you think is the easiest, by which you simple mean the most familiar

Harris's Principles of Algebraic Geometry is on my shellf begging my indulgence again, Cox's Primes of the form x^2+ny^2 for recreational purposes, Mumford's Red Book. Generally I don't read books much these days: I tend to read papers and use books for reference purposes. However I'm trying to get into a new area of research so there are a few of books lying around. I have a few non-technical books: Tim Gowers's Very Short Introduction to Mathematics is one I often reread and should be on everyone's shelf. Plus The Man Who Loved Only Numbers, Erdos's biography. Somewhere inbetween there's the Proofs from the Book book that often glance at. Less so these days since I no longer need to find interesting, relatively content free high level mathematics to talk to undergraduates about these days (no teaching for three years!)

You could theoretically use this idea, but it is practically useless. The main thing to remember is that computers only ever use rational numbers for (non-symbolic) calcuation since we can only store a finite amount of bit data. For integer bases, there is no real difference between any choice since we can easily transform between any two systems. There would be now way to translate between a transcendatal base, like pi, and ordinary base 10, say. What is 2 in base pi? it is an infinitely long pi-imal string. That is not a good thing. Better altogether is simply to use symbolic manipulation: [math]\pi[/math] is pi.

cantor-scroeder bernstein, or whatever the correct spelling is, is an obviously true theorem, though that is different from a trivially true theorem. it states that if we partially order sets by X <= Y if X injects to Y then X <=Y and Y<=X implies X=Y, which we all agree is 'obviously' true. the proof is not hard to understand but it is very clever. however, it is easy to write down a bijection directly, as i did, as long as you remember the tricks and that you've got to stop thinking of 'nice' functions.

Evidently no continuous function will do as any continuous image of a compact set is compact, and I think that is your problem with trying to figure this out. Existence proof: consider the maps i:[0,1] to R the inclusion and j:R to [0,1] which is the composition of the inverse tan function, which maps R bijectively with (-pi/2,pi/2) followed by adding pi/2 to map it to (0,pi) bijectively, followed by dividing by pi. j is an injection from R to [0,1] thus we have injections between the two sets and hence there exists some bijection by cantor-schroeder bernstein. writing one directly. notice that we can easily get a bijection between (0,1) and R using the tan trick. so we just need one from [0,1] to (0,1) and that is the standard trick: map 0 to 1/3, 1/3 to 1/5, 1/5 to 1/7 etc ..... map 1 to 1/2, 1/2 to 1/4, 1/4 to 1/6...etc leave x fixed if x has not been mentioned this is a bijeciton from [0,1] to (0,1)

I don't think you're on the right track, but that's because you are needlessly trying to reason by analogy. if you're not a mathematician, fine, but this is a theorem about/in mathematics and only applies to mathematics. I don't like analogies: there is no such thing as a good one. Perhaps if you phrased it as "forms a class of animals" and then took as models "mammals" and "reptiles" it might look better. There are things that can be said about one but not the other, call one of these a phenomenon. Thus merely being a class of animals is not sufficient to force that phenomenon, nor its negation to be automatically satisfied.

Do you understand what an axiomatic system is? It is simply a set of (not inconsistent) rules. A model for the axioms is a mathematical gadget in which the rules are true. There are many such ways of realizing the rules. If the list of rules to satisfy is finite, and if it requires you to have a copy of the natural numbers in any model of it, then godel states that there are statements that are true in the model but cannot be deduced from the rules alone, and that there will be some other model of the rules in which this statement is false. Here's a psuedo example. Let R be the axioms for Groups, a model of R is just a group (note this doens't force us to define the natural numbres as a subset of G, but that is not important). Suppose that the model is the integers, then the statement "G is abelian" is true in that model. Yet in the model "invertible 2x2 matrices" the statement is false. Thus the statement "G is abelian" is independent of the axioms.

it is? but that would require infinity to be a real number, which it isn't. (that is to say this result is only true if both gradients are defined, and infinity is not acceptable in this context)

here let me indent and clarify: why have you written this? this is what we want to deduce, but you shouldn't write it here as if it were true [math]2(n+1)+1=(2n+1)+2\le{2^n+2}\le{2^{n+1}}[/math], since for [math]2^n+2\le{2^{n+1}}[/math] we have [math]n\ge{1}[/math] Here I don't particularly have the energy to follow what you're doing, and more importantly you don't even say what you're doing or why.

I did not say "assume A(n) is true" is wrong. i said that writing the statement of A(n+1) as the first line in the proof as if it were true is wrong. it is bad mathematics to do that. you start at A(n) and deduce A(n+1), that's how proofs like this go. i didn't check through what your strings of symbols without explanations in the first post might or might not mean. if you don't explain the steps then you can not expect anyone to mark them for correctness.

That doesn't appear to make sesne. Firstly, induction goes: given P(n) then P(n+1). Thus you must start with assuming n^2 < 2^n and deducing (n+1)^2<2^(n+1), however your first line of the proof is to write out the thing you wish to prove as if it were true, which is not how one does a proof. I also see no explanation of how you deduce A(n+1) from A(n) just a series of inequalities at the end of which you decide n>1. It is sufficient to prove that (n+1)^2 < 2n^2, for then (n+1)^2<2n^2<2.2^n =2^(n+1) by induction on n. That is it suffices to show n^2-2n-1 = (n-1)^2 -2>0, which is true for all n>2 and we are done (the initial case is obviously true)

I think you'll find it was the bit in post 5 where you don't appear to know what "from first principles" means that is causing the head scratching by me (and I'd guess Tom). You see you appear to have initally thought it was ok to use the formula, but now you're saying you agree it isn't acceptable.

How do you know that your formula holds? You need to prove it when asked to prove it.... Ie, when asked to prove that P is true you cannot simply say since P is true, P is true, which is exactly what you just did.

it's a volume of revolution problem. google for many worked examples.

No, I won't show you step by step, since it is elementary manipulation of variables, let me just remind me of what you need to do you need to somehow work out: ((x+e)^1/3 - x^1/3)/e It may help you to bear in mind the identity s^3-t^3 = (s-t)(s^2+st+t^2) now, what happens if you let s^3=x+e and t^3=x? and put it all together and let e tend to zero... (if done right you should just be able to let e=0 and get the answer.)

i disagree, i don't see it as neat, or clever. but then we all have different rationales about what we think constitutes a nice proof: one from the book.

probability????

look at the definitions and have some faith in you ability (since the answer is correct)