infinity, and the confusion it is causing me

Any interval and the real line have a bijection between them, yes.

An infintesimal is not a real number, they are not in the real interval [0,1]. No matter what set you are mapping into the real interval [0,1] you will not somehow 'create' any more elements in [0,1].

 

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Not create, just illustrate that they're already there. It's a logical argument. If it's logically sound, there's nothing wrong with it, right?

 

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They aren't there, hence your argument is logically flawed. There is no real number, x, in the interval (0,1] with the property that an infintessimal has, that for every positive integer n, x<1/n. This follows from the so-called "Axiom of Archimedes" of the reals: for every real number there is a larger integer.

I'll try to sort out where your thinking goes wrong, when you said "Then a are mapped onto a number in A, nonzero, yet infinitely small," this sentence doesn't parse properly. What exactly did you mean? What do you hope maps onto an "infinitely small" number? You realize that this bijection maps into [0,1], which is a real interval, and there are no "infinitely small" numbers in it.

You should also realize that the existance of a bijection from one set to another says nothing about any other properties these sets may have (besides cardinality), like some kind of natural order, or some kind of inherent group operation on them. Indeed, there is no order preserving bijection from [0,1] to R. If there were, then the real numbers would be bounded from above and below!

 

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Ok. Is it possible to demonstrate? I'm quite new at this.

I meant that for that number a in R, there is a mapping to a number in A. Which number? I thought it must be less than (assuming order) any number in R. But as I'm uttering these words in realize that it becomes circular, if it were to be true.

I was hoping for good counterarguments. Thanks for setting me straight =) Well it was an interresting idea. Certainly it helps learning the properties of numbers.

 

Sure. Will you believe the Axiom of Archimedes?

"If x is any real number, then there exists an integer n where n>x"

If so, let x be in (0,1]. Then 1/x is a (positive) real number. Apply the Axiom above to get an integer n (also positive) such that n>1/x and therefore x>1/n. So x does not satisfy the nice property of infintessimals and probably won't fit any intuitive idea you have of "infinitely small".

Now it's named the Axiom of Archimedes, but it's not really an axiom (it can be deduced from the usual axioms for real numbers) and I don't think it's attributed to Archimedes (it's Eudoxus I think). It's usually something that doesn't take much convincing, but I can provide a proof of it if you wish (are you familiar with the least upper bound axiom?).

 

I will believe the Axiom of Archimedes! =)

I'd like to see the proof for educational purposes (not convincing). Least upper bound axiom. Hmm I'm guessing it has to do with converging sequences.

 

A couple definiteioins first:

Upper Bound: L is an upper bound of a set S if for all x in S we have L>x

Least Upper Bound: L is a least upper bound of a set S if for all upper bounds L' of S we have L'>=L. Equivalently, L is a least upper bound of S if for any L>L', then L' is not an upper bound of S.


Least Upper Bound Axiom:Let S be any nonempty set of real numbers with an upper bound, then S has a least upper bound.

This is a bonafide Axiom of the reals, sometimes called the Completeness Axiom. You can rephrase it in terms of sequences if you like.


Now for the Axiom of Archimedes.

Let x be a real number. If 0>x, n=0 will work.

If 0>=x, let S=set of all integers less than or equal to x. S is non-empty (-1 is in S at least), and has an upper bound, x, so it has a least upper bound, call it L. Since L is the least upper bound of S, anything smaller is not an upper bound. So there will be an integer m in S where m>L-0.5. Then m+1 is an integer and m+1>L+0.5, so m+1 is not in S (as it's larger than an upper bound for S). Hence let n=m+1 and we have our integer n where n>x, by the definition of n not being in S. (Note this proof is from my memory of Royden).

 

Hockeywings,

I would like to point out that in the case of the set of integers Z and even integers Z+; you say that if you add anything to them they should not be equal.

1. The only thing you can add to the set of even integers is even integers. By adding an even number to the set Z+ you are also adding an integer to the set of integers Z. So their totals don't change, no matter how you think you can change the size of the sets.
NOTE: I have italicised those words to indicate that sets of infinite size don't have regular total sizes as we understand and you cannot add an element to an infinite set. (Then it wouldn't be infinite would it?).

 

well oxy, then that makes me more confused about the greater infinity, because any decimal you add, just add another integer, eh?

i wasnt looking at this problem as a function base, just as a comparative one, i didnt see how having even numbers matching up with integers could be equal and then adding the odd numbers to all the even numbers could make that still equal with the same set as the first, seems to me like it shouldnt work out, i understand what people are saying and i guess it is just gonna have to be the fact that it seems to be contradictory to me but thats just me i guess

 

try a slightly simpler example. How about we compare the cardinality of A={0,1,2,3,4,...} and B={1,2,3,4,...}? A is the set of non-negative integers and B is the set of positive integers. We have a bijection from A to B defined by f(x)=x+1, so the sets have the same cardinality. This might seem more reasonable than the even integers to the integers?

Remember that cardinality says nothing about any structure the sets may have. We don't care that A has more elements less than 10 than B does (for example). All that matters is the existance of this bijection from A to B. You could think of this bijection as a relabelling of the elements of A using the labels from B as determined by this map.

I guess you could ask if this bijection business is a reasonable way to measure the "size" of a set. For finite sets it's exactly what you've used your entire life and if you go to blindly use this on infinite sets, you do get some interesting results that may run counter to your intuition, like an infinite set may have the same cardinality as a proper subset. But how far do you trust your intuition when dealing with the infinite when this intuition was built by working with the finite?

 

Maybe a little motivation for the definition will help:

Assume for a minute that there are infinitely many (visible) stars in the universe, and we want to name them. We choose the simplest naming system possible to do so: we just number them, so we pick one star and call it 1, a second and call it 2, and so on. Since we have infinitely many, this goes on forever. So how many stars are there? Well, infinitely many! But can we get more precise? We notice that we've assigned one and exactly one positive integer to each star, so it would probably make sense to say there are the same number of stars as there are positive integers.

Now let's try a different system. With the same universe, we instead choose to name the stars with just the even positive integers. So we call the first one 2, the second 4, and so on. How many stars are there? By the same reasoning, the number of stars should be equal to the number of even positive integers (and note how loosely I'm using the term "number" here, we're actually talking about cardinalities, which aren't real numbers). But this is the same universe, so there are definitely the same number of stars under either naming system!

We conclude that there must be the same number of even positive integers as there are positive integers. That's why we define cardinality in the way we do.

 

so how would you find the cardinality of the set of all functions mapping the reals into rationals?