comparing infinities

[Lord Antares]

I've given you a lot more info than you asked for. The tl;dr is that

 

Why would you think that? You've given me exactly what I was looking for and earned an upvote in the process

 

I had the same ideas as this George person but wanted to check what educated people think.

 

Oops, terminology strikes again. I meant to say ''aren't then all intervals of any real numbers the same cardinality'', instead of sets, which you answer to be true.

However, I'm glad I mixed this up because it brings up an even more interesting question:

 

By which logic can infinite sets of real numbers have different cardinality, as opposed to intervals of reals?

They both deal with the same issue and that is different-sized infinities.

 

This wikipedia article states that the cardinality of real numbers is higher than the cardinality of integers. The cardinality of both is infinite.

The cardinality of both my A and B intervals is also infinite, but apparently equal.

 

How can this be? The only difference between the two is that the set of all reals vs. all integers have the same range as opposed to my intervals where the B interval has double the range, but why should that matter; there is no difference since the increment in all reals is infinitely smaller than the increment of all integers. If anything, it should point to the opposite.

[studiot]

 

lord antares

By which logic can infinite sets of real numbers have different cardinality, as opposed to intervals of reals?

They both deal with the same issue and that is different-sized infinities.

 

This is why I started with the smallest infinity, that of the whole numbers or integers.

 

An infinite set, chosen from the reals, (ie an infinite subset of the reals) can be chosen to only include the whole numbers.

 

This has already be shown to be of smaller size than the set of reals itself.

[Lord Antares]

Yes but how? Or rather, why doesn't the same apply to intervals? I assume you are referring to this:

 

So in considering just the set of positive whole numbers we have found an infinity.

 

But we haven't included any of the fractional numbers in between, let alone those that can't be expressed as fractions.

 

So we are forced to the conclusion that more transfinite numbers are needed to place infinite sets into one-to-one correspondence.

 

Why aren't we forced to the same conclusion for these A and B intervals?
Also, how do sets compare with intervals in this regard?

[studiot]

Yes but how? Or rather, why doesn't the same apply to intervals? I assume you are referring to this:

 

 

Why aren't we forced to the same conclusion for these A and B intervals?

Also, how do sets compare with intervals in this regard?

 

First terminology again, sorry.

An 'interval' is a technical term for a set of real numbers which contains every point between the end points.

We don't get to pick and choose.

For other technical reasons not of interest here we should only apply 'interval' to the real numbers.

Another technical point of interest is that intervals come in two types.

Closed intervals include their end points, open intervals do not. This may become useful and relevant if this discussion develops.

 

So

 

The set of real numbers

[math]\left\{ {x:1 \le x \le 11,x \in R} \right\}[/math]

Is an closed interval between 1 and 11

and the set

[math]\left\{ {x:1 < x < 11,x \in R} \right\}[/math]

Is a open interval between 1 and 11.

 

However the sets

[math]\left\{ {x:1 \le x \le 11,x \in N} \right\}[/math]

and

[math]\left\{ {x:1 \le x \le 11,x \in Q} \right\}[/math]

are not intervals since they refer to integers (denoted N) and to rational numbers denoted Q).

Both these sets are subsets of the real number interval.

The set of integers is finite , the set of rationals is infinite, and has a lower cardinality than the interval set in R.

This is just one way to approach the meaning of numbers, however.

We realise that we need different types of number to satisfy different equations.

For instance consider the equation 2x = 5

There is no integer that satisfies this equation.

In fact there is an infinity of such equations, one for each odd number.

[Sriman Dutta]

Now I see that infinitesimals cannot be used as an independent number, but are instead used for making some definitions simple-like this:

 

It [math]f(x)[/math] is a function of [math]x[/math] then -

[math] \frac{d}{x} f(x) = \lim_{h -> 0 }\frac{f(x+h)-f(x)}{h} [/math]

where h is an infinitesimal.

Edited December 27, 2016 by Sriman Dutta

[Lord Antares]

/cut

 

Thank you. This clarifies a few things, primarily of terminology but not of the logical issue I want to know about.

 

For a moment I thought this was just a misunderstanding and that it's very simple UNTIL I read this:

 

 

the set of rationals is infinite, and has a lower cardinality than the interval set in R.

 

Can you explain why it has a lower cardinality than the interval in R? This is what I'm trying to find out.

Both are infinite. If you're saying it has a lower cardinality because not every element of R can be expressed by any element of Q, that does not make any sense to me.

 

A quick google says ''While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary (in particular infinite) sets.''

 

So that might, in fact, be what you mean to say. It's a matter of definition of cardinality then, isn't it?

 

Thinking about the things in this link https://en.wikipedia.org/wiki/Cardinality makes my brain hurt. It contains so many paradoxes when it comes to infinite sets and intervals and things which I wouldn't say are inherently logical but in a way also are.

Edited December 27, 2016 by Lord Antares

[studiot]

 

Thank you. This clarifies a few things, primarily of terminology but not of the logical issue I want to know about.

 

For a moment I thought this was just a misunderstanding and that it's very simple UNTIL I read this:

 

 

 

Can you explain why it has a lower cardinality than the interval in R? This is what I'm trying to find out.

Both are infinite. If you're saying it has a lower cardinality because not every element of R can be expressed by any element of Q, that does not make any sense to me.

 

A quick google says ''While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary (in particular infinite) sets.''

 

So that might, in fact, be what you mean to say. It's a matter of definition of cardinality then, isn't it?

 

Thinking about the things in this link https://en.wikipedia.org/wiki/Cardinality makes my brain hurt. It contains so many paradoxes when it comes to infinite sets and intervals and things which I wouldn't say are inherently logical but in a way also are.

 

 

Once again some preliminary work is needed.

 

It is important to realise that transfinite numbers do not obey the same rules of arithmetic as finite numbers.

 

In particular if you remove (subtract) a finite number or a lesser infinity of numbers from an infinite set you still have the original cardinality.

 

So if you remove an infinite subset of R made up of all the rational numbers you are still left with larger infinity of irrational ones.

 

It is far easier, though, to follow the route of the pioneers and construct the number system, starting with the simple counting numbers and adding new types of number as they become necessary and then exploring as fully as possible the properties of these new types of numbers.

That way you prove the rationals have the same cardinality as the integers.

 

http://math.stackexchange.com/questions/12167/the-set-of-rationals-has-the-same-cardinality-as-the-set-of-integers.

Edited December 27, 2016 by studiot

[Lord Antares]

This is what I've been saying all along.

 

If we go back to considering my intervals A and B, A contains all real numbers in between 3 and 4. B contains all real numbers in between 3 and 5. If you removed the range of A from B, you will still be left with an infinity. How can then the cardinality of these two intervals be the same if what you said is true? Why would it be different in your example if both use the same exact logic?

THIS is what doesn't make sense to me.

 

Interval B is the higher infinity (comparable to R in your example) and A is the lesser infinity (comparable to Q in your example).

 

Also, if you change my A and B to be rational numbers (i.e. a set instead of an interval, right?), nothing should change logically, since the same kind of infinity is present, no?

Edited December 27, 2016 by Lord Antares

[wtf]

If we go back to considering my intervals A and B, A contains all real numbers in between 3 and 4. B contains all real numbers in between 3 and 5. If you removed the range of A from B, you will still be left with an infinity. How can then the cardinality of these two intervals be the same if what you said is true?

If you start with the counting numbers 1, 2, 3, 4, ... and you remove all the even numbers, how many numbers are left?

[Lord Antares]

∞

 

But that supports what I said, no?

 

EDIT: That is one abomination of an infinity sign.

Edited December 27, 2016 by Lord Antares

[wtf]

][/b]

 

But that supports what I said, no?

No. Each of the sets of even numbers and odd numbers are bijectable to the natural numbers. Right?

[studiot]

If you start with the counting numbers 1, 2, 3, 4, ... and you remove all the even numbers, how many numbers are left?

 

 

 

∞

 

But that supports what I said, no?

 

EDIT: That is one abomination of an infinity sign.

 

Just to make wtf's question a little more difficult

 

How many are left if you only leave every 3rd number instead of every second?

 

What about only leaving every 4th number?

 

What happens if we only leave every nth number?

 

And what happens if we let n tend to infinity?

[Lord Antares]

No. Each of the sets of even numbers and odd numbers are bijectable to the natural numbers. Right?

 

Oooooh, I think I finally understand.

For even numbers, you can pair each new even number with the new whole number, i.e. (2,4,6,8,) paired with (1,2,3,4,) and go into infinity. But you can't do this for rational and real numbers because which number would you pair it with, since there is always a smaller one than the one you consider pairing. Is this the point?

 

This simple fact somehow eluded me.

 

@studiot - yes, the answer is always the same. You can't do this for my A and B intervals. That expands upon wtf's example in a helpful way because I didn't see what he was getting at.

[wtf]

Oooooh, I think I finally understand.

For even numbers, you can pair each new even number with the new whole number, i.e. (2,4,6,8,) paired with (1,2,3,4,) and go into infinity. But you can't do this for rational and real numbers because which number would you pair it with, since there is always a smaller one than the one you consider pairing. Is this the point

Counterintuitively, you CAN pair the natural numbers with the rationals. But you can't pair the naturals to the reals.

 

http://www.homeschoolmath.net/teaching/rational-numbers-countable.php

 

https://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Edited December 27, 2016 by wtf

[Lord Antares]

Correct again.

Disregarding that error, was that the point you two were trying to make?

[wtf]

Correct again.

Disregarding that error, was that the point you two were trying to make?

Sorry, which two? I speak only for myself here. And sometimes not even that

[Lord Antares]

I believe studiot was making the same point.

I would think it wouldn't be hard to guess as you are the only who have replied in the last several posts

[wtf]

I believe studiot was making the same point.

I would think it wouldn't be hard to guess as you are the only who have replied in the last several posts

In fact that's what I was trying to get at.

 

Without having read everything studiot wrote, I can't say whether I agree with it or not. I speak only for myself. That was my only point.

Edited December 28, 2016 by wtf