Thread: Set theory

This topic brings to mind something similar I've though about. Consider how many points there are on a number line between 1 and 2. You can always find a point between two points, so it follows that there are infinitely many points between 1 and 2. But then how many between 1 and 3? Also infinite... but is it twice as infinite as between 1 and 2? lol

You can always find a point between two points, so it follows that there are infinitely many points between 1 and 2.

Oh yeah, that sounds valid. Saying that you need to assume something called the completeness axiom, to make sure you get every point.

But then how many between 1 and 3? Also infinite... but is it twice as infinite as between 1 and 2? lol

That sounds like the continum hypothesis. Basically, it says that there is no highest infinity. However, nobody knows if this is true or false and you can't prove it true or false.

Basically, the problem is unprovable. Which, is very strange.

Originally Posted by wendylove

That sounds like the continum hypothesis. Basically, it says that there is no highest infinity. However, nobody knows if this is true or false and you can't prove it true or false.

Basically, the problem is unprovable. Which, is very strange.

Even the universe itself doesn't want to deal with the problem, so everything is quantized (countable infinity). Should it be any surprise, then, that something with literally no physical analog seems strange?

Originally Posted by Forsaken

This topic brings to mind something similar I've though about. Consider how many points there are on a number line between 1 and 2. You can always find a point between two points, so it follows that there are infinitely many points between 1 and 2. But then how many between 1 and 3? Also infinite... but is it twice as infinite as between 1 and 2? lol

Are you sure? When I was I kid, I was playing around with a calculator, and I divided 1 by 2. It of course was 0.5. But I wondered how far it could go on. You know. 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625, 0.001953125, 0.000976563, I went on and on and on, 0.000000954, 0.0000000477, 0.000000238, 0.000000119, 0.000000060, 0.000000030, 0.000000015, 0.000000007, 0.000000004, 0.000000002, 0.000000001, 0.000000000, 0.000000000

And no matter how much I divided by 2, my calculator said it was still zero. So, I guess there is a limit to how much you can divide the distance between two points. Right? After a while it becomes nothing, according to math (thx to my calculator).

Originally Posted by Neruo

Are you sure? When I was I kid, I was playing around with a calculator, and I divided 1 by 2. It of course was 0.5. But I wondered how far it could go on. You know. 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, 0.00390625, 0.001953125, 0.000976563, I went on and on and on, 0.000000954, 0.0000000477, 0.000000238, 0.000000119, 0.000000060, 0.000000030, 0.000000015, 0.000000007, 0.000000004, 0.000000002, 0.000000001, 0.000000000, 0.000000000

And no matter how much I divided by 2, my calculator said it was still zero. So, I guess there is a limit to how much you can divide the distance between two points. Right? After a while it becomes nothing, according to math (thx to my calculator).

Positive. There are numbers smaller than your calculator can show. There is no limit in math to how many zeros you can have after the decimal before your first non zero, hence, no smallest number. You simply ran out of decimal places on your calculator, you didn't find a limit to the number of decimal places in math.

Originally Posted by Forsaken

Positive. There are numbers smaller than your calculator can show. There is no limit in math to how many zeros you can have after the decimal before your first non zero, hence, no smallest number. You simply ran out of decimal places on your calculator, you didn't find a limit to the number of decimal places in math.

I think he was joking.

Originally Posted by drewmandan

I think he was joking.

Never know anymore....

However, you would think that this is all you need to express all the numbers forgetting complex numbers. However, it is not.

Disregarding the whole die thing, what about irrationals? :\

Most roots, and various constants like pi and e, can't be expressed as a simple fraction of two whole numbers.

Originally Posted by Xei

Disregarding the whole die thing, what about irrationals? :\

Most roots, and various constants like pi and e, can't be expressed as a simple fraction of two whole numbers.

That's what he said...

Where exactly?

As far as I can see the only number that was talked about was an infinite string of random digits. That seems like a very dodgy definition of a number to me, and quite strange considering there are many other properly defined numbers which could have been used instead.

But the digits aren't actually random with irrationals, they're a consequence of a logical numerical process (irrationals can actually be defined using infinite strings of fractions with a precise pattern).

And in any case... I don't see what the point of this thread is. What's the question here?

Originally Posted by Xei

Where exactly?

Originally Posted by wendylove

Now, you can count Q, however you cannot count the real numbers or R.

Originally Posted by Xei

And in any case... I don't see what the point of this thread is. What's the question here?

There isn't one.

Right then... :\

That quote doesn't seem relevant to what I was saying btw. Never mind tho.

Originally Posted by Forsaken

Positive. There are numbers smaller than your calculator can show. There is no limit in math to how many zeros you can have after the decimal before your first non zero, hence, no smallest number. You simply ran out of decimal places on your calculator, you didn't find a limit to the number of decimal places in math.

I was indeed joking

Disregarding the whole die thing, what about irrationals? :\

Most roots, and various constants like pi and e, can't be expressed as a simple fraction of two whole numbers.

Thats what I said.

The point of the die example is that there is no repeating pattern and so you can't express it as a fraction.

But the digits aren't actually random with irrationals, they're a consequence of a logical numerical process (irrationals can actually be defined using infinite strings of fractions with a precise pattern).

Yeah, thats basic. However, I was trying to point out that there are more irrationals then rational numbers and what this says in general.

As far as I can see the only number that was talked about was an infinite string of random digits. That seems like a very dodgy definition of a number to me, and quite strange considering there are many other
properly defined numbers which could have been used instead.

Actually its a good description of irrational numbers. The dice example is given in the book Yet another introduction to analysis to illustrate that there are more irrational numbers then rational numbers.

There isn't one.

It was mean't to be a thread about how the film Pi is wrong.

Note Q is the set of rational numbers. You can count Q, but you can't count R, which is just all the numbers on the number line. This shows that the size of the irrational numbers against the rational numbers.

Since this topic seems to be about strings of random numbers:

Lets say you have a random number generator, pumping out 1's and 2's. So, lets say the first numbers two are 112221. The chance that this pattern will repeat is 0.5^6, 0.015625. Right? Okay, maybe a few other numbers could show up that would count as patterned. (for instance 112221222112221222112221). Anyhow. As long as there is a form of repetition I guess.

What would happen if you kept the random 10-number generator running infinitely, after the first 100 numbers, would you ever get a repeating pattern? If you have 123456789, the chance of those numbers repeating right behind it are 0.1^9, right? With one 100 numbers, that chance would be 0.1^100, 10^-100, that's a pretty small number. Would, given infinite time, a repetition even occur (to repeat, like this: start---numbers---same_numbers---'end')? I mean, theoretically given enough time everything with a chance greater than 0 happens, but in a way this looks like Zeno's paradox. (since when you finally manage to repeat those 100 numbers, you probablly need to repeat 10000000)

Maybe it's wrong, also like Zeno's paradox.

Any 1337-math people that could say what MATH would say of this, if anything?

[edit]just realized this is quite similar to the above question, sorry about that[/edit]

An infinite random string tickles my imagination... I'm thinking that because it is infinite, EVERY possible pattern will come up in it, and not only that, but those patterns will come up an infinite number of times. No matter how long a piece of it you select and call a "pattern", it eventually repeats again and again an infinite number of times. To my way of thinking, it is impossible to go on to infinity without repetition.