this doesn't make sense to me, smart people? (infinity)

[givememyleg]

the part about an infinite set of numbers between 0-1 being larger than an infinite set of integers. an infinite set of integers can just keep going to fill the void he claimed is made... right?

[oskar]

http://www.youtube.com/watch?NR=1&v=...ture=endscreen

this explains it better.

There are a lot of fun things you can do with infinity. I like this one: http://en.wikipedia.org/wiki/Hilbert...he_Grand_Hotel

[kiwiMark]

"The first thing I want to recommend is if you haven't again, listen to the previous segment"

guess again, buddy!

[kiwiMark]

holy fuck 3 minutes in and he still hasn't said anything. think I'm gonna need breakfast before tackling this one

[kiwiMark]

ummm this seems dumb I don't think his X exists

[kiwiMark]

Okay so I'm dumb and he's smart and probably right and all the rest of it but I learn best by assuming that I am king of the world and arguing. As such:

He seems to be saying:

let's take allllllll the real numbers between 0 and 1. (0.4, 0.7235762355 etc.)
let's take allllllll the natural numbers (1, 2, 3 etc.)

So far these two sets are equal. If you like, imagine that you're counting the first set, this automatically generates the second. He's saying if we were to list these out to infinity (also seems troubling that he keeps saying that like it's a logical possibility) then we get all of them.

Then he says, now imagine a real number X that isn't equal to any of the real numbers "already listed".

So now we're rocking it like this:

1. We've already "listed" "all" the natural numbers.
2. Before this whole X thing, we have a "list" of real numbers that is exactly as long as the "list" of "all" the natural numbers.
3. So once we add X to that second list, the second list becomes longer than the first.

This is all well and good but he fails to prove that X can exist.

To my mind, any possible value that X could be, we've already covered since we kept listing real numbers out to infinity.

If you want to say that there's a magical real number X that's not equal to any of the "other" real numbers, why not a magical natural number Y, that's not equal to any of the other natural numbers and restores the 1:1 equilibrium?

[kiwiMark]

So for some reason I watched oskar's video before carl's. If you're just watching Carl's (prolly recommendable), the bit I have an issue with, is the claim that we can always create some magical new decimal number - just saying it doesn't make it true.

EIGHTEENTH POST IN A ROW POWER

[givememyleg]

another dumb guy here, i think that's good analysis.

in a way it sounds like he's saying one set of infinity is larger than the other? but that sounds impossible, (infinity) = (infinity * 100) or whatever you want. infinity is even the same as infinity * infinity.

[spoonitnow]

another dumb guy here, i think that's good analysis.

in a way it sounds like he's saying one set of infinity is larger than the other? but that sounds impossible, (infinity) = (infinity * 100) or whatever you want. infinity is even the same as infinity * infinity.

Some infinities are larger than others. They're considered to be different numbers. What's fun is that you can prove that some infinities are different from each other without being able to prove which one is larger.

The easiest place to start to wrap your head around it is getting an idea between what's countably infinite and uncountably infinite. The natural numbers {1, 2, 3....} are countably infinite, and all of the reals between 0 and 1 are uncountably infinite. You can prove that these two sets have different sizes with a little bit of logic. Without getting too deep into the math:

Step 1: Assume that they have the same size.
Step 2: Do some shit.
Step 3: You get a massive contradiction that proves the assumption in step 1 is false.

I'm pretty sure the details of step 2 are covered in first-level real analysis classes, though you might be able to find an easy-to-digest version online somewhere.

[a500lbgorilla]

Describe infinity.

You'll never run out of ways to describe it because the concept is a symptom of inherent human reasoning.

It's a symptom of mathematical reasoning through induction. If you say 0 exists, and you say 1 exists, and you say 1 is the immediate successor of 0, then you're allowed to say there is an immediate successor to the immediate successor of 0.

It's inherent to the concept of recursion which underlies human thought. Any time you construct a thought that places an object in a setting, you're nesting one thought inside another thought.

Or the recursion of placing the idea of a successor to 0 inside the idea of a successor to 0.

0=0
1=1=0+1
2=1+1=(0+1)+(0+1) = (0+(0+1))+(0+(0+1))
3=2+1=1+1+1=...
.
.
.

It's a symptom of human thought that when you try to formalize it or describe it, infinities keep popping up. Because in our minds, we have no trouble seeing that because 0 and 1 exist and their relation is defined, 100 exists, 1000 exists, 1000000000000000000000000...0000 exists.

And that's why people who try to describe reality don't like it when infinities pop up because it's a sign that our thinking is getting in the way.

[daviddem]

This stuff is part of an area of mathematics called "set theory" which, despite being a real headache, is pretty interesting and also has philosophical implications etc.

You can't really apply the notion of "size" or the adjectives "larger" or "smaller" in the conventional sense of the term when talking about infinite sets. That is why mathematicians have come up with a concept called "cardinality" to generalize the notion of size to infinite sets.

Basically, the cardinality of two sets is the same if there exists a bijective function mapping the sets onto each other.

What the guys in the videos really are saying is that the cardinality of R (the set of all real numbers) is strictly greater than the cardinality of N (the set of all natural numbers). Additionally:
- the cardinality of R is the next bigger cardinality after the cardinality of N. This means that you cannot find any set the cardinality of which is between that of N and that of R. (interestingly, this one cannot be proven or disproven)
- the cardinality of N x N (all pairs of natural numbers) is the same as the cardinality of N. Same for NxNxN or NxNxNxN etc.
- the cardinality of Z (set of all integer numbers) is the same as the cardinality of N.
- the cardinality of Q (set of all rational numbers) is the same as the cardinality of N.
- the cardinality of R can be shown to be equal to 2^N0, where N0 represents the cardinality of N.
- the cardinality of any interval [a,b[ or [a,b] of real numbers where b > a is the same as the cardinality of R
- the cardinality of RxR, RxRxR, etc is the same as the cardinality of R
- what set has a cardinality greater than the cardinality of R then? It can be shown for example that the set of all subsets of real numbers has a cardinality greater than that of R.

So in the end, when you say that an infinite set "is bigger" than another, what you really mean is that its cardinality is greater than that of the other set.

Fun fact: intuitively, you might think that the set of all real numbers in the interval [1, 3] is "bigger" than the the set of all real numbers in the interval [1, 2] because [1, 3] includes [1, 2] and you can easily find extra numbers in [1, 3] that are not in [1, 2]. However this is wrong in the mathematical sense: both intervals have the same cardinality: that of R.

This can get really complicated for non mathematicians... here is a simple article about it:
http://www.sciencenews.org/view/gene...y_Big_Infinity

And some wikipedia and other references which require more math:
http://en.wikipedia.org/wiki/Cardinality
http://en.wikipedia.org/wiki/Cardina..._the_continuum
http://en.wikipedia.org/wiki/Cantor%...gonal_argument
http://www.ma.utexas.edu/users/mwill...ardinality.pdf

[oskar]

@rilla
That's a good point. Our brain tends to interpret it as a sequence rather than a condition. So it probably confuses us for the same reason that it confuses us that 1/3 is 0.3333... and 3/3 is 0.999... and 0.999... = 1 because it's not like it's forever trying to get to 1, it's actually just 1.

The proof that there are infinitely more real than natural numbers is very neatly explained in the video I posted, you just have to sit through it :P

[MadMojoMonkey]

If you say 0 exists, and you say 1 exists, and you say 1 is the immediate successor of 0, then you're allowed to say there is an immediate successor to the immediate successor of 0.

I smell a degree in mathematics.

That sentence is, at its core, the foundation of all math.

I mean, historically, it was the postulate of 1, and the successor was 2 (It was 100 years later that someone "discovered" the number 0), but the successor function was the first mathematical idea.

So it probably confuses us for the same reason that it confuses us that 1/3 is 0.3333... and 3/3 is 0.999... and 0.999... = 1 because it's not like it's forever trying to get to 1, it's actually just 1.

Actually, all of the rational numbers are dually stated. for example, 0.299999 (repeating 9) = 0.3 ; .012319999 (repeating 9) = .01232.

Every number has (at least) 2 names.

*note, this is not an artifact of the decimal system, in any base, this phenomenon exists.

[kiwiMark]

The proof that there are infinitely more real than natural numbers is very neatly explained in the video I posted, you just have to sit through it :P

nope :P

[Lukie]

.999... = 1

true or false.

[kiwiMark]

Gotta go with false. Doesn't everything equal everything otherwise?

[MadMojoMonkey]

0.999 (repeating 9) = 1

PROOF:
1/3 = 0.333 (repeating 3)
Multiply both sides of the equation by 3.
3/3 = 0.999 (repeating 9)
by definition, 3/3 = 1, therefore
1 = 0.999 (repeating 9)

QED

just for fun:
1/12 = 0.08333 (repeating 3)
multiply by 3
3/12 = 0.24999 (repeating 9)
3/12 = 1/4 = 0.25
so then
0.25 = 0.24999 (repeating 9)

All rational numbers have 2 "names" in this manner.

[kiwiMark]

1/3 = 0.333

isn't this not true, though? 0.333 recurring isn't actually = 1/3, just the closest decimal approximation

[oskar]

If you think of it in base 12: 1/3 is 0.4,
3/3 is 1

0.333... is exactly 1/3

in that case it is an artefact of 10 base, but you get repeating numbers in all bases that converge to a number, and they are that number. But thanks for proving my point, kiwi
Your brain wants to push that last 0.000000000001 out to infinity, but it doesn't exist. It isn't infinitely chasing that one decimal, it is actually just 1/3

[kiwiMark]

I dun get it. Just because there's an accurate base 12 representation of 1/3, the inaccurate base 10 representation becomes accurate?

[oskar]

It's not inaccurate!
0.333... in base 10 is the same as 0.4 in base 12 is the same as 0.1 in base 3 and so on.

[kiwiMark]

Seems like it is though, on account of when you multiply it by 3 you still get a recurring number rather than a nice round 1.

I mean I totally get that for practical purposes there's no difference because the difference is infinitely small, but to me there still seems to be a leap to then say in absolute terms that there's no difference, if that makes sense?

[kiwiMark]

Especially given that in math I was always (like when talking about asymptotes for example) required to say that stuff tends towards zero rather than stuff is zero.

[oskar]

Sorry, it doesn't. The only remaining artefact is in your brain. Like rilla said, the problem is how your brain works, and not in the number. You get a nice round 1, and whenever you are adding repeating numbers that add up to a whole number, you just put in the whole number because that's what it is.

[kiwiMark]

Numbers suck.

[Lukie]

Seems like it is though, on account of when you multiply it by 3 you still get a recurring number rather than a nice round 1.

I mean I totally get that for practical purposes there's no difference because the difference is infinitely small, but to me there still seems to be a leap to then say in absolute terms that there's no difference, if that makes sense?

One way you can look at it is to try to find a number between the two that are listed.

so for example, does .99999999999999 = 1? No. .9999999999999999999999 is between those 2 numbers. When you get to .9 (repeating) i.e. .999..., there is nothing between that number and 1. Obviously that's not the most technical explanation.

[kiwiMark]

That makes the most sense so far...my immediate thought was that that just shows that 1(.000...) is the next number in the sequence not that they're the same, but I guess the point is if the number line is continuous rather than discrete that doesn't make sense. And in fact 1 isn't any more a nice round number than 0.99999... because it's actually 1.00000... it's just that my brain likes it when stuff ends in zero.

Thanks guys!

edit: I'm less happy with the above than I was a second ago. My head hurts.

[oskar]

They should make it an exercise in math class to make kids comfortable with that kind of stuff to just complicate everything into infinity, like:

2=1+1
2 = (0.999...)+((0.666)+(0.333...))
2 = (0.999...)+((0.666)+(0.333...)) / ((0.333...)+(0.666...)
(sqrt2)^2 = (0.999...)+((0.666)+(0.333...)) / ((0.333...)+(0.666...)
sqrt[[sqrt2]^2] = sqrt[(0.999...)+((0.666)+(0.333...)) / ((0.333...)+(0.666...)]
.
.
.

[kiwiMark]

I think zero is causing my headache. Why should .1 ≠ .11 and .9 ≠ .99 but .0 = .00? Just seems like jesus messed up.

[oskar]

I think you need a break
You should sign up to the duodezimal society, and have them parade you around like the elephant man and have them go: Look what base 10 did to this man! Only the patients genitals remain intact and unaffected!

[kiwiMark]

They should make it an exercise in math class to make kids comfortable with that kind of stuff to just complicate everything into infinity, like:

To be honest I don't think it's missing content in math class but rather exactly because of the content that was in math class. As I say, in linear algebra it was always drilled into us that a curve and its asymptote never touch, because while 0.99999 gets closer to 1 with the more 9s you add on, it never actually reaches it. Now we're turning around and saying in other branches of maths this is handled differently.

You're right on the break front for sure

[Lukie]

That makes the most sense so far...my immediate thought was that that just shows that 1(.000...) is the next number in the sequence not that they're the same, but I guess the point is if the number line is continuous rather than discrete that doesn't make sense. And in fact 1 isn't any more a nice round number than 0.99999... because it's actually 1.00000... it's just that my brain likes it when stuff ends in zero.

Thanks guys!

edit: I'm less happy with the above than I was a second ago. My head hurts.

lol

Mathematics isn't always intuitive, and most often people ask questions like the one above or various problems/paradoxes specifically because the actual proof is counter-intuitive.

Sometimes I am in the same boat. I am extremely good at general math and specifically arithmetic and to a slightly lesser extent algebra and geometry, and I know some calculus. But I really don't have extensive formal education in mathematics and can't really explain high level concepts and that sort of thing. It is practical but I am not a maths professor.

[seven-deuce]

0.999 (repeating 9) = 1

PROOF:
1/3 = 0.333 (repeating 3)
Multiply both sides of the equation by 3.
3/3 = 0.999 (repeating 9)
by definition, 3/3 = 1, therefore
1 = 0.999 (repeating 9)

QED

just for fun:
1/12 = 0.08333 (repeating 3)
multiply by 3
3/12 = 0.24999 (repeating 9)
3/12 = 1/4 = 0.25
so then
0.25 = 0.24999 (repeating 9)

All rational numbers have 2 "names" in this manner.

Ok, i understand that 3/3=1 and 3 x 1/3 = 0.99999... so therefore 0.99999... must also equal 1.

Does this just mean any number that ends in a recurring 9 is the same as the next integer up like 5.999999... = 6?

Also - See in your 1/12 example the way 1/12 = 0.08333 there is a recurring 3 and you can multiply to end up with a recurring 9, what if you had a number that ended in recurring 2 or 4 like 0.082222 or 0.084444 you cant multiply them to get a recurring 9?

What does that mean then?

[MadMojoMonkey]

The thing is that there is no asymptote in the naming issue. There is no "limit as x goes to infinity" here. There's just a dual name.

Anyone who can do "long" division can show that 1/3 = 0.333 (repeating 3).
My examples above don't involve any functions, just simple grade-school math.

It is funny and counter-intuitive that rational numbers should have 2 names, but it doesn't really change anything. It's just unexpected, and cool.

[spoonitnow]

There is no "limit as x goes to infinity" here.

This is what confuses people on this particular topic 0.999... times out of 1 in my experience.

[daviddem]

Ok, i understand that 3/3=1 and 3 x 1/3 = 0.99999... so therefore 0.99999... must also equal 1.

Does this just mean any number that ends in a recurring 9 is the same as the next integer up like 5.999999... = 6?
Yes, but not just the integers. Like also 0.17999999999... = 0.18
Also - See in your 1/12 example the way 1/12 = 0.08333 there is a recurring 3 and you can multiply to end up with a recurring 9, what if you had a number that ended in recurring 2 or 4 like 0.082222 or 0.084444 you cant multiply them to get a recurring 9?
Yes, you can. Multiply 0.084444... by 2.25 and you get 0.1899999...=0.19.
What does that mean then?
Nothing. It's just two different ways of writing the same number or designating the same quantity. You could also write it in binary or in octal or hexadecimal: it would still be the same number.

*

[seven-deuce]

It is funny and counter-intuitive that rational numbers should have 2 names, but it doesn't really change anything. It's just unexpected, and cool.

What do you mean two names?

Like 0.999... has the name 0.999... and also 1?

It feels like my whole world just exploded.

[MadMojoMonkey]

Does this just mean any number that ends in a recurring 9 is the same as the next integer up like 5.999999... = 6?

Yes. any number that terminates, like 0.25 = 0.24999... has this form.

Also - See in your 1/12 example the way 1/12 = 0.08333 there is a recurring 3 and you can multiply to end up with a recurring 9, what if you had a number that ended in recurring 2 or 4 like 0.082222 or 0.084444 you cant multiply them to get a recurring 9?

I can multiply them by 9/2 and 9/4, respectively.

What does that mean then?

That, in math, we start with remarkably simple principles (like the successor function) and by application of logic, we find vast worlds of unexpected, interesting things.

Or it means nothing. Meaning is in the mind of the beholder.

So numbers have 2 names? Numbers are an invention of a mind, a symbolized representation of abstract ideas like single-ness and identity, and counting. Numbers are fabrications that, really, only describe the world to us, they do not exist in the world, as such.

[seven-deuce]

@ Daviddem

So 0.99.. = 1 but does it also equal 0.99..?

If 0.99.. = 1 then is '0.99..' not really a number in it's own right just another way of saying '1'.

0.99.. doesn't exist? It's just another way of saying 1?

Does this mean 0.111... is the same as saying 0?

[MadMojoMonkey]

Why would 0.999... be not really a number? Why can't 1 be not really a number, as it's just a short-hard way to write 0.999...? They are just 2 equivalent ways of describing one quantity.

I think what you're trying to ask is if 0.000...01 is equal to zero, and the answer is yes.

[daviddem]

Yes, two different ways of writing the same number.

Neither 1 or 0.99... exist, no more than the letter a. They're just symbols.

We could have chosen the symbol z for the sound 'a' instead. Wouldn't change anything. If you look at different handwritings, you will see several different shapes for the same letter. Same story.

It's like a caveman when numbers did not exist. One day he saw a mammoth. And he called him "mammoth". The the next day he saw two mammoths and he wondered how he was gonna call that. He decided that he was going to call that 2 mammoths, and the sighting of the day before was going to be 1 mammoth to differentiate. Then the 10th day he was sick of inventing new symbols so he got a brainwave and decided to start reusing the same symbols in sequence to designate even more mammoths.

And no wtf 0.11111... is not 0.

0.11111... is greater than 0.1 so it's not even close.

0.0000{insert infinite number of zeroes}1 is 0. You can write it 0.0...01

[Savy]

I think what you're trying to ask is if 0.000...01 is equal to zero, and the answer is yes.

erm, that would never be written. In fact I'm pretty sure you can't do it, because it'd still be >0.

0.0000 to infinity then you add 0.0....01 and therefore it's no longer 0.

[MadMojoMonkey]

erm, that would never be written. In fact I'm pretty sure you can't do it, because it'd still be >0.

0.0000 to infinity then you add 0.0....01 and therefore it's no longer 0.

1 = 0.999...

1 - 0.999... = 0

[seven-deuce]

Why would 0.999... be not really a number? Why can't 1 be not really a number, as it's just a short-hard way to write 0.999...? They are just 2 equivalent ways of describing one quantity.

I think what you're trying to ask is if 0.000...01 is equal to zero, and the answer is yes.

0.000...01 = 0

0.999.. = 1

1.000...01 = 1

0.57999... = 0.58

I wouldn't say i understand this, how can they be equivalent quantites if there's 0.0000...01 of a difference? Surely they are different quantities no matter how minute the difference?

[daviddem]

erm, that would never be written. In fact I'm pretty sure you can't do it, because it'd still be >0.

0.0000 to infinity then you add 0.0....01 and therefore it's no longer 0.

yes, because zero plus zero is zero.

As long as you agree that the symbol "0..." represents an infinite number of zeroes in succession then 0.0...01 equals 0. And also 0.0...99 is equal to zero and also 0.0...099... or 0.0...5793

[daviddem]

0.000...01 = 0

0.999.. = 1

1.000...01 = 1

0.57999... = 0.58

I wouldn't say i understand this, how can they be equivalent quantites if there's 0.0000...01 of a difference? Surely they are different quantities no matter how minute the difference?

since 0.0000...1 equals zero, then there is zero difference

The difference is not "minute". The difference is infinitely small, which is another way of saying that there is no difference at all. It's not the same thing as a difference tending to zero.

[seven-deuce]

1 = 0.999...

1 - 0.999... = 0

Strong logic, logic is strong.

@ Daviddem only realized how dumb that was after i posted it the 0.11.. thing lol

How do you do multiple quotes in a reply like if i wanted to quote davvidem in here as well instead of writing @davvidem.?

[daviddem]

Just press (select) the multi-quote buttons to the right of the quote button in the posts you want to quote, then press the quote button in one of those same posts.

[seven-deuce]

since 0.0000...1 equals zero, then there is zero difference

The difference is not "minute". The difference is infinitely small, which is another way of saying that there is no difference at all. It's not the same thing as a difference tending to zero.

The penny just dropped, the difference is infinitely small so there is no difference, well explained.

Also using a bit of logic; 0.00....01 = 0 so that can be rewritten as 0 = 0 which shows a lot more clearly that there is no difference.

[MadMojoMonkey]

[daviddem]

Fun fact to complete the thread derail, with a surprise twist at the end to get it back on track:

Infinitely small does not exist in the real world. Quantum physics show that it is impossible to differentiate between two points of space less than a particular distance apart. This distance is called the Planck length. It can be proved that the Planck length is the shortest measurable length, and no improvement in measure apparatus could ever change that. Not that we're any close to be able to measure anything that small (1.6*10^-35 meters or 10^-20 times smaller than a proton). Actually at this scale, the very concepts of size and distance loose their meaning entirely.

One possible implication is that, at that scale, space-time could be discrete or granular, as opposed to being a continuum. In other words, the cardinality of space-time would be N0...

Now we can go back to cardinality...

[seven-deuce]

Fun fact to complete the thread derail:

Infinitely small does not exist in the real world. Quantum physics show that it is impossible to differentiate between two points of space less than a particular distance apart. This distance is called the Planck length. It can be proved that the Planck length is the shortest measurable length, and no improvement in measure apparatus could ever change that.

Remember seeing this elsewhere it's pretty cool.

So back to infinity, the infinity of real numbers between 0 - 1 is greater than the infinity of integers from 1 to infinity because you can keep generating new real numbers that lie between 0 - 1 forever, so that it exceeds the integers. This is what i got from the video is that right?

[daviddem]

Sort of. The steps are:
1) Assume that the real numbers between 0 and 1 are countable, just like the natural numbers. If we can prove this wrong, and the real numbers between 0 and 1 cannot be counted then by obvious extension the entire set of real numbers also cannot be counted. But for now, assume the real numbers between 0 and 1 are countable.

2) Since you assumed they are countable, it implies that you should be able to come up with a list that includes them all, just like the list
0
1
2
3
.
.
.
represents the list of all natural numbers.

So let's say that our list of all real numbers between zero and one is:

0.a11a12a13a14...
0.a21a22a23a24...
0.a31a32a33a34...
0.a41a42a43a44...
.
.
.

where aij is a figure 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. For example if the first number in the list is 0.5678000... then a11=5, a12=6, a13=7, a14=8, a15=0 and so on.

3) Now let's show that whatever this list is (for whatever possible aij) we can come up with a real number between zero and one which is not included in this list. Since this number is not included in the list, it ensues that this list cannot be the list of all real numbers between 0 and 1. And since we can do this for any aij and hence for any list of real numbers between 0 and 1, this proves wrong our assumption that the real numbers between 0 and 1 are countable: no list can represent all of them, since for every possible list, I can find a real number between 0 and 1 that is not included in it.

A number not included in the above list is 0.b1b2b3b4... where:
b1 can be any figure not equal to a11.
b2 can be any figure not equal to a22.
b3 can be any figure not equal to a33.
b4...
etc
This number is not in the list because by its structure it differs from every number in the list by at least one decimal place. So we just found a real number between 0 and 1 (actually a bunch of them) which is not in this supposed list of all real numbers between 0 and 1. So obviously the list was not really the list of all numbers as we wrongly assumed. And since we can do that for any list we like, it means that no list at all can include all the real numbers between 0 and 1. And so the real numbers are not countable as we first assumed.

[Lukie]

since 0.0000...1 equals zero, then there is zero difference

The difference is not "minute". The difference is infinitely small, which is another way of saying that there is no difference at all. It's not the same thing as a difference tending to zero.

I think most people kind of, sort of understand the concept of infinity. It seems like less understand the concept of infinitely small, or 1/infinity. For example those questions about having a positive bankroll and an edge at a game, you will not go broke a certain percentage of the time (depending on the specifics) even given infinite chances.

[Savy]

1 = 0.999...

1 - 0.999... = 0

After giving it some thought, this is horribly wrong.

1-0.999.... = 0

It does not equal 0.00...01

Therefore the two aren't comparable, we'll just conclude that 0.00...01 isn't something which exists.

[seven-deuce]

Sort of. The steps are:
1) Assume that the real numbers between 0 and 1 are countable, just like the natural numbers. If we can prove this wrong, and the real numbers between 0 and 1 cannot be counted then by obvious extension the entire set of real numbers also cannot be counted. But for now, assume the real numbers between 0 and 1 are countable.

2) Since you assumed they are countable, it implies that you should be able to come up with a list that includes them all, just like the list
0
1
2
3
.
.
.
represents the list of all natural numbers.

So let's say that our list of all real numbers between zero and one is:

0.a11a12a13a14...
0.a21a22a23a24...
0.a31a32a33a34...
0.a41a42a43a44...
.
.
.

where aij is a figure 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. For example if the first number in the list is 0.5678000... then a11=5, a12=6, a13=7, a14=8, a15=0 and so on.

3) Now let's show that whatever this list is (for whatever possible aij) we can come up with a real number between zero and one which is not included in this list. Since this number is not included in the list, it ensues that this list cannot be the list of all real numbers between 0 and 1. And since we can do this for any aij and hence for any list of real numbers between 0 and 1, this proves wrong our assumption that the real numbers between 0 and 1 are countable: no list can represent all of them, since for every possible list, I can find a real number between 0 and 1 that is not included in it.

A number not included in the above list is 0.b1b2b3b4... where:
b1 can be any figure not equal to a11.
b2 can be any figure not equal to a22.
b3 can be any figure not equal to a33.
b4...
etc
This number is not in the list because by its structure it differs from every number in the list by at least one decimal place. So we just found a real number between 0 and 1 (actually a bunch of them) which is not in this supposed list of all real numbers between 0 and 1. So obviously the list was not really the list of all numbers as we wrongly assumed. And since we can do that for any list we like, it means that no list at all can include all the real numbers between 0 and 1. And so the real numbers are not countable as we first assumed.

10/10 Post.

[daviddem]

Strong logic, logic is strong.

@ Daviddem only realized how dumb that was after i posted it the 0.11.. thing lol

How do you do multiple quotes in a reply like if i wanted to quote davvidem in here as well instead of writing @davvidem.?

since 0.0000...1 equals zero, then there is zero difference

The difference is not "minute". The difference is infinitely small, which is another way of saying that there is no difference at all. It's not the same thing as a difference tending to zero.

After giving it some thought, this is horribly wrong.

1-0.999.... = 0

It does not equal 0.00...01

Therefore the two aren't comparable, we'll just conclude that 0.00...01 isn't something which exists.

See here, under "argument from arithmetic": http://www.purplemath.com/modules/howcan1.htm

I will concede that 0.0...1 is not a common or conventional way of writing zero, but if you adopt the convention that 0... represents an infinite number of zeroes, then yes, 0.0...1 equals 0. Just like 0.999...8 could also represent 1 even though nobody does it. It's just that, why bother writing that last number, since there isn't a last number... So yeah, writing it is wrong, it's pretty much an implicit contradiction. Just do:
1 - 0.999... = 1 - 1 = 0. Period.

[daviddem]

I think most people kind of, sort of understand the concept of infinity. It seems like less understand the concept of infinitely small, or 1/infinity. For example those questions about having a positive bankroll and an edge at a game, you will not go broke a certain percentage of the time (depending on the specifics) even given infinite chances.

There is never an absolute 0 probability that you will loose all your bankroll, no matter how high your average winrate is, unless you play some game where your chance of winning is exactly 100% every time. You can have a bankroll of a million buy ins, and be dealt AA 1 million times in a row and get it in a million times preflop with your opponent holding 27o and loose every single time.

While very, very highly improbable, it is not impossible.

[Lukie]

There is never an absolute 0 probability that you will loose all your bankroll, no matter how high your average winrate is, unless you play some game where your chance of winning is exactly 100% every time. You can have a bankroll of a million buy ins, and be dealt AA 1 million times in a row and get it in a million times preflop with your opponent holding 27o and loose every single time.

While very, very highly improbable, it is not impossible.

Right, which is basically the same thing I am saying, only I wasn't all that clear.

My point was that some people think that if you have a positive edge, you still have to go broke 100% of the time if you play any non-100% edge infinite times, which is not true.

[a500lbgorilla]

This is a good thread.

[a500lbgorilla]

One implication is that, at that scale, space-time is discrete or granular, as opposed to being a continuum.

There's one fun addition to this.

'to us'

That we can't measure it, doesn't mean it isn't. It just means it isn't to us.

[MadMojoMonkey]

Yah, if you're gonna highlight that quote, then as a physicist I can pretty much squash it.

The Planck length is just a number. It's achieved by taking some other constants and multiplying /dividing them via dimensional analysis to come up with a number whose units are length only.

It does not represent anything physical, and any notion of what it might represent is purely hypothetical. As was mentioned, there is no physical way to measure anything that small.

On the point that space is granular on some scale. I take it you mean that on some scale a thing can be at point A or point B, but nowhere in between.

IF this were true, then there would be diffraction planes in space (albeit immeasurably close together, and not necessarily stationary). These planes would alter the way waves travel through space and would have very measurable effects. For one, certain frequencies of light would not travel in certain directions.

Also, quantum mechanics shows that a particle is not a pinpoint-localized phenomenon, but a kind of fuzzy bordered area of probability density. There is no way for a particle to be in a "single place".

Recall the link between position and momentum. The more defined the momentum is, the less defined the position is. Since there are always reasonable limits to place on the momentum, there is implicit uncertainty in the position.

If that's what you meant by granular space, it's pretty well disproved.

[daviddem]

Yah, if you're gonna highlight that quote, then as a physicist I can pretty much squash it.

The Planck length is just a number. It's achieved by taking some other constants and multiplying /dividing them via dimensional analysis to come up with a number whose units are length only.

It does not represent anything physical, and any notion of what it might represent is purely hypothetical. As was mentioned, there is no physical way to measure anything that small.

On the point that space is granular on some scale. I take it you mean that on some scale a thing can be at point A or point B, but nowhere in between.

IF this were true, then there would be diffraction planes in space (albeit immeasurably close together, and not necessarily stationary). These planes would alter the way waves travel through space and would have very measurable effects. For one, certain frequencies of light would not travel in certain directions.

Also, quantum mechanics shows that a particle is not a pinpoint-localized phenomenon, but a kind of fuzzy bordered area of probability density. There is no way for a particle to be in a "single place".

Recall the link between position and momentum. The more defined the momentum is, the less defined the position is. Since there are always reasonable limits to place on the momentum, there is implicit uncertainty in the position.

If that's what you meant by granular space, it's pretty well disproved.

No I was not really talking about a thing being at point A or point B because by the uncertainty principle, the position of "a thing" a this scale is definitely fuzzy. I was more talking about the very nature of space-time itself.

I think that the truth is we have no idea, since we have no complete theory of quantum gravity as of yet. While I have not studied string theory, I read that it seems to hint that the universe at such a short scale is neither continuous, nor discrete but something else entirely which we don't understand yet. Other theories point to a discrete structure, and others yet seem to disprove it.

http://physics.stackexchange.com/que...-or-continuous

Wish I had time to go back to university and study all that properly, or even study it by myself. I have an engineering degree but sadly we went nowhere near deep enough in theoretical physics. In particular quantum physics and the relativity theory are absolutely fascinating to me.

If there is one thing that annoys me, it is that I will probably die before humanity gets to the bottom of this and finalizes a theory of everything..

[surviva316]

Skipping a lot of posts from a lot of people who are way the fuck smarter than me, especially in this subject area, but as someone who argues theology a lot, I can say that understanding that inf != inf has a lot of philosophical applications.

The most basic explanation is that infinity isn't a number; it literally just applies to any fucking thing that isn't finite, which is an entire class things.

The best way for me to think of it is to say that if a "God" is eternal, then it is necessarily infinite. It exists for an infinite length of time, so it's impossible to count every single instance of God. But if this God's just an eternal speck of dirt on a rock on Mars, then it's very easy to imagine how you can increase God's instances in time and space. If God is all of Mars, or all of the solar system, etc, then God is WAAAAY larger than the speck of dirt, even though neither has a finite existence.

In fact, in Christian beliefs, any soul that makes it into heaven is infinite (even though they have a beginning point, they supposedly have no end point, so like a ray, it is also not finite), but clearly this soul has a shorter existence than God himself who always was and always will be.

A misunderstanding of this leads to a lot of logical shit shows. Even a supposed genius like St Augustine really cunted this up good in the Summa Theologica by making conclusions about how since God is infinite, he has to be everything everywhere all the time, which means that {insert some crappy conclusion that I forget} and BINGO this infinite thing must exist!

Anyway, didn't mean to make this about religion, but I thought that it might be interesting to bring a non-mathematic perspective to the importance of this concept.

[oskar]

There are some very entertaining plays on that in The God Delusion, if you like that sort of stuff. St Augustine seems to start with the premise that something with the properties of god must exist, and [...] therefore god exists, with the middle part being a clever misdirection that we already accepted that god exists in the premise, but I'll look into that if sunday goes slow :P

Seems to be fairly similar to the ontological argument.

[oskar]

Okay so I'm dumb and he's smart and probably right and all the rest of it but I learn best by assuming that I am king of the world and arguing. As such:

He seems to be saying:

let's take allllllll the real numbers between 0 and 1. (0.4, 0.7235762355 etc.)
let's take allllllll the natural numbers (1, 2, 3 etc.)

So far these two sets are equal. If you like, imagine that you're counting the first set, this automatically generates the second. He's saying if we were to list these out to infinity (also seems troubling that he keeps saying that like it's a logical possibility) then we get all of them.

Then he says, now imagine a real number X that isn't equal to any of the real numbers "already listed".

So now we're rocking it like this:

1. We've already "listed" "all" the natural numbers.
2. Before this whole X thing, we have a "list" of real numbers that is exactly as long as the "list" of "all" the natural numbers.
3. So once we add X to that second list, the second list becomes longer than the first.

This is all well and good but he fails to prove that X can exist.

To my mind, any possible value that X could be, we've already covered since we kept listing real numbers out to infinity.

If you want to say that there's a magical real number X that's not equal to any of the "other" real numbers, why not a magical natural number Y, that's not equal to any of the other natural numbers and restores the 1:1 equilibrium?

I'll just come back to that because it's super easy, and I can feel smart by answering it.
If you want to start representing natural numbers by real numbers, you can just take a natural number that is

2164984351684684684....
where each digit is
1a2a3a4a5a6a
And represent it by a real number that is
0.1a2a3a4a5a6a...
or
0.0001a2a3a4a5a6a...
or
99.123bacon1a2a3a4a5a6a...

You can already see that intuitively you can fit infinite infinites of natural numbers in a real number any which way you like, so proving that there are infiniely more numbers in any arbitrary interval of real numbers than there are in all of N is just a formality.

[surviva316]

There are some very entertaining plays on that in The God Delusion, if you like that sort of stuff. St Augustine seems to start with the premise that something with the properties of god must exist, and [...] therefore god exists, with the middle part being a clever misdirection that we already accepted that god exists in the premise, but I'll look into that if sunday goes slow :P

Seems to be fairly similar to the ontological argument.

After looking back over the piece, I am confused, but my confusion isn't with Descartes' ontological proof. The problem with that proof is purely syntactical. Basically, if I say that there exist no good players on the US national team, then this sentence is descriptive of what? It's describing the US national team as shitty, right? Surely it's not describing good players, and making a point about how they're always so decidedly not US national teamy, right?

Well the problem is that that's exactly what Descartes tried to argue. He tried to argue that a non-existence of God in this universe is somehow an imperfection in God, and since God is perfect that's not possible, so BAMO he exists! All that Descartes really proved, though, is that a godless universe does not contain the best and most perfect thing that could possibly be conceived, which is not a terribly interesting comment.

My confusion was that the Summa Theologica addresses an objection to the existence of God (not a proof of it) that involves this so-called cunting of the concept of infinity: "It seems that God does not exist; because if one of two contraries be infinite, the other would be altogether destroyed. But the word "God" means that He is infinite goodness. If, therefore, God existed, there would be no evil discoverable; but there is evil in the world. Therefore God does not exist." Basically, I remember being frustrated with St. Augustine's rebuttal of this argument because he didn't simply say "inf != inf." I am perfectly willing to admit that this is petty on my part as an atheist to say that he didn't find the best way to refute an atheist argument.

Anyway, Augustine has 5 proofs of God and they all have their own problems, but they mostly boil down to the fact that metaphysical arguments that predate quantum physics generally suck ass. It's kinda like, "As far as my Dark Ages eyes have seen, an arrow has never flown through the air without there being some intelligent being pulling back a bow, so clearly everything ever has its root cause in something intelligent." Science has since made this simplistic premise a little more *ahem* complicated.

In any case, I hope that I've shown an example of how inf != inf can be an important philosophical concept.

[Savy]

The most basic explanation is that infinity isn't a number; it literally just applies to any fucking thing that isn't finite, which is an entire class things.

Infinity isn't a number, it's a concept.

[surviva316]

Infinity isn't a number, it's a concept.

Yes.

[daviddem]

Right, which is basically the same thing I am saying, only I wasn't all that clear.

My point was that some people think that if you have a positive edge, you still have to go broke 100% of the time if you play any non-100% edge infinite times, which is not true.

You are right. The problem you are mentioning is called "risk of ruin" and can be simply described as follows:
- assume a gambler starts with an initial number of buy-ins BI
- assume he endlessly (unless he goes broke) plays a game where his probability of winning is P and his probability of loosing is 1-P, with P > 0.5
- his chance of going broke is ((1-P)/P)^BI. In the other cases he will become infinitely rich.
- as an aside, if P<=0.5, the gambler always eventually goes broke

So for example if a gambler has $5 and he repeatedly plays a $1 game where he has 60% chance to win (P=0.6), then his chance of going broke is:
(0.4/0.6)^5 = (2/3)^5 = 32/243 = ~13.2%

The mathematical proof can be found here:
http://www.columbia.edu/~ks20/FE-Not...7-Notes-GR.pdf

You can do further calculations to apply this to a poker player who has a given winrate, standard deviation and some number of buy-ins. See here: http://archives2.twoplustwo.com/show...st682045683150
and here for calculators based on these formulas:
http://www.castrovalva.com/~la/bank.htm
http://www.reviewpokerrooms.com/poke...uirements.html
(note that the winrate and standard deviation are in bb/100 and the bankroll is in bb, not in BI. A typical poker player's standard deviation is 80bb/100, check yours out in PT or HEM).

[rong]

I'll just come back to that because it's super easy, and I can feel smart by answering it.
If you want to start representing natural numbers by real numbers, you can just take a natural number that is

2164984351684684684....
where each digit is
1a2a3a4a5a6a
And represent it by a real number that is
0.1a2a3a4a5a6a...
or
0.0001a2a3a4a5a6a...
or
99.123bacon1a2a3a4a5a6a...

You can already see that intuitively you can fit infinite infinites of natural numbers in a real number any which way you like, so proving that there are infiniely more numbers in any arbitrary interval of real numbers than there are in all of N is just a formality.

I get this, but I don't think it has any value whatsoever, and maybe isn't even right, or perhaps the problem is that the mathematical language we have created is finite. It's basically a chicken and an egg thing. Because for any real number that you can come up with that is between two natural numbers I can just add another natural number.

I think it becomes a problem with a definition of infinity. I don't think any infinity can be greater than any other infinty, because doesn't infinity mean no limit? In which case there is no difference between the two. Trying to compare two infinitys is stupid because they are exactly the same thing, quite simply unlimited, so one can't possibly be larger than the other because they aren't a size, they are an absence of one.

[rong]

Or another way of putting it is you're saying that

infinitelyxinfinity > Axinfinity > infinity (where A>1)

And I'm saying they're all equal.

[daviddem]

I get this, but I don't think it has any value whatsoever, and maybe isn't even right, or perhaps the problem is that the mathematical language we have created is finite. It's basically a chicken and an egg thing. Because for any real number that you can come up with that is between two natural numbers I can just add another natural number.

I think it becomes a problem with a definition of infinity. I don't think any infinity can be greater than any other infinty, because doesn't infinity mean no limit? In which case there is no difference between the two. Trying to compare two infinitys is stupid because they are exactly the same thing, quite simply unlimited, so one can't possibly be larger than the other because they aren't a size, they are an absence of one.

No they are not the "same" infinity. Let's start with an example with finite sets.

The cardinality of a finite set is simply the number of elements in the set. It is pretty trivial to see that when two finite sets have the same cardinality, you can always create some relation between the two sets so that one and only one element of one set corresponds to one and only one element of the other. Example:

set A
a
b
c

set B
d
e
f

and you can always create a relationship simply by pairing one element of one set with one element of the other, for example a-e, b-d and c-f. Each element in set A has one and only one buddy in set B and each element of set B has one and only one buddy in set A.

But if the sets have different cardinality:
set A
a
b
c

set B
d
e
f
g

Then whatever you do there is always one element of set B which won't have a buddy in set A...

Well now for infinite sets it's exactly the same story: you can't find a natural number buddy for every single real number...

Consider the set of all natural numbers:
0, 1, 2, 3, ...
and the set of all even natural numbers:
0, 2, 4, 6, ...

Both these sets are infinite, and we can say that they are the same kind of inifinity (in mathematical language, you say that both sets have the same cardinality) because you can find a function that puts in correspondence one element of one of these sets with one and only one element of the other set. In this example, the function would be
f(x) = 2*x where x belongs to the set of natural numbers. With this function:
f(0) = 0
f(1) = 2
f(2) = 4
f(3) = 6
...
So via this function, to each element of the set of natural numbers corresponds one and only one element of the set of natural even numbers. The inverse of the function is of course invf(y) = y/2 where y belongs to the set of natural even numbers:
invf(0) = 0
invf(2) = 1
invf(4) = 2
invf(6) = 3

So via these two functions:
- if you give me any natural number, I can tell you which is the corresponding even natural number
- if you give me any even natural number, I can tell you which is the corresponding natural number

And that is why we say that these two sets have the same cardinality.

Another way of saying it is that you can unequivocally and non-ambiguously "map" one set onto the other and vice-versa.

You can try for your entire life to find such a function (and its inverse) that would unequivocally put in correspondence one and only one real number with one and only one natural number: you won't find one because it does not exist.

So while it is not really correct to say that there are "more" real numbers than natural numbers because "more" is an adjective that applies only in the world of finite sets, mathematicians needed a way to express the existence or non-existence of such an unequivocal relation between two sets, and the way they did that was to say that the cardinality of two sets is the same if such a function can be found and the cardinality of two sets is different if such a function cannot be found.

Additionally, it can be demonstrated that the cardinality of the set of real numbers is strictly greater than the cardinality of the set of natural numbers, because there exist injective functions that map N on R but no bijective functions (which means that you can create a function that finds a real number corresponding to each and every single natural number, but its inverse won't cover all the real numbers: for example f(x) = x with x being a natural number finds a real number for every natural number, but its inverse, which is invf(y) = y with y being a real number does not work for example for 0.5 because to the real number 0.5 would supposedly correspond the natural number 0.5 but since 0.5 is not a natural number, that doesn't work. In other words, via these two functions:
- if you give me any natural number, I can tell you which is the corresponding real number
- but for some real numbers (actually most of them), I wouldn't be able to tell you which is the corresponding natural number, because there isn't one (they don't have a natural number buddy).

er... clear?

[CoccoBill]

My logic says 0.999... is infinitesimally close to 1 but not quite there. Smells like something that's just a close enough approximation to be used for practical purposes, not something worthy of a = in my book.

[daviddem]

My logic says 0.999... is infinitesimally close to 1 but not quite there. Smells like something that's just a close enough approximation to be used for practical purposes, not something worthy of a = in my book.

Your logic is flawed because the convention is that 9... represents an infinite number of 9's, not a number of 9's tending to infinity.

If 9... was a number of 9's tending to infinity, then your assertion would be correct.

A number that is infinitesimally close to 1 is equal to 1. A number that tends to be infinitesimally close to 1 is "not quite 1", as you said.

If you wished though, you could adopt for yourself the convention that 9... represents a number of 9's tending to infinity, however this would not generally be recognized by the community of mathematicians at large. Conventionally, the way mathematicians would represent a number smaller than 1 that tends to 1 is:
lim
x->1-
(with x->1- in subscript below the "lim") like here: http://en.wikipedia.org/wiki/One-sided_limit

or in calculus you can write for short
1-dx
where it is conventionally understood that dx is a positive quantity tending to zero.