/sci/ - Science & Math

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Anonymous Tue Jul 7 22:39:44 2015 No.7379503 [Reply] [Original]

Who the fuck came up with the idea that the reals are not countable? If you spent infinite time counting, you could potentially count to infinity (inb4, you couldn't count to a "large infinity," well you could because you could potentially count at a faster rate).

Also the idea that if you pick a random point on the line of reals "it has no width" makes no sense because even differential width *is* measurable width when taken over a certain period (particularly large infinity).

>>	Anonymous Tue Jul 7 22:43:28 2015 No.7379506 File: 41 KB, 724x524, sad-count.png [View same] [iqdb] [saucenao] [google] >>7379503

Anonymous Tue Jul 7 22:46:20 2015 No.7379514 [DELETED]

Countable does not imply that you can reach an arbitrary large value. Countable implies there is an effective way to list the elements of an infinitely large set. For instance:

\bbN = {1, 2, 3, ...}[/spoiler]
\bbZ = {0, 1, -1, 2, -2, ...}[/spoiler]
\bbQ = {0, 1, -1, 2, 1/2, -2, -1/2, 3, 1/3, 3/2, 2/3, ...}[/spoiler]

The point is, using any of the methods I gave, I will eventually reach any arbitrary element of the given set. The same cannot be said for the reals. There is no effective way to list them.

>>	Anonymous Tue Jul 7 22:48:11 2015 No.7379518 Nobody! It was discovered. You just don't understand mathematics yet. Finish calc 1 first.

>>	Anonymous Tue Jul 7 22:48:26 2015 No.7379520 >>7379503 Terrible bait, 2/10 for making me answer

Anonymous Tue Jul 7 22:49:25 2015 No.7379522

Countable does not imply that you can reach an arbitrary large value. Countable implies there is an effective way to list the elements of an infinitely large set. For instance:

\mathbb{N} = {1, 2, 3, ...}[/spoiler]
\mathbb{Z} = {0, 1, -1, 2, -2, ...}[/spoiler]
\mathbb{Q} = {0, 1, -1, 2, 1/2, -2, -1/2, 3, 1/3, 3/2, 2/3, ...}[/spoiler]

The point is, using any of the methods I gave, I will eventually reach any arbitrary element of the given set. The same cannot be said for the reals. There is no effective way to list them.

>>	Anonymous Tue Jul 7 22:49:40 2015 No.7379523 >>7379518 >>7379514 that didnt work; please clarify

>>	Anonymous Tue Jul 7 22:50:41 2015 No.7379526 >>7379522 OK, so using the "technical" mathematical definition of "counting" is different from the intuitive act?

Anonymous Tue Jul 7 22:52:31 2015 No.7379529

>>7379526
Not really, a mathematically countable set can still be ordered so as to be intuitively counted (like Cantor diagonalization), you go one by one

The reals just can't be ordered like that. This expands on the idea of infinity: it's a larger infinity in the sense that three dimensions are larger than two (it's a basic analogy, don't read too much into it).

>>	Anonymous Tue Jul 7 22:52:40 2015 No.7379530 >>7379526 yes countable implies there's a bijection between the natural numbers and the set i.e. you can order them so there's clearly a first, a second, a third, and so on

Anonymous Tue Jul 7 22:52:49 2015 No.7379531

>>7379526
A set is countable if it is finite or if there is a objection between it and the set of natural numbers.

If you can't do this, the set is called uncountable.

R being uncountable means it is larger than N in the sense that if you have a list of real numbers (if you can list them, it implies countable) of any size, there will always be ones not on the list.

>>	Anonymous Tue Jul 7 22:53:40 2015 No.7379533 >>7379529 isn't there a bijection from R^3 -> R^2?

>>	Anonymous Tue Jul 7 22:53:54 2015 No.7379535 >>7379529 >>7379530 OK well math has a logical progression and is either/or...

>>	Anonymous Tue Jul 7 22:55:29 2015 No.7379538 >>7379533 Riemann sphere

>>	Anonymous Tue Jul 7 22:56:29 2015 No.7379540 >>7379533 Argh, you're probably right. The analogy doesn't work on the basic level, then ;-; Cardinality is what I mean. The reals have a greater cardinality than the natural numbers.

Anonymous Tue Jul 7 22:56:39 2015 No.7379542

>"it has no width" makes no sense because even differential width *is* measurable width "it has no width" makes no sense because even differential width *is* measurable width
It's not because people often treat differential operators as something that can be separated that they actually are.

If you remember, everything is defined in terms of epsilons and deltas at this point, there is no such thing as differential width or separation of the differential operators, even it's often explained this way in basic classes or in physics.

Anonymous Tue Jul 7 22:56:40 2015 No.7379543

>>7379535
the intuitive act of counting isn't
one person can consider ordering the fractions and labeling them 1sst, 2nd, 3rd, counting and another may not

you are counting the number fractions but you aren't counting them the same way you'd count natural numbers based on their size

the point is we have the same mathematical definition of countable

>>	Anonymous Tue Jul 7 22:56:43 2015 No.7379544 >>7379533 I forget if there is, but so? Neither of those have a bijection with the naturals, so they're both still uncountable.

>>	Anonymous Tue Jul 7 22:57:47 2015 No.7379546 >>7379522 What does "Q" stand for? N is integers, Z is...whole?

>>	Anonymous Tue Jul 7 22:59:48 2015 No.7379549 >>7379542 then why can you divide and multiply differentials like discrete variables when using separable equations

Anonymous Tue Jul 7 22:59:59 2015 No.7379550

>>7379544
the analogy fails because the cardinality of the reals is greater than the cardinality of natural numbers, but R^2 and R^3 have the same cardinality

>>7379546
Q is the rationals or fractions
Z is integers (includes negatives)
N is natural numbers (just 1,2,3....)

Anonymous Tue Jul 7 23:01:20 2015 No.7379554

>>7379550
>the analogy fails because the cardinality of the reals is greater than the cardinality of natural numbers, but R^2 and R^3 have the same cardinality

WHAT
How the fuck do R^2 and R^3 have the same cardinality?! How do you define that mathematically?

>>	Anonymous Tue Jul 7 23:02:21 2015 No.7379557 >>7379550 Thanks. This is the first math thread I've seen on /sci/ that doesn't suck.

>>	Anonymous Tue Jul 7 23:02:47 2015 No.7379559 >>7379549 You can't in all cases. It's just so happens that you can in most cases.

Anonymous Tue Jul 7 23:03:28 2015 No.7379561

>>7379554
if you can show R^2 has the same cardinality as R, you can do the same for R^3 and R^2

I don't know one off the top of my head but this stackoverflow question seems to have some answers
http://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr

>>	Anonymous Tue Jul 7 23:04:03 2015 No.7379564 >>7379559 >It's just so happens that you can in most cases As a chemist, I have to think of the idea that N=1 is statistically not a coincidence...

Anonymous Tue Jul 7 23:04:25 2015 No.7379566

Have you even ever looked at a proof? The reals aren't countable because it is impossible to order them. You can neatly order infinitely long decimals and still produce a decimal that should be within the list. You can spend eternity counting all the numbers between 0 and 1. And then after that, you can still construct a new number that you missed. The proof is based off the assumption that you can count the reals but it results in a logical contradiction.

>>	Anonymous Tue Jul 7 23:04:30 2015 No.7379567 >>7379546 Q is quotients, or rationals. N is naturals. Z is zahlen, which is German for integers.

>>	Anonymous Tue Jul 7 23:05:05 2015 No.7379568 >>7379561 Whoah....so does R^2 and R^4, R^5 or R^n and R^n+k etc have the same cardinality too?

>>	Anonymous Tue Jul 7 23:05:43 2015 No.7379570 >>7379554 (0,1) has the same cardinality as R. So does (0,1/2). So does (0,0.000000001). Etc

>>	Anonymous Tue Jul 7 23:06:13 2015 No.7379571 >>7379566 >The proof is based off the assumption that you can count the reals but it results in a logical contradiction. I followed you until that contradictory statement. It sounds like your definitions are incorrect at root.

>>	Anonymous Tue Jul 7 23:06:44 2015 No.7379573 >>7379568 Yes. R^R is bigger than R^n though.

>>	Anonymous Tue Jul 7 23:07:54 2015 No.7379578 >>7379573 Wait, how do you do R^R if the reals aren't countable? How do you count to the largest value of R, such that R > n?

Anonymous Tue Jul 7 23:08:14 2015 No.7379579

>>7379571
No, that's method of proof by contradiction. It works by trying to prove the negative statement but then you find out the negative statement is false so the original statement is true. Its an very precise procedure. Clearly you haven't taken basic proofs.

>>	Anonymous Tue Jul 7 23:08:19 2015 No.7379580 >>7379571 Assume reals are countable. Then derive a contradiction.

Anonymous Tue Jul 7 23:10:25 2015 No.7379585

>>7379568
yeah
if f(x) maps from R^2 -> R, then you can have an ordered n-tuple (x1,x2,x3,...xn) and map it to (f(x1,x2),x3,x4) which is an n-1 tuple and repeat until you have just R

>>7379573
R^R is the set of functions from R->R, right?

Hm
Anyone know the cardinality of R^R vs the cardinality of the powerset of R?

>>	Anonymous Tue Jul 7 23:11:19 2015 No.7379586 >>7379578 P(R), the set of all subsets of R, also has a bigger cardinality than R.

>>	Anonymous Tue Jul 7 23:14:38 2015 No.7379595 >>7379585 P(R) has the same cardinality as R^R has the same cardinality as the power set of the power set of naturals. Yes, R^R is functions from R to R.

Anonymous Tue Jul 7 23:16:15 2015 No.7379600

>>7379564
Well, the main reason I said at this point is because you can work define it in terms of infinitesimals, or other things, which justify why it happens.
But naturally, that's not explained in basic calculus classes, thus why I said at this point.

I think the most basic example of it failing is with partial derivatives.

>>	Anonymous Tue Jul 7 23:16:43 2015 No.7379601 >>7379595 >has the same cardinality as the power set of the power set of naturals >the continuum hypothesis >decidable in ZFC pick one and only one

>>	Anonymous Tue Jul 7 23:17:27 2015 No.7379604 >>7379579 >>7379580 What the fuck? and no, why would anyone take proofs if it wasn't a requirement for their studies.

>>	Anonymous Tue Jul 7 23:18:59 2015 No.7379610 >>7379600 >But naturally, that's not explained in basic calculus classes, thus why I said at this point. surprise

>>	Anonymous Tue Jul 7 23:21:13 2015 No.7379612 >>7379604 If it results in a contradiction, then it must be false. Come on homie even if you didn't take proofs this is basic logic.

>>	Anonymous Tue Jul 7 23:21:58 2015 No.7379613 >>7379595 interesting btw I found a page with a ton of proofs for similar questions http://legacy.earlham.edu/~peters/writing/infapp.htm

>>	Anonymous Tue Jul 7 23:22:34 2015 No.7379615 >>7379612 If you can contradict a hypothesis it does not mean that the reverse is true, it simply means that you need to your adjust your model and get more data to determine if the reverse is indeed false.

Anonymous Tue Jul 7 23:23:28 2015 No.7379618

>>7379604
Because logic applies to all fields. The correct application of logic is called reasoning, and logic is how you reason properly.
The way it works is like so.
When you want to Prove something, you connect your claim and your conclusion with a bunch of true statements. So if you follow the logically true statements to the end and they are all true then the conclusion is true.
How does proof by contradiction work? It exploits this. Suppose your claim is false.
Then you follow a series of correct steps just like in a normal proof. If you arrive at a contradiction ie somethinf you know is false, then at least one of your previous statements must have been wrong. But we know that all the statements are true except the first one: the original claim. So, your claim was wrong which means the opposite is true!
We do this with the proofs of the cardinalities here.

>>	Anonymous Tue Jul 7 23:23:35 2015 No.7379619 >>7379613 Thanks bro. I have looked for something like this for awhile but I never know what to search.

>>	Anonymous Tue Jul 7 23:25:19 2015 No.7379622 >>7379615 Mathematical propositions are either true or false. There's no in-between. That's the difference between a hypothesis and a proposition.

>>	Anonymous Tue Jul 7 23:27:04 2015 No.7379627 File: 20 KB, 514x197, image_thumb[27].png [View same] [iqdb] [saucenao] [google] contradiction works because saying "if p is true, then q is true" means the same as saying "if q is false, then p is false"

Anonymous Tue Jul 7 23:27:15 2015 No.7379628

>>7379618
I just clearly explained to you where "classical logic" breaks down.

let me make it even more empirical:

you run a reaction in which you expect a site-specific substitution of a chloride for an iodide. the data shows that you substituted the iodide at 50% of total yield, with the other half being an unknown product. After running the experiment multiple times under different conditions, with different solvents, and with the presence of catalysts, you have realized that the neighboring bromide is prone to substitute for the chloride first. This is not expected, but you accept the empirical evidence and the statistical mixture of products remains.

The initial hypothesis was not disproven, nor was it proven. There was insight into deeper mechanistic reality, however. Not all reasoning/logic is cut and dry.

>>	Anonymous Tue Jul 7 23:30:00 2015 No.7379635 >>7379615 >it simply means that you need to your adjust your model and get more data to determine if the reverse is indeed false. that's not math there buddy, that's modeling in logic, if p is false, then ~p (the negation of p) must be true.

>>	Anonymous Tue Jul 7 23:31:14 2015 No.7379638 >>7379628 Your hypothesis is either true or not true. Logic applies.

Anonymous Tue Jul 7 23:33:00 2015 No.7379642

Cantors diagonalization argument is very straightforward. You don't understand the definition of countability, it's a technical term that's more specific than what you have in mind.

For the random number thing, measure theory was invented to answer questions like you have.

Anonymous Tue Jul 7 23:34:24 2015 No.7379650

>>7379628

Basically what >>7379638 said. You're misinterpreting what "negation" means in this case. The proposition is "there will be full substitution of chloride for iodide", the negation of which is "there will not be full substitution of chloride for iodide". It was confirmed in the experiment that the proposition was false, and thus its negation was true.

>>	Anonymous Tue Jul 7 23:48:53 2015 No.7379665 >>7379601 CH has nothin to do with >>7379595

>>	Anonymous Tue Jul 7 23:53:36 2015 No.7379669 >>7379635 then why did i use mathe the whole time

>>	Anonymous Wed Jul 8 00:10:24 2015 No.7379680 >>7379622 >>7379635 There are people that reject the principle of excluded middle. For example, intuitionists.

Anonymous Wed Jul 8 00:17:39 2015 No.7379688

>>7379503
But that's wrong OP. If you spent infinite time counting (i.e. saying each number one by one), you wouldn't even get from 0 to 1. What's the real "after" 0? There is none. Once you get past high school math you'll learn that there are different kinds of infinities.

>>	Anonymous Wed Jul 8 00:22:01 2015 No.7379696 Everyone here should read about what a "net" is!

>>	Anonymous Wed Jul 8 00:30:40 2015 No.7379708 >>7379550 Well, totally skipped over that analogy. Thought you were just pointing that out for some reason and I didn't really know why.

>>	Anonymous Wed Jul 8 00:58:20 2015 No.7379759 >>7379680 That's not really a concern here. Intuitionists can't prove nearly as many things. Besides, despite having a potentially uncountable amount of states between true and false, the end bounds are still trueband false.

>>	Anonymous Wed Jul 8 01:02:44 2015 No.7379768 >>7379503 > Who the fuck came up with the idea that the reals are not countable? Cantor > you could potentially count to infinity No, you can'tor.

>>	Anonymous Wed Jul 8 02:00:28 2015 No.7379883 >>7379768 kek That's a good one

>>	Anonymous Wed Jul 8 10:41:16 2015 No.7380644 >>7379533 assuming the axiom of choice, any infinite set is in bijection with a finite product of that set with itself.

>>	Anonymous Wed Jul 8 13:50:38 2015 No.7380865 Sure is Summer in here.

>>	Anonymous Wed Jul 8 17:49:33 2015 No.7381270 >>7379578 R^R is, in some sense, the set of all functions f: R -> R.

>>	Anonymous Wed Jul 8 17:52:47 2015 No.7381279 >>7379627 No, that's not why. What you're describing is the contrapositive, and a genuine proof by contradiction is different.

>>	Anonymous Wed Jul 8 18:38:41 2015 No.7381348 Just read the first chapter of Munkres you cretins

>>	Anonymous Thu Jul 9 10:43:42 2015 No.7382537 File: 156 KB, 400x240, Phoenix-Wright-Ace-Attorney-coffee-spill.jpg [View same] [iqdb] [saucenao] [google] >>7379531 >objection

>>	Anonymous Thu Jul 9 20:48:38 2015 No.7383625 >>7379538 could you explain a bit more please?

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