My Math Forum If you remove the decimal from [ 0 - 1 ) what does that set match?

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May 9th, 2017, 01:35 PM   #21
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 Originally Posted by v8archie Your definition of A isn't very clear then. And there is no element of A that maps to zero, so you still don't have the bijection you were looking for. Perhaps a little more consideration and a little less aggression?
First off, he's (we're) not using function $f$ to go from $A$ to [0, 1), so it doesn't matter whether there is an element $a \in A$ such that $f(a) = 0$. I told him to find a function $h$ that is what we want because function $f$ doesn't suffice, so quit spouting garbage.

Second, be considerate? Ahem...

Quote:
 Originally Posted by v8archie No you don't, fool.
Quote:
 Originally Posted by v8archie Learn some mathematics.
Quote:
 Originally Posted by v8archie Learn some mathematics!
Zylo, et al., is obviously struggling. There is something more important here than just trying to get Zylo to admit that he/she doesn't have a valid disproof of Cantor's Theorem. IMHO, you need to stop ridiculing and back off. Until you do, you'll never get Zylo to listen to you (I wouldn't either).

 May 9th, 2017, 02:10 PM #22 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,274 Thanks: 2435 Math Focus: Mainly analysis and algebra Zylo has been spouting that nonsense for the best part of a year. He has ignored every argument that doesn't fit what he wants to believe not just inthat thread, but in at least a dozen like it. He's not struggling to understand because he's not attempting to do so. He is just trying to muddy the waters for other students. I apologise for misunderstanding that a thread including $[0,1)$ in the title, every interval definition and your definition of A isn't supposed to include zero. Perhaps more clarity would help. Establishing that the cardinality of the set of infinite digit strings is the same as that of the reals seems to me to be much easier than trying to establish a bijection.
 May 9th, 2017, 02:29 PM #23 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,274 Thanks: 2435 Math Focus: Mainly analysis and algebra Right on cue, he's started another one.
May 9th, 2017, 05:43 PM   #24
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Quote:
 Originally Posted by Maschke My own opinion is that Cantor's uncountability results are the very first example anyone ever encounters of a result in higher math that's deeply counterintuitive, yet accessible to beginners. So a lot of people get stuck there.
Quote:
 This argument is often the first mathematical argument that people meet in which the conclusion bears no relation to anything in their practical experience or their visual imagination. Compare it with two other simple facts of cardinal arithmetic...
Thanks for the link. It was worth the read, and quite a few chuckles to be had there.

I really like his mention of assessing unformalised arguments, and i think the following sums everything up quite nicely...

Quote:
 Let us prove, he said, that Cantor’s argument is invalid. We start by assuming that it is valid. If it is valid we are entitled to use it; and so we do, down to the point where we get a contradiction. But since we have reached a contradiction, our original assumption must have been wrong. That is to say, Cantor’s argument is invalid. There is a quick though slightly dishonest refutation of this critique...
Quote:
 Originally Posted by v8archie I think that many (such as Zylo) struggle to cope with the shattering of the idea that "infinity" isn't a number, the idea that arbitrarily large finite numbers do exist and that there is something even bigger than the "bigger than all the numbers" that they always thought of as a number.
Thanks.

May 9th, 2017, 05:59 PM   #25
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 Originally Posted by InkSprite I'm just the other crazy guy that jumped in for no apparent reason because I don't like .9repeating==1. Speaking of converging. can you converge on an irrational number?
The real numbers are the limits of all convergent sequences of natural numbers. It's a definition of them.

The representation $0.999\ldots$ means $$\sum_{k=1}^\infty \frac9{10^k}= \lim_{n \to \infty} \sum_{k=1}^n \frac9{10^k} = 1$$
or, equivalently, $0.999 \ldots$ is the limit of the sequence $\left( \frac9{10}, \frac{99}{100}, \frac{999}{1000}, \ldots, \frac{10^k-1}{10^k} \ldots \right)$.

Obviously we can do the same - finding the limit - with $1$. $1 = 1.000\ldots$ means $$1+ \sum_{k=1}^\infty \frac0{10^k}= \lim_{n \to \infty} \left(1 + \sum_{k=1}^n \frac0{10^k} \right) = 1$$
or, equivalently, $1.000 \ldots$ is the limit of the sequence $(1, 1, 1, \ldots)$.

Limits can also be irrational. $$\pi = \sum_{k=0}^\infty (-1)^k \frac4{2k+1} = \lim_{n \to \infty} \sum_{k=0}^n (-1)^k \frac4{2k+1}$$
or $\pi$ is the limit of the sequence $(3, 3.1, 3.14, 3.141, \ldots)$.

Indeed, one definition of Euler's number $e = 2.71828\ldots$ is $$e = \lim_{n \to \infty} \left(1 + \frac1n\right)^n$$

Last edited by v8archie; May 9th, 2017 at 06:16 PM.

May 9th, 2017, 06:15 PM   #26
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 Originally Posted by InkSprite I'm just the other crazy guy that jumped in for no apparent reason because I don't like .9repeating==1.
In addition to v8archies post, you might want to watch this. Much more basic examples are given in the video compared with archies explanation, but sometimes that helps.

May 9th, 2017, 06:35 PM   #27
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 Originally Posted by Maschke A math professor wrote a paper about that. It's here in ps form. http://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps It's called, An Editor Recalls Some Hopeless Papers, by Wilfred Hodges.
Note to Zylo - from page 4:
Quote:
 Other authors, less coherently, suggested that Cantor had used the wrong positive integers. He should have allowed integers which have infinite decimal expansions to the left, like the p-adic integers. To these people I usually sent the comment that they were quite right, the set of real numbers does have the same cardinality as the set of natural numbers in their sense of natural numbers; but the phrase ‘natural number’ already has a meaning, and that meaning is not theirs... How does anybody get into a state of mind where they persuade themselves that you can criticise an argument by suggesting a different argument which doesn’t reach the same conclusion?

Last edited by v8archie; May 9th, 2017 at 06:37 PM.

May 10th, 2017, 05:41 AM   #28
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So this is where I picked up the hint about space filling fractal curves.
javascript - I need to make my function return a more organic collection of results - Stack Overflow
Quote:
 Mathematicians usually uses definitions to reproduce a class of ideas that can be brought and thought together. To their surprise, the definition of continuity did include really weird functions to be curves: space-filling-curves, fractals!!! They called them monsters at the time.
Unfortunately he didn't expound upon them much beyond that, so I asked him to give me more information about them. My next questions will be "Who were these mathematicians?", "What do the formulas of these curves look like?". 1, to make sure these things exist, 2 to make sure they are applicable to a function the can describe a sequence of string combinations.

If the space can be filled continuously by the curve, then my novice intuition tells me the "potential space" of string combinations is also filled. The only thing left would be to find the right filling curve function.

Last edited by InkSprite; May 10th, 2017 at 05:44 AM.

May 10th, 2017, 06:12 AM   #29
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 Originally Posted by InkSprite So this is where I picked up the hint about space filling fractal curves. javascript - I need to make my function return a more organic collection of results - Stack Overflow Unfortunately he didn't expound upon them much beyond that, so I asked him to give me more information about them. My next questions will be "Who were these mathematicians?", "What do the formulas of these curves look like?". 1, to make sure these things exist, 2 to make sure they are applicable to a function the can describe a sequence of string combinations. If the space can be filled continuously by the curve, then my novice intuition tells me the "potential space" of string combinations is also filled. The only thing left would be to find the right filling curve function.
The prototypical class of fractals that you have probably seen is the Mandelbrot set (Benoit Mandelbrot). But there's loads of others, check out this list.

Most of the fractals in the list are 'pretty' looking fractals, a special sort of fractal. But in general, the key feature is 'self-similarity'. The idea that, regardless of scale, and object may exhibit the same structure.

If you read the thread you posted, the first response lists various mathematical topics that are relevant here. Personally i think a good place to start, formally, would be a book by Steven Strogatz - "Non-linear dynamics and chaos". From there you could progress to Edward Ott - "Chaos in dynamical systems".

 May 11th, 2017, 05:27 AM #30 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,329 Thanks: 94 If you remove the decimal from [ 0 - 1 ) what does that set match? The set of natural numbers. Proof: Assume there is a natural number not in the set. Put a decimal point in front of it. By definition, the numbers in [0,1) are the real numbers minus the integral part. The real numbers are countable. Ref: Cantors Diagonal Argument, Logic Real Numbers and Natural Numbers

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