My Math Forum Rudin Theorem 2.15 and Countability of the Real Numbers

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July 27th, 2016, 05:42 AM   #11
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Quote:
 Originally Posted by zylo Every irrational number in [0,1) is given numerically by a countably infinite decimal, which, along with the rationals in [0,1), are countably infinite by 2.15.
I accept that each irrational in [0, 1) is represented by a countably infinite decimal. I do not accept the validity of 2.15 as you use it. The n in that theorem is a natural number, not infinity of any sort.

Quote:
 I have yet to see a correct proof the reals are uncountable.
There is a difference between denying the premises of a proof (such as denying ontologically the existence of transfinite numbers) and denying the validity of a proof.

Quote:
 Another point of view is, since the reals are defined by the natural numbers, there can't be more of them than you can define with the natural numbers. That is off the cuff.
It is off the wall. As you are using the word "defined," the natural numbers are defined using 10 (or 2) digits. It does not follow that there are only 10 (or 2) natural numbers.

July 27th, 2016, 06:08 AM   #12
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Quote:
 Originally Posted by zylo which, along with the rationals in [0,1), are countably infinite by 2.15.
No they aren't. Your argument is based on the same misunderstanding as always, which you are too stubborn to address.

Quote:
 Originally Posted by zylo I have yet to see a correct proof the reals are uncountable.
No. You have yet to understand a proof that the reals are uncountable because you cling stubbornly to the fiction that if you count for long enough, you will get to "infinity".

Quote:
 Originally Posted by zylo Another point of view is, since the reals are defined by the natural numbers, there can't be more of them than you can define with the natural numbers. That is off the cuff.
As often turns out to be the case with your statements. This statement is trivially true but you don't have any idea what it means. In particular you have no knowledge of how many reals that can be defined using the natural numbers.

July 27th, 2016, 10:27 AM   #13
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Quote:
 Originally Posted by v8archie As often turns out to be the case with your statements. This statement is trivially true but you don't have any idea what it means. In particular you have no knowledge of how many reals that can be defined using the natural numbers.
Only countably many, as it turns out. It's a subtle question of mathematical logic https://en.wikipedia.org/wiki/Definable_real_number

It's true that there is no bijection from the naturals to the reals. But that's a lot different than the question of how many reals are definable.

"Definable" is a technical term and it would be far better to avoid it in the context of the present discussion. There's a fantastic discussion of the issue of definable reals in Joel David Hamkins's green-checked answer in this thread. http://mathoverflow.net/questions/44...definable-numb

Mathoverflow is a site for professional mathematicians and it's not necessary for any of us mere mortals to understand everything he wrote. But at the very least I hope we'll banish the word "definable" from this thread. And that even professional mathematicians have to be set straight by a world-class expert on the subject of definability.

As I understand it, Zylo is essentially correct that we can only define countably many real numbers using natural numbers. That does not affect the uncountability of the reals. In fact even in countable models of the reals (and there are such things) it's still true that there's no bijection from the naturals to the reals. That's because uncountability is not really about cardinality, which is a relative notion; but rather because the bijection, which we can see from "outside" the model, is not part of the model.

I don't mean to throw too many irrelevant notions into this discussion; but I do mean to point out that the question of how many real numbers we can define is a lot different than the question of whether the reals are uncountable.

Last edited by Maschke; July 27th, 2016 at 10:44 AM.

July 27th, 2016, 01:23 PM   #14
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Quote:
 Originally Posted by Maschke As I understand it, Zylo is essentially correct
Zylo is not correct because he is not talking about what you are. Moreover, if by "define", we mean that in principal there exists an infinite string of mathematical symbols that defines a unique number, then there are infinitely many of them.

Math Overflow is used by many non-professionals too although many respondants forget that in the tone and depth of their answers. I'd love Zylo to try posting his nonsense there.

July 27th, 2016, 01:48 PM   #15
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Quote:
 Originally Posted by v8archie Zylo is not correct because he is not talking about what you are. Moreover, if by "define", we mean that in principal there exists an infinite string of mathematical symbols that defines a unique number, then there are infinitely many of them.
Then you would be using "define" in a decidedly nonstandard manner. The definition you gave is simply not what the word means. Personally I have never seen a definition requiring infinitely many characters. That's precisely the problem here. You can't define all that many real numbers. You can define at most countably many. Definability can only confuse the issues of this thread.

Quote:
 Originally Posted by v8archie I'd love Zylo to try posting his nonsense there.
It would be better, in my opinion, to stick to math and leave out the invective. If a certain poster has gotten into your head, why not just leave the subject alone? Insults don't support your mathematical argument or convince anyone of anything.

I guess I'd add that when I'm interacting online with someone who "just doesn't get it," I also have a tendency to want to get upset. Lately I'm challenging myself to be more clear, more specific, drill down to smaller and smaller points of agreement. I think if I can't do that, it's probably better to remain silent. I'd just like to put in my two cents that civility should be the number one rule here. For everyone. Online math forums attract people who have "alternative" views of math; and those (myself included) who love to argue with them.

We should strive to remember that just as the "alternative" person is compelled to express their views; we are compelled to respond. Seven billion other people on the planet have chosen NOT to express themselves here. Maybe they're on to something.

So I say that if someone expresses ideas you don't like, then try to come up with clearer ideas. Or go read a math book, that would actually be useful! (I'm talking to myself here).

But there should never be any excuse nor any tolerance for disrespect. Zylo has violated no forum rules. On the contrary, he's provides a steady stream of content that some people love to hate. That's a service; because without "alternative" math people, what would the "counter-alternative" types do for fun?

Ok thank you for listening. I think this is a pretty cool forum and I don't like to see people insulted for their views. Online or in the real world.

Last edited by Maschke; July 27th, 2016 at 02:43 PM.

 July 27th, 2016, 02:43 PM #16 Senior Member   Joined: May 2016 From: USA Posts: 744 Thanks: 301 Well, I have previously accepted Zylo's proposition that every real in [0, 1) could be represented as a decimal number with a countably infinite number of decimal places. I merely objected to Zylo's failure to show, or even attempt to show, how the set of such numbers could be put into one to one correspondence with the natural numbers. Zylo just makes that assertion. I am being a little slow here today, but I think that M is making two new assertions. First, the proposition that every real number in [0, 1) can be represented as a decimal number with a countably infinite number of decimal places is asserted to be false. Certainly Zylo has not tried to demonstrate it to be true. I am agnostic about this assertion but admit I am no mathematician. The transfinite realm is strange. Second, M is asserting that Zylo's set can be put into one to one correspondence with the natural numbers even though Zylo has not done so. This assertion seems to be refuted by Cantor's diagonal argument. (I am now beginning to think I was quite wrong in disputing with Archie what was a useful definition of the word "list.") I am more than a bit dubious about this second assertion, but have been wrong many times. Of course, the two new assertions do not salvage Zylo's argument that the number of real numbers is countably infinite. They merely change the locus of the flaw in his argument. Whether the flaw lies in his assertion that his set includes ALL the reals or in his assertion that his set can be put into one to one correspondence with the natural numbers or in both assertions, he has to prove both and has done neither.
 July 27th, 2016, 04:07 PM #17 Math Team   Joined: Dec 2013 From: Colombia Posts: 6,876 Thanks: 2240 Math Focus: Mainly analysis and algebra I haven't re-read anything on describable numbers for a while, but I believe the idea is that a description of a number must be of finite length. So every real number has (at least) one decimal representation, but we have no way to write down most of these because the only representation we have is infinite and so we can never actually write it down. Since the number of symbols one could use is finite, the number of terminating orderings (of any finite length) of those symbols is (at most) countably infinite. This means that we have no way to exactly describe the bulk of the real numbers. After that, the subject gets very complicated because it is difficult to define a description of a number. The word "describe" has quite a wide definition here, including such things as 0.345, $\pi$, $\sqrt2$, ˝the unique real number $x$ such that $\int \limits_1^x \frac1t\,\mathrm d t=1$, "the decimal expansion of $\sqrt3$ with every numeral except 9 incremented by one", etc.. But it is limited. For example "the smallest positive number that is not describable" turns out not to describe a number. Probably "the smallest positive describable number" doesn't describe a number either. So Maschke's claim is the we can't describe more than a countably infinite number of reals using a finite number of symbols. Although this claim is true (by definition) this is definitely not the claim that Zylo is making even though one could read his claim that way. I hope this helps your understanding.
 July 27th, 2016, 04:53 PM #18 Senior Member   Joined: May 2016 From: USA Posts: 744 Thanks: 301 Archie I shall try to avoid getting into another unnecessary and probably silly argument with you, but it does not help me to understand Mashke's argument by your being condescending. Frankly, I am not quite sure what Mashke is saying, and I certainly cannot resolve what appear to be his contradictory citations. (The one says that the other is, or perhaps used to be, wrong.) I took Mashke to mean that all the real numbers in [0, 1) cannot be expressed by a countably infinite number of digits, not saying what I believe even Zylo accepts, namely that all the real numbers in [0, 1) cannot be expressed in a finite number of digits. Mashke gave no proof, and quite probably I would not understand a valid proof if it were given. But it would not surprise me at all if it takes at least a continuum of decimal places to express all the real numbers in [0, 1). That contention is consistent with (though not I suspect required by) Cantor's diagonal argument. Without giving it any thought, I had accepted Zylo's construction of the set of all real numbers in [0, 1) using decimals with a countable infinity of places. Zylo has not demonstrated that his construction does that, and I believe that Mashke is saying that Zylo cannot make that demonstration. If I understand Mashke's second point, he is saying that Zylo's set using a countably infinite number of decimal places can be put into one to one correspondence with the natural numbers. This is consistent with the proposition that the reals cannot be put into one to one correspondence with the natural numbers (because Mashke is saying that the set does not contain all the reals). However, it appears so far to be inconsistent with Cantor's diagonal argument, and so I am dubious about its validity.
July 27th, 2016, 06:19 PM   #19
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Quote:
 Originally Posted by JeffM1 it does not help me to understand Mashke's argument by your being condescending.
I have no idea why you would read that into my post, nor why you are so keen to take offence at whatever I write.

Quote:
 Originally Posted by JeffM1 I took Mashke to mean that all the real numbers in [0, 1) cannot be expressed by a countably infinite number of digits
I'm pretty sure that he isn't. Not least because that would be wrong.

Quote:
 Originally Posted by JeffM1 it would not surprise me at all if it takes at least a continuum of decimal places to express all the real numbers in [0, 1).
Decimal places are countably infinite. Each represents the number of units of size $10^{-n}$ in the number. The $n$ here serves to form a bijection between the decimal places and the natural numbers.

Quote:
 Originally Posted by JeffM1 If I understand Mashke's second point, he is saying that Zylo's set using a countably infinite number of decimal places can be put into one to one correspondence with the natural numbers.
I'm sure he isn't saying that because it flatly contradicts Cantor (as you pointed out).

The set that Maschke is talking about is the set of real numbers that can be described using a finite number of symbols (mathematical and in natural language). It is bigger than the algebraic numbers, but still countable.

Last edited by v8archie; July 27th, 2016 at 06:22 PM.

July 27th, 2016, 06:56 PM   #20
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In an earlier post, you said

Quote:
 Originally Posted by v8archie In particular you [meaning Zylo] have no knowledge of how many reals that can be defined using the natural numbers.
Maschke responded with

Quote:
 Originally Posted by Maschke Only countably many, as it turns out.
Now I do not find that to be clearly talking about finite representations. The previous discussion had been about creating a set of all the reals in [0, 1) represented by decimals with an infinite number of places. In context, what he seems to be saying is that a countably infinite number of decimal places will leave some real numbers unrepresented. You say he can't be saying that because that assertion is wrong, but offer no proof that the assertion is wrong.

I hope this helps your understanding.

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