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July 18th, 2016, 01:03 PM   #11
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Originally Posted by v8archie View Post
The suggestion that a number/sequence that does not terminate is the same as (in fact is) a number/sequence that terminates is very obviously untrue in general however you list them unless you have a proof to the contrary.

Zylo's process has no irrational numbers in it. It has few of the rationals in fact because every number in his list terminates.
As I said, Zylo's use of language is extremely imprecise. I do not interpret Zylo's list as intended to be finite. Given his imprecision, it is hard to be sure. But I doubt even Zylo believes that any finite list contains all the reals in the interval [0, 1). I believe that he is positing an infinite list of sums, each with infinite summands. Such a list might include

$\displaystyle \pi - 3 = -\ 3 + 4 \sum_{n=1}^\infty \left ((-\ 1)^{i+1} * \dfrac{1}{2i - 1} \right).$

There exist irrational numbers in the interval (1, 2). One such is the square root of 2. Take its reciprocal and put it in the list. In the interval (2, 3), there is the irrational number 1 plus the square root of 2. Take its reciprocal and put it into the list. In the interval, (3, 4) .... That is a conceptually non-terminating process for generating a countably infinite list of unduplicated irrationals. So it is certainly conceptually possible to set up a list containing a countably infinite number of irrationals in the interval (0, 1). I just gave a constructive proof. But I did not show that the list contains every irrational in the interval (0, 1).

Dan has pointed out that the process indicated by Zylo will generate in a finite number of steps a finite number of ever better approximations to $\pi - 3.$

Dan may also be saying that it is indeterminate whether treating the process as non-terminating means that the resulting non-terminating list of sums with non-terminating summands will include $\pi - 3.$ But let's assume that it will (though I agree with you that Zylo has not proved it will).

It still does not do what Zylo thinks it does. He has not shown that his non-terminating list contains all the reals in (0, 1). He is never going to show that because of Cantor's diagonal argument. Any process such as mine shown above that generates a denumerably infinite list of irrationals in the interval (0, 1) does not generate them all.

Where Zylo gets annoying is not that he always fails to describe non-terminating processes. It is that when he does describe a non-terminating process, he just assumes the final step, the one linking each member of the list to the natural numbers in a defined way. Which natural number is $\pi - 3$ to correspond to? He can't just say that is in the list. He has to say where. I do not think he really understands the rules of the game.

EDIT While I was writing this Zylo put up a new post. Here he does seem to be advocating a purely finitistic approach. In that case he should not be dealing with Cantor's argument at all, but saying that mathematics should deal with what is physically observable, which is always a finite number of rationals, and putting that on a solid logical foundation.
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July 18th, 2016, 04:27 PM   #12
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I doubt even Zylo believes that any finite list contains all the reals in the interval [0, 1).

While I was writing this Zylo put up a new post. Here he does seem to be advocating a purely finitistic approach. In that case he should not be dealing with Cantor's argument at all, but saying that mathematics should deal with what is physically observable, which is always a finite number of rationals, and putting that on a solid logical foundation.
You are vastly overestimating Zylo's abilities and understanding. He defines finite lists of terminating numbers of $n$ digits. He then attempts, wrongly, to let $n$ become infinite while remaining a natural number. This is because he doesn't understand the difference between potential and actual infinities and he doesn't know what a natural number is. He falsely claims then to have an infinite list of non-terminating numbers.

He then claims to unify his finite and infinite lists of terminating and non-terminating numbers to get results that are nonsense.

He doesn't understand any of the reasons why his guesswork is nonsense, instead believing that assumption is a rigorous form of proof. He's too stubborn to attempt to learn any of this stuff properly.

What he is doing is nothing to do with finitism. He claims to use infinite structures. Real finitists don't bother with Cantor because he uses an axiom that they don't wish to accept. Zylo just doesn't know what he is talking about.
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July 19th, 2016, 06:01 AM   #13
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I'll try once more despite the ad hominem attacks.

Cantor HYPOTHESIZES a denumerably infinite list in his diagonal proof. People who accept that proof do not ask for photographs of the list. They do not ask for notarized affidavits from qualified witnesses that they confirmed either actual or potential infinities. Infinity is a mental concept dealt with in the imagination.

I am not sure whether the process described by Zylo at the start of this thread would construct all real numbers in the interval [0, 1) if continued without termination. It seems plausible that it does, but plausibility about the infinite is close to contradictory and is certainly not proof. However, whether that process continued without termination constructs all real numbers in [0, 1) is completely irrelevant. Zylo could just as well start with a HYPOTHESIZED list of all real numbers in [0, 1) without any fuss about a process for constructing it as follows:

$\mathbb S = \{s\in\ \mathbb S \iff s \in \mathbb R\ and\ 0 \le s < 1\}$. By definition then

$0 \in \mathbb S,\ 0.25\in \mathbb S,\ \sqrt{2} - 1 \in \mathbb S,\ e - 2\in \mathbb S,\ and\ 3 - \pi \in \mathbb S.$

And here is where, every time, Zylo misses the boat. He thinks because he has in his imagination a set of every real in [0, 1), he has somehow disproved Cantor. But of course he has not. He must show how to make each element in his set correspond one to one with the natural numbers.

The trick is not to imagine the reals in [0, 1). Mathematicians do it every day. The hard part (actually the impossible part) is to show how to bring them into 1-to-1 correspondence with the natural numbers.
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July 19th, 2016, 09:26 AM   #14
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I'll try once more despite the ad hominem attacks.
I haven't attacked you at all. I'm not looking for an argument. Please stop trying to pick one with me.

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Originally Posted by JeffM1 View Post
Cantor HYPOTHESIZES a denumerably infinite list in his diagonal proof. People who accept that proof do not ask for photographs of the list. They do not ask for notarized affidavits from qualified witnesses that they confirmed either actual or potential infinities.
It's a never-ending list. It has an infinite number of elements. It is what is known as an actual infinity. It is not the limit of something as some parameter grows without bound. That would be a potential infinity. Potential infinities do not exist, even in the mathematical imagination. They are just the limit as some finite parameter grows arbitrarily large but remains finite.

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I am not sure whether the process described by Zylo at the start of this thread would construct all real numbers in the interval [0, 1) if continued without termination.
It wouldn't. His decimals are all of finite length $n$ where $n \in \mathbb N$.

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Zylo misses the boat. He thinks because he has in his imagination a set of every real in [0, 1), he has somehow disproved Cantor.
He doesn't. He thinks he has, but that is because he doesn't know the difference between actual infinities and potential infinities. He doesn't understand that a limit is not a part of the sequence (or function) of which it is a limit. It is a separately defined mathematical object.
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July 19th, 2016, 10:15 AM   #15
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It wouldn't. His decimals are all of finite length $n$ where $n \in \mathbb N$.
If that is indeed what he is trying to say, you and I are in total agreement. He hasn't even defined something with all the rationals, and it cannot be put into 1-to-1 correspondence with the set of natural numbers because it is finite.

Where you and I seem to differ is this. I think (but am not sure because he is not a careful writer) he is imagining the set of infinite decimal representations, repeating or non-repeating, of all real numbers in [0,1).

If that is correct, there is no problem with the set (though there may be with how he describes it). That set certainly does include $3 - \pi$ or any other real number in the designated interval by definition.

He then needs to show (to disprove Cantor) how to put that set into 1-to-1 correspondence with the natural numbers. He never does this; he assumes it.

He wastes everyone's time with constructions of a set that, for those who are not finitists, needs no construction.
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July 19th, 2016, 10:34 AM   #16
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He is "imagining" the set of reals and trying to produce a list (or in this case, a countably infinite set of lists) of them. But his constructions never contain all the real numbers and they nwver will while he persists in trying to create a list because such a list impossible (Cantor).
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July 19th, 2016, 11:11 AM   #17
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Analysis can be dealt with very nicely with the concepts of finite and limit as n -> infinity (countable), based on the natural numbers and fractions (continuum).
The rational numbers are not the continuum. I'm not even sure that they are even a continuum. They certainly aren't complete which is required for general analysis - specifically limits. But since analysis uses numbers, it naturally uses only finite things. There is no infinity in real analysis except as a linguistic shorthand for unboundedness. No interval ever includes infinity because infinity does not exist.

But that doesn't really matter, because you aren't talking about analysis. You are talking about set theory. In set theory infinities do exist (but they are not called infinities).
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July 19th, 2016, 11:47 AM   #18
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He is "imagining" the set of reals and trying to produce a list (or in this case, a countably infinite set of lists) of them. But his constructions never contain all the real numbers and they nwver will while he persists in trying to create a list because such a list impossible (Cantor).
Oh we do disagree there. (Not in a bad tempered way.)

What is the difference between an imaginary mental list with infinite lines and a set with infinite elements?

Cantor did not prove that the set of reals was impossible so he did not prove that an imaginary list of them was impossible. (Why does the mind boggle at an imaginary list with infinite lines if the infinity is the number of the continuum but not boggle at an imaginary list with infinite lines if the infinity is aleph null.) The list is just a visualizing metaphor for a set. But of course each line of Cantor's "list" implicitly contained two numbers, a natural number and an irrational number. What he proved was that such a list would not contain all the irrational numbers in (0, 1).

Cantor's proof does not depend in any formal way on a list. It depends on hypothesizing a one-to one mapping of an arbitrary set of irrationals onto the set of natural numbers, and then showing how to construct an irrational that is not in the first set, thereby showing that the irrationals cannot be put into one to one correspondence with the natural numbers.

Now I do not know what is in Zylo's mind. If he is thinking of a finite set, he clearly cannot prove it to be infinite of any variety. So it seems to me more charitable to assume that he is indeed ULTIMATELY thinking of the set of all reals in [0, 1) arrived at through whatever psychological tricks of visualization, which he mistakenly thinks are relevant.

Granting that, he simply does not even try to put that set into one to one correspondence with the natural numbers.
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July 19th, 2016, 03:28 PM   #19
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What is the difference between an imaginary mental list with infinite lines and a set with infinite elements?
A list is automatically in 1-1 correspondence with the natural numbers because it has a first element, a second element, ..., an $n$th element. Some mathematicians use the word "listable" in preference to "countable".
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July 19th, 2016, 05:48 PM   #20
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A list is automatically in 1-1 correspondence with the natural numbers because it has a first element, a second element, ..., an $n$th element. Some mathematicians use the word "listable" in preference to "countable".
If that is the common definition of "list," then clearly Cantor did prove that real numbers cannot be put into a "list" as defined. I must admit that when I think of a list without an end, I see no reason to suppose that it must have a beginning either, but it is silly to argue over definitions of words. So I concede the point.

In any case, it is really irrelevant. Nothing in Cantor's proof depends on a "list" however defined. What he posits is a one-to-one correspondence between an arbitrary set of irrational numbers in the interval (0, 1) and the set of all natural numbers. The crux of his proof is an explanation of how to construct an irrational number not in the first set, which proves that the first set does not contain all irrational numbers in (0, 1). It is an unbelievably spare and elegant proof.

So where you and I seem to disagree is on a relatively small point. You do not believe that Zylo ever has in mind an infinite set, let alone a set of all the real numbers in (0, 1). If you are correct, his argument is infantile because a finite set cannot by definition be put into one-to-one correspondence with an infinite one. On the other hand, I believe that Zylo has in mind (at least some of the time) the set of all real numbers in the interval [0, 1) or (0, 1). He does not, however, even attempt (and would necessarily fail if he did attempt) to show that the set is in one-to-one correspondence with the natural numbers.

So all I am saying is that his efforts to construct the set of real numbers in [0, 1) may or may not be nonsense. I do not care because I am willing to accept either set conceptually. What strikes me is his absence of any understanding that, to prove his point, he must construct a one-to-one correspondence between that set and the set of natural numbers.

I'll let you have the last word on this because ultimately we are both saying that Zylo's demonstrations fail, albeit for different reasons.
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