Cantor Diagonal Argument disproof

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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Fri Jun 02, 2017 10:01 am

JeffJo wrote:
robert 46 wrote:
JeffJo wrote:They do with parametric equations such as the functions x(t), y(t), z(t) producing the position of a point in time.
In physics, yes, the PARAMETRIC EQUATIONS define paths. Points themselves don't. And besides, this is Math Class.

What part of parametric equations is not mathematical?
There is no disagreement that points can be put on a line.
There certainly is. Points are not "put" anywhere. They ARE the endpoints of line segments.

If a line is y=m*x+b then if p1=(x1,y1) and p2=(x2,y2) are two points which satisfy the equation then they are put on the line to define a line segment.
Naïve set theory was loaded with contradictions. You can define a fantasy as rigorously as you like, but that doesn't mean it is sensible.
Then it's a good thing that the Axiom of Infinity, and the Diagonalization Theorem, belong to Axiomatic Set Theory,, isn't it?

Not considering that there are set theories with and without the Axiom of Choice.
I was showing that JeffJo was setting up a trick question.
robert was, and still is, evading the question. Are there, or are there not, more line segments in [0,1] than in [0,10]? Explain all you want, but provide a yes or no with it.

You have said that "yes" and "no" are both valid answers, so I have no interest in choosing one.
Where an answer is "yes and no" there must be an equivocation.
You mean, like you are doing now? To evade the question?

I invite you to explain the equivocation which gives both "yes" and "no" answers.
So why does a system need to have infinite sets???
Because when you don't have them, you get your nonsensical answers.

JeffJo's projection. Finitists disagree.
Thank you for finally expressing that thought,

Ambiguous as to which thought JeffJo is referring.
Only if you refuse to grasp the point that infinite segments, and "infinite+1" (whatever that is supposed to mean) endpoints is the nonsensical part.

Well, I said it was nonsensical.
and showing why a Mathematics based on "actually producing" every element t of such a set is "nonsensical".

Recall that we are interested in producing an example, or producing an algorithm which would produce an example.
No, recall that what we are interested is defining an example,

Well, anyone can define a unicorn however they like.
and that robert refuses to define what he thinks "producing" and "algorithm" mean, or why these undefined concepts are superior to a Mathematically valid algoritm that defines an infinite string.. He just insists that Cantor didn't do it, but won't say what was not done.

What was not done is get to the end of an endless list to produce a string ~D. What was done is define having gotten to the end of an endless list to define a string ~D. Baum defined a tornado taking Dorothy and Toto to the land of Oz to meet the wizard.
The nth character of ~D is the opposite character of the nth character of the nth string in the mapping we assume exists. There is no part of ~D that is not "produced" by this well-defined algorithm; it is just a random-access algorithm instead of the sequential one robert seems to think is needed, but refuses to explain why.

Because there is no reason to think that an endless parallel access is any more valid than an endless sequential access.
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Re: Cantor Diagonal Argument disproof

Postby JeffJo » Fri Jun 02, 2017 5:03 pm

Robert uses many ploys to evade a point he can't address. One is misdirection:
robert 46 wrote:What part of parametric equations is not mathematical?
The equations themselves? No part. What is being modeled? All of it. Math is the tool the Physics uses to model the universe.

A other ploy is misinterpretation:
There is no disagreement that points can be put on a line.
There certainly is. Points are not "put" anywhere. They ARE the endpoints of line segments.

If a line is y=m*x+b then if p1=(x1,y1) and p2=(x2,y2) are two points which satisfy the equation then they are put on the line to define a line segment.
If y1=m*x1+b, then it is on that line whether or not you express it in an equation.

And the point here, that robert is trying to brush under the rug, is that the parametric form y=m*x+b does define every possible point, in the Mathematical sense. An infinity of them in a continuum, on the line. His point was that Physics may not recognize all of them, which is irrelevant. As was bringing up space-time.

Here's misinterpretation mixed with another ploy, contradicting positions he has claimed before:
Not considering that there are set theories with and without the Axiom of Choice.
So is robert finally admitting that Mathematics is not a single field of study, but is based on the axioms one choses to accept? But note that the Axiom of Choice is the "C" in ZFC, or Zermelo–Fraenkel set theory with the axiom of choice. Which includes the Axiom of Infinity, and Cantor Diagonalization as a theorem. As does plain ZF.

And then there is pure evasion:
You have said that "yes" and "no" are both valid answers, so I have no interest in choosing one.
I have said they are both valid in Cantor's mathematics. Robert uses a different Mathematics, which he won't define. In it, the answer to the question has to be one, or the other. So robert uses any excuse he can to not answer.

Then there's parroting, with changed meaning:
I invite you to explain the equivocation which gives both "yes" and "no" answers.
I have. In modern set theory, an infinite set can be matched 1:1 with a strict subset of itself. This allows it to be considered, by robert's implied definitions, any of larger, smaller, or the same size as itself. The flaw isn't the theory of infinite sets, it is trying to use robert's unstated, but necessarily flawed definitions of what '"larger, smaller, or the same size" means with them. And this is why robert needs to answer the question with either yes or no, or explain how both can be true (which he has denied).

Yet another ploy is ignoring facts by calling them "projection"
So why does a system need to have infinite sets???
Because when you don't have them, you get your nonsensical answers.

JeffJo's projection. Finitists disagree.
Finitists are the ones who can't answer the question I asked, so their mathematics is flawed.

More evasion:
Well, I said it was nonsensical.
And I said it was the form of robert's question, not the concept robert was criticizing, that was non-sensical.

And yet another ploy is evasion by hyberbole:
Well, anyone can define a unicorn however they like.
The issue robert evades here is that he refuses to define the procedures he insists Cantor's theorem must follow. The definitions he compares to imaginary creatures are the quite real, and valid, definitions that robert doesn't want to accept.

What was not done [by Cantor] is get to the end of an endless list to produce a string ~D.
What is done, is the provision of an algorithm that is capable of "producing" any character in the string. What is not done by robert, is justifying why it has to be "produced" sequentially, what he means by reaching the end of an endless list, or how he can assume the existence of the endless list that is required as input to the algorithm, when he insists on this absurd requirement.

I'll ignore the hyperbole in that part. It's just more evasion.

Because there is no reason to think that an endless parallel access is any more valid than an endless sequential access.
There is also no reason to think it is less valid. At least, robert hasn't even attempted to provide one, only to assert that sequential access is required.
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Re: Cantor Diagonal Argument disproof

Postby Markket » Thu Jun 15, 2017 4:40 am

Very Thanks.
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Re: Cantor Diagonal Argument disproof

Postby robert 46 » Thu Jun 15, 2017 7:59 am

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