## Can you prove these are countable?

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**2**of**2**• 1,**2**### Re: Can you prove these are countable?

Also:

"The set of symbols {"1","1/2","1/3","2/3","1/4","2/4","3/4",...} is countable. The set of unique rational numbers, which corresponds to a subset of it, is therefore also countable. How you might "find" the duplicates is irrelevant this way."

That's only the set of rationals (Q) in the range (0,1], not the whole set of rationals.

You need to extend to { 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ... },

and exclude all reducible terms (such as 2/2 = 1/1),

to get all the positive rational numbers, then thread in the negatives and zero

to get the full 1-1 correspondence between the rationals and the counting numbers.

"The set of symbols {"1","1/2","1/3","2/3","1/4","2/4","3/4",...} is countable. The set of unique rational numbers, which corresponds to a subset of it, is therefore also countable. How you might "find" the duplicates is irrelevant this way."

That's only the set of rationals (Q) in the range (0,1], not the whole set of rationals.

You need to extend to { 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ... },

and exclude all reducible terms (such as 2/2 = 1/1),

to get all the positive rational numbers, then thread in the negatives and zero

to get the full 1-1 correspondence between the rationals and the counting numbers.

- davar55
- Intellectual
**Posts:**728**Joined:**Tue Jun 13, 2006 4:24 pm**Location:**New York City

### Re: Can you prove these are countable?

A set is implemented in computer science as a one-dimensional array of bits, where there are sufficient bits for every possible element in the set.

Thus for a set having a maximum of 256 elements (for example), the set is stored in 32 bytes. Element n is in the set if bit n is 1 and is not in the set if bin n is 0. The set can be tested for element n by testing bit n for 1. The set operations intersection, union, and complement can be implemented for sets A and B with A*B, A+B, ~A (A and B, A or B, not A).

Whereas a one-dimensional array of bits is equivalent to a list, all elements of a set are necessarily countable independently of what the elements represent. However, it is impossible to implement an infinite set, so as far as it concerns computer science: infinite sets do not exist. But as an abstract object, an infinite set is equivalent to an infinite one-dimensional array, so the elements of an infinite set are necessarily countable; and this applies to the infinite counting numbers and any infinite subset of the counting numbers of necessity.

Thus for a set having a maximum of 256 elements (for example), the set is stored in 32 bytes. Element n is in the set if bit n is 1 and is not in the set if bin n is 0. The set can be tested for element n by testing bit n for 1. The set operations intersection, union, and complement can be implemented for sets A and B with A*B, A+B, ~A (A and B, A or B, not A).

Whereas a one-dimensional array of bits is equivalent to a list, all elements of a set are necessarily countable independently of what the elements represent. However, it is impossible to implement an infinite set, so as far as it concerns computer science: infinite sets do not exist. But as an abstract object, an infinite set is equivalent to an infinite one-dimensional array, so the elements of an infinite set are necessarily countable; and this applies to the infinite counting numbers and any infinite subset of the counting numbers of necessity.

- robert 46
- Intellectual
**Posts:**2841**Joined:**Mon Jun 18, 2007 9:21 am

### Re: Can you prove these are countable?

Wrong.robert 46 wrote:A set is implemented in computer science as a one-dimensional array of bits, where there are sufficient bits for every possible element in the set.

But thank you for demonstrating how you prefer to make up you own definitions in order to appear right, rather that admit what everybody can plainly see. That you are wrong.

And even if that was the CompSci definition, which it isn't, we are talking about set theory. Use this definition. You even changed the discipline from one where infinity is accepted, to one where it is not. And you expect to be taken seriously? Really?

Why don't you just try to find a flaw in the actual proof I showed to you? Is it because you can't? It's been about 14 months since you started claiming to have disproved Cantor, and you still have not addressed a single issue that is relevant to the proof.

+++++

Davar, do you now see why I don't like to even start new threads with similar topics? He will use anyway possible to divert attention from what he is ignoring.

- JeffJo
- Intellectual
**Posts:**2609**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Can you prove these are countable?

JeffJo wrote:Wrong.robert 46 wrote:A set is implemented in computer science as a one-dimensional array of bits, where there are sufficient bits for every possible element in the set.

I suggest you read the article; particularly "type theory". The method I described is exactly how sets were implemented in Turbo Pascal.

And even if that was the CompSci definition, which it isn't, we are talking about set theory.

But I was talking implementation.

You even changed the discipline from one where infinity is accepted, to one where it is not. And you expect to be taken seriously? Really?

Infinity is not accepted in computer science set implementations.

Why don't you just try to find a flaw in the actual proof I showed to you? Is it because you can't?

Why don't you just continue to resort to denial?

+++++

Davar, do you now see why I don't like to even start new threads with similar topics? He will use anyway possible to divert attention from what he is ignoring.

JeffJo regularly projects his own faults. His response to your question was definitely "half-fast". Perhaps you recognize that now.

- robert 46
- Intellectual
**Posts:**2841**Joined:**Mon Jun 18, 2007 9:21 am

### Re: Can you prove these are countable?

Maybe you should. Try the first sentence: "In computer science, a set is an abstract data type that can store certain values, without any particular order, and no repeated values. It is a computer implementation of the mathematical concept of a finite set."robert 46 wrote:JeffJo wrote:Wrong.robert 46 wrote:A set is implemented in computer science as a one-dimensional array of bits, where there are sufficient bits for every possible element in the set.

I suggest you read the article; particularly "type theory". The method I described is exactly how sets were implemented in Turbo Pascal.

I suggest you stick to fields of mathematics, and not outdated Computer Programming Languages.

And Cantor's proof is not, making what you talk about irrelevant.But I was talking implementation.

Which was my point?You even changed the discipline from one where infinity is accepted, to one where it is not. And you expect to be taken seriously? Really?

Infinity is not accepted in computer science set implementations.

Just try to show where I have "been in denial." Like that quote just above this one.Why don't you just continue to resort to denial?

- JeffJo
- Intellectual
**Posts:**2609**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Can you prove these are countable?

So what's the status of my OP question?

- davar55
- Intellectual
**Posts:**728**Joined:**Tue Jun 13, 2006 4:24 pm**Location:**New York City

### Re: Can you prove these are countable?

Nice argument Jeffjo and Davar. I am afraid that I have to agree with Jeffjo and Davar on this one robert.

love creation machine

- bill
- Intellectual
**Posts:**1262**Joined:**Sat Apr 22, 2006 2:09 pm

### Re: Can you prove these are countable?

I'm satisfied that a proof that

any subset of the integers is either finite or countably infinite

can be found in this thread.

any subset of the integers is either finite or countably infinite

can be found in this thread.

- davar55
- Intellectual
**Posts:**728**Joined:**Tue Jun 13, 2006 4:24 pm**Location:**New York City

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