Are the number of operations countable or uncountable?

[agentredlum]

Georg Cantor also gave examples of subsets of the real line with unusual properties

This is exactly what i was talking about in all messages but was unable to word it so eloquently.

I would loooooooove to see these unusual subsets!

Check this out http://en.wikipedia.org/wiki/Cantor_set

Here is a sentence from this article

"It has been conjectured that all algebraic irrational numbers are normal. Since members of the Cantor set are not normal, this would imply that all members of the Cantor set are either rational or transcendental"

OH MY GOD! If this conjecture is true then that means there is a subset of the Reals that collects the Rational with Transcendental by some method. To me this would be an AMAZING result because that method would uncover a PROPERTY shared ONLY by Rational and Transcendental numbers. Very fantastic!

 

[SteveL27]

Check this out http://en.wikipedia.org/wiki/Cantor_set

Here is a sentence from this article

"It has been conjectured that all algebraic irrational numbers are normal. Since members of the Cantor set are not normal, this would imply that all members of the Cantor set are either rational or transcendental"

OH MY GOD! If this conjecture is true then that means there is a subset of the Reals that collects the Rational with Transcendental by some method. To me this would be an AMAZING result because that method would uncover a PROPERTY shared ONLY by Rational and Transcendental numbers. Very fantastic!

Yes, but it's not even known if sqrt(2) is normal, let alone all of the algebraic numbers. Maybe all the algebraic numbers are normal, maybe not. It's not known.

http://en.wikipedia.org/wiki/Normal_number

On the other hand, here is a set that contains only the rationals and transcendentals: just take the union of the rationals and transcendentals. Membership in that union is a "property shared only by rational and transcendental numbers." And you can prove that today, not just conjecture it.

Perhaps that's less "profound," as you put it earlier, than the Cantor set. But membership in the union of the rationals and the transcendentals is a "property." So it counts. What do you think about that? All it really is, is the complement of the algebraic numbers. It's every real number that's not algebraic. Is that of interest to you?

The Cantor set is a very interesting set, but for other reasons. Among other things, it's an uncountable set of measure zero, so it's a standard example in real analysis.

 

[agentredlum]

Yes, but it's not even known if sqrt(2) is normal, let alone all of the algebraic numbers. Maybe all the algebraic numbers are normal, maybe not. It's not known.

http://en.wikipedia.org/wiki/Normal_number

On the other hand, here is a set that contains only the rationals and transcendentals: just take the union of the rationals and transcendentals. Membership in that union is a "property shared only by rational and transcendental numbers." And you can prove that today, not just conjecture it.

Perhaps that's less "profound," as you put it earlier, than the Cantor set. But membership in the union of the rationals and the transcendentals is a "property." So it counts. What do you think about that? All it really is, is the complement of the algebraic numbers. It's every real number that's not algebraic. Is that of interest to you?

The Cantor set is a very interesting set, but for other reasons. Among other things, it's an uncountable set of measure zero, so it's a standard example in real analysis.

HELLO STEVE!

We seem to have wiki articles that contradict themselves. The article i posted says the Cantor set is uncountable, so its uncountability must come from the transcendental elements. It also says all elements in the Cantor set are not normal. The article you posted says most numbers are normal.

They can't both be right...but they can both be wrong LOL

Rational numbers are algebraic. So 'every real number that's not algebraic' describes the set of all transcendental. Rational numbers are absent from your property.

http://en.wikipedia.org/wiki/Algebraic_number

Steve, membership in a set is not a property, you must provide a METHOD for inclusion in the set. That METHOD usually uncovers a shared property among all members of the set.

The Cantor set provides a METHOD for putting rationals with transcendentals if the conjecture is true.

You can't just put them together by DEFINITION, somebody is going to ask you WHY? They shouldn't be put together just cause someone said so, that's not a logical scientific reason, no one is going to take it seriously.

 

[SteveL27]

HELLO STEVE!

We seem to have wiki articles that contradict themselves. The article i posted says the Cantor set is uncountable, so its uncountability must come from the transcendental elements. It also says all elements in the Cantor set are not normal. The article you posted says most numbers are normal.

They can't both be right...but they can both be wrong LOL

The phrase "almost all" is defined in real analysis as "all except for a set of measure zero." The Cantor set has measure zero, so there's no inconsistency between the two articles.

And by the way, although we're both posting a lot of Wiki links here, it's worth mentioning in passing that Wiki is whatever anonymous people type into it. It's not an edited encyclopedia written by authorities in any field. You have to read Wiki with a critical eye.

Rational numbers are algebraic. So 'every real number that's not algebraic' describes the set of all transcendental. Rational numbers are absent from your property.

Yes of course, I misspoke myself and meant to say, "every real number that is not an irrational algebraic."

Steve, membership in a set is not a property, you must provide a METHOD for inclusion in the set. That METHOD usually uncovers a shared property among all members of the set.

The Cantor set provides a METHOD for putting rationals with transcendentals if the conjecture is true.

You are using a very different notion of set formation than is standard in mathematics. The axioms of set theory say nothing about "methods," and in fact there are many sets commonly used in math that can't be constructed at all. One can only prove their existence. The most common example is the Vitali set, which provides the standard example of a nonmeasurable set.

http://en.wikipedia.org/wiki/Vitali_set

Note that the article refers to the "construction" of the set, but that's a misnomer in my opinion. There is an existence proof but not a construction. You can't identify any of the elements of this set.

The Cantor set has a very simple description: It's the set of base-3 expansions of real numbers that don't contain the digit 1. That's no different than defining a set as the union of some other sets.

You will be hard-pressed to come up with a definition of the word "method" that would provide for very many interesting sets; and you would need to rewrite a new version of set theory if you were going to insist that there should be some kind of "method" to defined a set.

You can't just put them together by DEFINITION, somebody is going to ask you WHY? They shouldn't be put together just cause someone said so, that's not a logical scientific reason, no one is going to take it seriously.

I already mentioned that my example is trivial; but it has the virtue of having the property you said you were interested in: it contains only rationals and transcendentals, but no irrational algebraics.

You claim the Cantor set has that property, but that is only conjectured, not proven.

I already pointed out that my set is not very "profound," which is a word you used earlier. But it is formed by the legal rules of set formation; as is the Cantor set. The word "method" is vague. If you want to talk about it, you have to define it.

 

[agentredlum]

The phrase "almost all" is defined in real analysis as "all except for a set of measure zero." The Cantor set has measure zero, so there's no inconsistency between the two articles.

And by the way, although we're both posting a lot of Wiki links here, it's worth mentioning in passing that Wiki is whatever anonymous people type into it. It's not an edited encyclopedia written by authorities in any field. You have to read Wiki with a critical eye.
Yes of course, I misspoke myself and meant to say, "every real number that is not an irrational algebraic."You are using a very different notion of set formation than is standard in mathematics. The axioms of set theory say nothing about "methods," and in fact there are many sets commonly used in math that can't be constructed at all. One can only prove their existence. The most common example is the Vitali set, which provides the standard example of a nonmeasurable set.

http://en.wikipedia.org/wiki/Vitali_set

Note that the article refers to the "construction" of the set, but that's a misnomer in my opinion. There is an existence proof but not a construction. You can't identify any of the elements of this set.

The Cantor set has a very simple description: It's the set of base-3 expansions of real numbers that don't contain the digit 1. That's no different than defining a set as the union of some other sets.

You will be hard-pressed to come up with a definition of the word "method" that would provide for very many interesting sets; and you would need to rewrite a new version of set theory if you were going to insist that there should be some kind of "method" to defined a set.
I already mentioned that my example is trivial; but it has the virtue of having the property you said you were interested in: it contains only rationals and transcendentals, but no irrational algebraics.

You claim the Cantor set has that property, but that is only conjectured, not proven.

I already pointed out that my set is not very "profound," which is a word you used earlier. But it is formed by the legal rules of set formation; as is the Cantor set. The word "method" is vague. If you want to talk about it, you have to define it.

Man A: "Why did you put them in the same set?"
Man B: "Because they have the same property"
Man A: "What property is that?"
Man B: "The property of placing them in the same set"

This sounds like circular reasoning steve, set theory allows circular reasoning?


You already gave me a great answer with 'or equal to' so i still say you are brilliant.,let us not argue over semantics...Have a great day