# Are the number of operations countable or uncountable?

by agentredlum
Tags: countable, number, operations, uncountable
 P: 460 I haven't got the hang of this yet, the 1st paragraph under the quote is also a quote from steveL27
P: 800
 Quote by agentredlum I haven't got the hang of this yet, the 1st paragraph under the quote is also a quote from steveL27
I'm looking at your markup (by hitting Reply but not writing a reply to the earlier post) and you have an open QUOTE and a close QUOTE around your entire post. If you just surround the parts I say with QUOTE tags and leave your own text outside of QUOTE tags it all works.

In general I use the Preview Post button till I have the markup right. It usually takes me much longer to do the markup than it does to just write down the post itself.

I appreciate your crediting me with brilliance but there is no substantive difference between less-than and less-than-or-equal, leading me to think I'm not communicating the meaning of this example to you.

If $\{x_1, x_2, ...\}$ is a countable set of transcendentals, then the sets

$A_{x_n} = \{x \in R : x$ is transcendental and $x_n \lt x \}$

are each an uncountable set of transcendentals that differ from any other uncountable set of transcendentals.

The sets

$A_{x_n} = \{x \in R : x$ is transcendental and $x_n \leq x \}$

are also each an uncountable set of transcendentals that differ from any other uncountable set of transcendentals. It only differs from the other set because it includes $x_n$.

I posted that example in response to your request to show some property that can be used to distinguish one uncountable set of transcendentals from another.

But you can only form these sets for a countable number of transcendentals $x_n$, because you have no way of naming more than countably many transcendentals.

This point is valid no matter whether you illustrate it using $\lt$ or $\leq$.
P: 460
 Quote by HallsofIvy What do you mean by "distinct property"? Every number has a "distinct property" different from every other number- that of being itself. And the fact that sqrt(2) is algebraic is missing from the statement that it is irrational. I still don't see what point you are trying to make. Yes, of course, in any set of numbers, you can find some properties that distinguish some of them from others. But what, exactly, are you looking for?

Quote:every number has a "distinct property" different from every other number-that of being itself.

Forgive me but how is this a "distinct property" if all numbers have this property?

I do not like to play with words but you have stated the obvious. What are you saying? 2 is special because it's not 3? Well 5 is not 3 either or A is not B for any 2 different real numbers. I need a little more than that. What you have done is put every real number in a set with one element...ITSELF!!

I apologize for not saying sqrt(2) is algebraic. Personally i believe what makes sqrt(2) AMAZING is the fact it is irrational, algebraic comes later in importance to me. THE IRRATIONALITY OF SQRT(2) EXPELLED IT FROM THE PYTHAGOREAN PARADISE OF COMMENSURABLE RATIONALITY.

In my opinion pi is AMAZING because it is transcendental. An irrational number that is not like sqrt(2). If Pythagoras would raise an eyebrow at the irrationality of sqrt(2)...he would do a double-take at the transcendant nature of pi. What i'm saying is that in some sense pi has a 'higher' level of irrationality than sqrt(2)

Question:Are there only two levels of irrationality? sqrt(2) being in one level, pi in another level. Are there any levels between sqrt(2) level and pi level? Are there any levels above pi level?

A level can have many elements. All algebraic irrational are probably on the same level. However all irrational are NOT on the same level (notice the word algebraic is absent from the last sentence)

Thank you again for your time. Have a great day!
P: 800
 Quote by agentredlum Thank you very much for your reply. Please consider the following Quote:every number has a "distinct property" different from every other number-that of being itself.
I am in complete agreement with you on this point. Each real number has one property that completely characterizes that number as being unique among all the other reals: namely, the property of being itself.

I absolutely agree with you. $\forall x \in \mathbb{R}, x = x$. I believe that!

The problem is that we don't have names for all of them. Forgive me here for trying to stretch an analogy. You own dog boarding facility. You run the place based on the tag number of each dog. You know what cage they're in, what time they need to be fed, any special health or medical issues, any special instructions from the owner, etc. It's all keyed off the tags.

As it happens, you have exactly 100 tags, and your tag maker is on vacation. So you can't get any more tags. And one day you get a lot of business and you have 200 dogs.

You can keep track of 100 of them. But you have no way to say that #150 is different than dog #151, because you just have no idea. You have no names for them so even though you know they're there, because you can see them with your own eyes; nevertheless, it is still a meaningless question to say, "How many collies do I have?" or "Fido gets a vegan dinner." Since you can't name them, you can't deal with them at all.

That's the problem with the real numbers. We can prove there are uncountably many of them. But we only have names for countably many of them. If x and y are two distinct members of this set of undefinable numbers, then the only things we can say about them follow from the axioms for the reals.

We can say that either x < y or y < x (we already said they're distinct.). We can say that if we multiply x times y, then we multiply y times x, we will get the same answer.

Anything that's true about the reals, we can say about x and y. But we can't say anything else about them. Because we have no names for them.

What do you think about that?
P: 460
 Quote by SteveL27 I'm looking at your markup (by hitting Reply but not writing a reply to the earlier post) and you have an open QUOTE and a close QUOTE around your entire post. If you just surround the parts I say with QUOTE tags and leave your own text outside of QUOTE tags it all works. In general I use the Preview Post button till I have the markup right. It usually takes me much longer to do the markup than it does to just write down the post itself. I appreciate your crediting me with brilliance but there is no substantive difference between less-than and less-than-or-equal, leading me to think I'm not communicating the meaning of this example to you. If $\{x_1, x_2, ...\}$ is a countable set of transcendentals, then the sets $A_{x_n} = \{x \in R : x$ is transcendental and $x_n \lt x \}$ are each an uncountable set of transcendentals that differ from any other uncountable set of transcendentals. The sets $A_{x_n} = \{x \in R : x$ is transcendental and $x_n \leq x \}$ are also each an uncountable set of transcendentals that differ from any other uncountable set of transcendentals. It only differs from the other set because it includes $x_n$. I posted that example in response to your request to show some property that can be used to distinguish one uncountable set of transcendentals from another. But you can only form these sets for a countable number of transcendentals $x_n$, because you have no way of naming more than countably many transcendentals. This point is valid no matter whether you illustrate it using $\lt$ or $\leq$.
The equations you post my browser shows me as $\It$ I don't understand that...I'm using Playstation 3 so maybe thats why. I don't know how to fix this. PS3 browser does not decode 100% i've had problems elsewhere also.

I agree with most of your points you have made it abundantly clear that almost all numbers are an un-namable amorphous blur. I never disputed that.

Let me ask the question another way.

Question:Do you think that some day we will discover a new class of numbers within the transcendentals?

If you ask me what i mean by class i'm going to slap myself LOL

There are classes of numbers. Integers...Rationals..Irrationals ..Algebraic..Transcendental ..Reals...Complex

Classes can have subclasses and subclasses can have sub-subclasses.

Example:The rationals are a subclass of the Reals, the integers are a subclass of the rationals

Like i said, putting 'or equal to' makes it work, you don't think that's true?

What about the example i gave with 2e and 3e? doesn't that prove my point that 'or equal to' is necessary?

Under the stated property the set 2e and 3e are not different. They have the same defining property they both have all their elements greater than pi. Sure, they both have different elements but that does not make them dfferent sets as far as this property ALONE is concerned.

When you say all transcendentals greater than or equal to pi then there is only one set that has this property ALONE. You cannot find another set with this property, if you do it will be THE SAME SET, IDENTICAL. This is due to trichotomy law of ordering over the reals

Trichotomy law for Reals: A<B or A=B or A>B Given any A, B Real, only one of these conditions holds.
P: 460
 Quote by SteveL27 I am in complete agreement with you on this point. Each real number has one property that completely characterizes that number as being unique among all the other reals: namely, the property of being itself. I absolutely agree with you. $\forall x \in \mathbb{R}, x = x$. I believe that! The problem is that we don't have names for all of them. Forgive me here for trying to stretch an analogy. You own dog boarding facility. You run the place based on the tag number of each dog. You know what cage they're in, what time they need to be fed, any special health or medical issues, any special instructions from the owner, etc. It's all keyed off the tags. As it happens, you have exactly 100 tags, and your tag maker is on vacation. So you can't get any more tags. And one day you get a lot of business and you have 200 dogs. You can keep track of 100 of them. But you have no way to say that #150 is different than dog #151, because you just have no idea. You have no names for them so even though you know they're there, because you can see them with your own eyes; nevertheless, it is still a meaningless question to say, "How many collies do I have?" or "Fido gets a vegan dinner." Since you can't name them, you can't deal with them at all. That's the problem with the real numbers. We can prove there are uncountably many of them. But we only have names for countably many of them. If x and y are two distinct members of this set of undefinable numbers, then the only things we can say about them follow from the axioms for the reals. We can say that either x < y or y < x (we already said they're distinct.). We can say that if we multiply x times y, then we multiply y times x, we will get the same answer. Anything that's true about the reals, we can say about x and y. But we can't say anything else about them. Because we have no names for them. What do you think about that?
OH MAAAAAAN! That first quote was by someone else...I agree with it but I find it trivial. You didn't read my post carefully, you couldn't tell i was a little irritated and a little beligerant. Sorry HallsofIvy, my attitude could not be expressed as the quotient of two integers, it was irrational LOL
 P: 460 We can say that either x < y or y < x (we already said they're distinct.). We can say that if we multiply x times y, then we multiply y times x, we will get the same answer. It is very interesting that while i was writing about Trichotomy Law you had already posted it. This happened twice to me today. I was writing about PLANCK LENGTH and at the same time i was writing someone was posting in the same thread about PLANCK LENGTH. Somebody call Rupert Sheldrake cause this may be a good example of his Morphogenetic Field Theory. Once i could dismiss as a coincidence, but twice in a few hours?
P: 800
 Quote by agentredlum We can say that either x < y or y < x (we already said they're distinct.). We can say that if we multiply x times y, then we multiply y times x, we will get the same answer. It is very interesting that while i was writing about Trichotomy Law you had already posted it. This happened twice to me today. I was writing about PLANCK LENGTH and at the same time i was writing someone was posting in the same thread about PLANCK LENGTH. Somebody call Rupert Sheldrake cause this may be a good example of his Morphogenetic Field Theory. Once i could dismiss as a coincidence, but twice in a few hours?
Each subject has a natural progression. When you start thinking about the real numbers, you start thinking about the properties of the real numbers. So you think about their order, and the field axioms that let you add and multiply them, and even divide them provided you are not dividing by zero. It's not a cosmic coincidence. It's just the natural thing to think of when we ask ourselves the question: What can I know about an unnamable real number?

Once you ask that, you are pretty much going to end up with the axioms for a complete ordered field. Which is where we started ... with the algebraic notion of an element transcendental over a field.

So in fact it's inevitable to have these particular thoughts. No additional theories needed.

(ps) -- I did just see your other response, which I didn't read yet. I will respond to it when I get a chance.

I did notice that you mentioned you might be having some trouble reading my TeX markup. That's ironic because I'm in the process of learning TeX, so I use it every chance I get.

If you like, I would be glad to devolve back to ASCII math. If it will make any of this clearer I'll certainly do that.

And also someone said there's a browser bug where the TeX doesn't render; but if you refresh your browser window, then it renders properly. I see that all the time on my browser and refreshing always works.
P: 460

Maybe i should learn how to use these things...got the smiley faces down. Please keep using TeX, that way you will master it faster. I don't know what TeX is...is it on the toolbar like smiley face?

Your ideas are very clear to me. I agree with 100% about uncountability, un-namability and so on. Even though i agree with your presentation, i don't see how those facts prohibit my question.

You say we can't name them. BEFORE MANKIND DISCOVERED TRANSCENDENTALS WE DID NOT HAVE A NAME FOR THEM BUT NOW WE DO.

You say we can't describe them because we can't name them. Be careful man, history has shown we have been wrong on this issue many times. We were wrong when we thought Rationals describe number completely, we were wrong when we thought union of Rationals with Irrationals describe number completely, we were wrong when we thought Real numbers describe number completely, then we had to go back and look at irrational closely and discovered a different kind of irrational so we gave it a name, transcendental.

The discovery of transcendental did not threaten the closure property of complex numbers subject to 4 binary operations and the extraction of roots. Today mathematicians say Complex numbers are closed under these operations, but that does not mean that we don't have to go back and rearrange a few things in the future if more discoveries are made. We had to do this when transcendentals were discovered.

I AM NOT QUESTIONING THE CLOSURE PROPERTIES OF YOUR PRECIOUS NUMBER SYSTEMS

So let me ask the same question in yet another way. i have asked this question 10 different ways now.

Question:Do you think a NEW discovery will be made about the real numbers? In particular, the uncountable set of transcendentals?
P: 800
 Quote by agentredlum Question:Do you think a NEW discovery will be made about the real numbers?
Undoubtedly. The set theorists are busy working on various new axioms, many of which have implications for our understanding of the real numbers. Eventually some of these ideas will gain traction and consensus in mainstream math.

As far as your original question of whether there's some "interesting" class of reals larger than the definable numbers, it's hard to imagine what that could be, but I'm not in a position to speculate.
P: 460
 Quote by SteveL27 Undoubtedly. The set theorists are busy working on various new axioms, many of which have implications for our understanding of the real numbers. Eventually some of these ideas will gain traction and consensus in mainstream math. As far as your original question of whether there's some "interesting" class of reals larger than the definable numbers, it's hard to imagine what that could be, but I'm not in a position to speculate.

here are 2 excerpts from this wiki entry

"Georg Cantor also gave examples of subsets of the real line with unusual properties?these Cantor sets are also now recognized as fractals."

"A class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic (all the above) or stochastic (that is, non-deterministic). For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2.Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2?but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related frac"

When we began this conversation i had no idea that Cantor was involved with fractals but my search to make my question more precise happily led me to this article. This is exactly what i was looking for, a simple formula that leads to a new concept of number, this article doesn't have formulas but i've seen them elsewhere, although fractals are 'objects' as far as i know, are there fractals that are pure Real numbers?
 P: 460 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!
P: 460
 Quote by 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

"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!
P: 800
 Quote by agentredlum 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.
P: 460
 Quote by SteveL27 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, thats not a logical scientific reason, no one is going to take it seriously.
P: 800
 Quote by agentredlum 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.

 Quote by agentredlum 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."

 Quote by agentredlum 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.

 Quote by agentredlum 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, thats 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.
P: 460
 Quote by SteveL27 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

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