Are the number of operations countable or uncountable?

[agentredlum]

Let me explain further my question by what i mean 'operation'. An 'operation' can be on a single number, example sqrt(2). An 'operation' can be 'binary' between two numbers, example 1+2. An 'operation' can be performed on more than two elements, example placing vectors on a grid and finding 'sum' by 'tail to head' method. I know that my last example can be reduced to performing 'binary' operations on vectors, however, if you place THREE vectors on a grid tail to head and you connect the tail of the first vector to the head of the third vector, you now have a new vector which is the 'sum' of the 3 vectors you started with. This can be done for any amount of vectors. To me, this does not LOOK like a 'binary' operation when done this way

 

[micromass]

Hi agentredlum!

I'm really not sure what you're asking or why you are asking it.
But, judging from what you want to hear. There are uncountably many "operations". We can take the n'th root of any number x:

\sqrt[n]{x}

and we can do this for any real (nonzero) number n. Thus there are uncountably many operations...

 

[agentredlum]

Hi agentredlum!

I'm really not sure what you're asking or why you are asking it.
But, judging from what you want to hear. There are uncountably many "operations". We can take the n'th root of any number x:

\sqrt[n]{x}

and we can do this for any real (nonzero) number n. Thus there are uncountably many operations...

Well, then that means there are more DISTINCT 'operations' that you can perform on the set of integers than there are integers themselves... FANTASTIC! I was worried that my 'notion' of 'operation' was a little too loose but i am happy that there is someone out there that does not find it too offensive. Taking roots is one FORM of operation, and i agree with you (happily) that there is an uncountable number of ways to take roots if you allow TRANSCENDENTAL exponents. Would you say there might be an uncountable number of FORMS? Example trgonometric functions may be considered a different form than taking roots. I am worried because given ANY real number you can write a power series expansion for it and that only uses the basic 4 'binary' operations (A COLOSSALL ACHIEVEMENT) but that doesn't mean there aren't any other operations out there we are not aware of. I'm not writing a paper, I'm not in school, just curious and fascinated by INFINITY. Thanks so much for your time.

 

[agentredlum]

Here is what i think i know so far. If any of this is wrong, please, somebody correct me.
1. The set of real numbers is uncountable
2. The set of rational numbers and the set of irrational numbers are DISJOINT, union of both gives the set of real numbers
3. The set of rational numbers is countable, the set of irrational numbers is uncountable
4. The set of irrational numbers includes algebraic irrational and transcendental irrational, these 2 sets are DISJOINT
5. The set of algebraic irrational is countable, the set of transcendental irrational is uncountable

Conclusion:The uncountability of the Real Numbers is a consequence of the uncountability of the transcendentals

Question:are there irrational numbers that are not algebraic or transcendental?

 

[HallsofIvy]

No. A number is defined to be "transcendental" if and only if it is NOT algebraic.

 

[agentredlum]

No. A number is defined to be "transcendental" if and only if it is NOT algebraic.

Hmmm...let me reword my question more carefully since it does not capture the 'spirit' of my inquiry.
Are there irrational numbers that are not algebraic or ONLY transcendental.

Let me explain by way of example between irrational and transcendental irrational.
1. sqrt(2) is irrational but algebraic
2. e is irrational, NOT algebraic so there exist irrational numbers that have a distinct property that is not shared by 'proper' irrationals (such as sqrt(2)) These irrationals by definition called transcendental

Question:are there transcendental numbers that have a distinct property that is not shared by other 'proper' transcendentals such as pi, e?

Why do i ask? To say sqrt(2)) is irrational conveys all information about its properties. To say e is irrational is true but does NOT convey all information about its properties since the transcendental nature of e is absent from the statement.

Can a statement be made, "X is irrational, not algebraic, transcendental AND ALSO ________" fill in the blank so in this sense X differs from normal transcendentals like pi, e.

I am aware that the union of algebraic and transcendental gives Reals, also aware that the union of rational and irrational gives Reals, but the latter does not provide any information that there exists a very important class of numbers within the irrationals.

 

[gb7nash]

I'll try to dissect this as best as I can.

Are there irrational numbers that are not algebraic or ONLY transcendental.

I can't make sense of this question. There are irrational numbers that are algebraic. (sqrt(2)). There are also irrational numbers that are transcendental (pi, e). Since transcendental is the dual of algebraic, it's either one or the other.

Let me explain by way of example between irrational and transcendental irrational.
1. sqrt(2) is irrational but algebraic
2. e is irrational, NOT algebraic so there exist irrational numbers that have a distinct property that is not shared by 'proper' irrationals (such as sqrt(2)) These irrationals by definition called transcendental

What is your definition of a 'proper' irrational? From first glance, it's an algebraic irrational. But sure, irrationals that are not 'proper' you can define as transcendental.

Question:are there transcendental numbers that have a distinct property that is not shared by other 'proper' transcendentals such as pi, e?

I'm confused. I thought proper meant an algebraic irrational number? Ignoring that, that kind of properties are you talking about? The only properties that you've listed thus far are algebraic and transcendental.

Why do i ask? To say sqrt(2)) is irrational conveys all information about its properties.

Again, what properties are you talking about? The fact that it's algebraic? If so, I'll agree. It's very easy to show that sqrt(2) is algebraic.

To say e is irrational is true but does NOT convey all information about its properties since the transcendental nature of e is absent from the statement.

This is true. It's not an easy task to show an arbitrary irrational number is transcendental.

Can a statement be made, "X is irrational, not algebraic, transcendental AND ALSO ________" fill in the blank so in this sense X differs from normal transcendentals like pi, e.

Again, what kind of property are you looking for? I can easily fill in the blank and say greater than 0, less than 0, divisible by pi, etc.. Some other properties could be much more difficult to prove.

the union of rational and irrational gives Reals, but the latter does not provide any information that there exists a very important class of numbers within the irrationals.

This is true. Some properties of the irrational numbers are easy to verify ( > 0, < 0). Some are not (transcendental vs. algebraic). It just depends on the property that you're dealing with.

 

[agentredlum]

I'll try to dissect this as best as I can.
I can't make sense of this question. There are irrational numbers that are algebraic. (sqrt(2)). There are also irrational numbers that are transcendental (pi, e). Since transcendental is the dual of algebraic, it's either one or the other.

What is your definition of a 'proper' irrational? From first glance, it's an algebraic irrational. But sure, irrationals that are not 'proper' you can define as transcendental.
I'm confused. I thought proper meant an algebraic irrational number? Ignoring that, that kind of properties are you talking about? The only properties that you've listed thus far are algebraic and transcendental.
Again, what properties are you talking about? The fact that it's algebraic? If so, I'll agree. It's very easy to show that sqrt(2) is algebraic.
This is true. It's not an easy task to show an arbitrary irrational number is transcendental.

Again, what kind of property are you looking for? I can easily fill in the blank and say greater than 0, less than 0, divisible by pi, etc.. Some other properties could be much more difficult to prove.
This is true. Some properties of the irrational numbers are easy to verify ( > 0, < 0). Some are not (transcendental vs. algebraic). It just depends on the property that you're dealing with.

Thank you for your reply, talking about this helps me a lot.

Yes, I would call an algebraic irrational a 'proper' irrational but the word 'proper' i would not confine to mean only algebraic. i would also use it to describe transcendentals. Are all transcendentals 'proper'??

I am asking if there are transcendental numbers that can be put in a class with a property that is not shared by ANY other transcendental numbers.

Example:sqrt(2) is irrational, pi is irrational but pi 'enjoys' the property of being transcendental and this property is not shared by sqrt(2)) so pi is not a 'proper' irrational.

so if you put sqrt(2)) and pi in the same class somehow you are holding back information. They really don't 'belong' together because pi is a different kind of irrational. (My opinion)

Replace irrational with transcendental in the above example

Example:pi is transcendental, X is transcendental but X 'enjoys' the property of being ________ and this property is not shared by pi or any OTHER 'proper' transcendentals similar in properties to pi. Again, 'proper' here being used simply as an english word and NOT as a mathematical definition.

I think historically it went something like this..from the integers you went to the rationals, then irrationals, then reals. At some point you had to go back because a class of numbers was discovered within the irrationals that had a PROFOUND property that separated them from other irrationals.

So my question, by way of analogy, is there a class of numbers within the transcendentals, that has a PROFOUND property which separates them from other transcendentals?

I am not talking about 'trivial properties' like "6 is the only number that gives 3 when divided by 2"

 

[SteveL27]

So my question, by way of analogy, is there a class of numbers within the transcendentals, that has a PROFOUND property which separates them from other transcendentals?

No, there are too many of them.

If by "property" you mean something you could write down as a formula or algorithm, then there are only countably many of them. If a formula or algorithm is a finite string, and your alphabet is countable, then there are only countably many strings of length 1; countably many strings of length 2; dot dot dot; and taking the union over all n, there are only countably many possible finite-length strings.

But there are uncountably many reals. So most reals can't be named, described, computed, or characterized by any finite string of symbols, even if you have countably many symbols.

The vast majority ("almost all" is the technical phrase) of the real numbers are an amorphous, unnamable, indescribable blur.

 

[agentredlum]

No, there are too many of them.

If by "property" you mean something you could write down as a formula or algorithm, then there are only countably many of them. If a formula or algorithm is a finite string, and your alphabet is countable, then there are only countably many strings of length 1; countably many strings of length 2; dot dot dot; and taking the union over all n, there are only countably many possible finite-length strings.

But there are uncountably many reals. So most reals can't be named, described, computed, or characterized by any finite string of symbols, even if you have countably many symbols.

The vast majority ("almost all" is the technical phrase) of the real numbers are an amorphous, unnamable, indescribable blur.

I agree with everything you say except the first sentence in your reply. Why do you say no? Are you saying that ALL PROFOUND properties that distinguish one class from another become trivial because there is an UNCOUNTABLE infinitude of them?

The question still remains...is there an UNCOUNTABLE subset of transcendentals that has a property not shared by any other UNCOUNTABLE subset of transcendentals?

My interest is not in programming so i would allow infinite series, after all, no one can write down ALL polynomials, no one can write down a SINGLE infinite polynomial, no one can write down the COMPLETE diagonalization in Cantor's argument, yet these ideas are used 'freely' when describing properties of real numbers.

 

[SteveL27]

I agree with everything you say except the first sentence in your reply. Why do you say no? Are you saying that ALL PROFOUND properties that distinguish one class from another become trivial because there is an UNCOUNTABLE infinitude of them?

No, I didn't say that. What I'm pointing out is that there are only countably many finite-length strings, but uncountably many real numbers. So most real numbers can't possibly be named or characterized by a formula or algorithm.

There's a mathematical theory of what numbers can be defined.

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

There are other ways to get at this. For example the definable numbers are a little different than the computable numbers. But regardless, only countably many numbers can have names or descriptions. So we have no way to say "this real number is different from that real number because of such and so property," except for a countable set of reals. The rest have no names or descriptions.

The question still remains...is there an UNCOUNTABLE subset of transcendentals that has a property not shared by any other UNCOUNTABLE subset of transcendentals?

Well sure, the uncountable set of transcendentals greater than or equal to pi is different than every other uncountable set of transcendentals. But note that you could only do this trick with countably many transcendentals, because we can have at most countably many different names for numbers. So if you have two real numbers that can't be named, how can you distinguish between them?

My interest is not in programming so i would allow infinite series, after all, no one can write down ALL polynomials, no one can write down a SINGLE infinite polynomial, no one can write down the COMPLETE diagonalization in Cantor's argument, yet these ideas are used 'freely' when describing properties of real numbers.

Yes, we can describe properties of the real numbers. For example in set theory we can prove: "There are uncountably many real numbers." But we can name only countably many of them. That's a bit of a philosophical mystery. You might be interested in constructive mathematics, where a mathematical object is not said to exist unless we have a specific construction for it.

http://plato.stanford.edu/entries/mathematics-constructive/

 

[agentredlum]

No, I didn't say that. What I'm pointing out is that there are only countably many finite-length strings, but uncountably many real numbers. So most real numbers can't possibly be named or characterized by a formula or algorithm.

There's a mathematical theory of what numbers can be defined.

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

There are other ways to get at this. For example the definable numbers are a little different than the computable numbers. But regardless, only countably many numbers can have names or descriptions. So we have no way to say "this real number is different from that real number because of such and so property," except for a countable set of reals. The rest have no names or descriptions.

Well sure, the uncountable set of transcendentals greater than or equal to pi is different than every other uncountable set of transcendentals. But note that you could only do this trick with countably many transcendentals, because we can have at most countably many different names for numbers. So if you have two real numbers that can't be named, how can you distinguish between them?
Yes, we can describe properties of the real numbers. For example in set theory we can prove: "There are uncountably many real numbers." But we can name only countably many of them. That's a bit of a philosophical mystery. You might be interested in constructive mathematics, where a mathematical object is not said to exist unless we have a specific construction for it.

http://plato.stanford.edu/entries/mathematics-constructive/

WOW! How come i didn't think of that? Very nice trick sir, the "or equal to" is what makes it work, it makes it BRILLIANT! Also accept your note very seriously and heed your warning about this trick. Hopefully the trick can be extended and used elsewhere. I came close...i considered uncountable transcendental subsets GREATER THAN any given transcendental but i did not come up with "or equal to" THANX!

Now having said that i must also say that i am a little disappointed...is this the best that we can do? Can't we find a more PROFOUND property? For instance the property that distinguishes transcendental from algebraic is PROFOUND (AMAZING) and not just a simple 'order relation' Thanx again because i have learned something very important today, the links you provided were also very helpfull. I've got some thinking to do, have a GREAT day!

PLEASE POST MORE TRICKS!

 

[micromass]

Maybe you'll like the following two kinds of numbers:

Computable numbers: http://en.wikipedia.org/wiki/Computable_number
Definable numbers: http://en.wikipedia.org/wiki/Definable_number

In general we have

Rational --> Algebraic --> Computable --> Definable --> Real

So these might be the classes you're looking for...

 

[agentredlum]

Maybe you'll like the following two kinds of numbers:

Computable numbers: http://en.wikipedia.org/wiki/Computable_number
Definable numbers: http://en.wikipedia.org/wiki/Definable_number

In general we have

Rational --> Algebraic --> Computable --> Definable --> Real

So these might be the classes you're looking for...

Thanx for the links, i think i have a rudimentary understanding but i want to learn more. It occurred to me, after my previous post, to start a thread on MATH TRICKS. Tricks are awsome! (trix are for kidz) I hope you can participate. It is in Number Theory category but the trick could be about anything you find interesting, even physics. I hope you post, 4000 posts you must have a few tricks under your thinking cap? I hope everyone posts a lot so many can benefit.

 

[agentredlum]

Rational --> Algebraic --> Computable --> Definable --> Real

Hmmmm...is there any class between Definable and Real?

Let me give an 'IMPORTANT' example of each class, so you can get an indication of how much i understand, hopefully i won't make a mistake.

Rational, 1/3

I will send $10 to the person who figures out why i picked 1/3

Algebraic, sqrt(2)

sorry, no prize here...LOL

Computable, e

Definable, Chaitins omega

Real, all numbers (division by zero not allowed)

 

[SteveL27]

WOW! How come i didn't think of that? Very nice trick sir, the "or equal to" is what makes it work, it makes it BRILLIANT! Also accept your note very seriously and heed your warning about this trick. Hopefully the trick can be extended and used elsewhere. I came close...i considered uncountable transcendental subsets GREATER THAN any given transcendental but i did not come up with "or equal to" THANX!

Now having said that i must also say that i am a little disappointed...is this the best that we can do? Can't we find a more PROFOUND property? For instance the property that distinguishes transcendental from algebraic is PROFOUND (AMAZING) and not just a simple 'order relation' Thanx again because i have learned something very important today, the links you provided were also very helpfull. I've got some thinking to do, have a GREAT day!

PLEASE POST MORE TRICKS!

I'm a little concerned that I haven't been clear enough to explain this. There is no difference between using greater-than or greater-than-or-equal. Either way, you get an uncountable set of transcendentals that differs from any other uncountable set of transcendentals.

But you can only do this with a countable number of transcendentals. You can't distinguish among the rest of them using any finite string of symbols.

There is no "trick" in using \lt versus \leq.

If you can explain why you think these are different or that one gives a great insight as opposed to the other one, then perhaps I'll be able to explain what I'm talking about more clearly.

Perhaps you could defined what you mean by "property." I'm taking it to mean something you can describe in a finite number of symbols. What do you mean by property?

 

[HallsofIvy]

Hmmm...let me reword my question more carefully since it does not capture the 'spirit' of my inquiry.
Are there irrational numbers that are not algebraic or ONLY transcendental.

Let me explain by way of example between irrational and transcendental irrational.
1. sqrt(2) is irrational but algebraic
2. e is irrational, NOT algebraic so there exist irrational numbers that have a distinct property that is not shared by 'proper' irrationals (such as sqrt(2)) These irrationals by definition called transcendental

Question:are there transcendental numbers that have a distinct property that is not shared by other 'proper' transcendentals such as pi, e?

What do you mean by "distinct property"? Every number has a "distinct property" different from every other number- that of being itself.

Why do i ask? To say sqrt(2)) is irrational conveys all information about its properties. To say e is irrational is true but does NOT convey all information about its properties since the transcendental nature of e is absent from the statement.

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.

Can a statement be made, "X is irrational, not algebraic, transcendental AND ALSO ________" fill in the blank so in this sense X differs from normal transcendentals like pi, e.

I am aware that the union of algebraic and transcendental gives Reals, also aware that the union of rational and irrational gives Reals, but the latter does not provide any information that there exists a very important class of numbers within the irrationals.

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?

 

[agentredlum]

I'm a little concerned that I haven't been clear enough to explain this. There is no difference between using greater-than or greater-than-or-equal. Either way, you get an uncountable set of transcendentals that differs from any other uncountable set of transcendentals.

But you can only do this with a countable number of transcendentals. You can't distinguish among the rest of them using any finite string of symbols.

There is no "trick" in using \lt versus \leq.

If you can explain why you think these are different or that one gives a great insight as opposed to the other one, then perhaps I'll be able to explain what I'm talking about more clearly.

Perhaps you could defined what you mean by "property." I'm taking it to mean something you can describe in a finite number of symbols. What do you mean by property?

Well sure, the uncountable set of transcendentals greater than or equal to pi is different than every other uncountable set of transcendentals. But note that you could only do this trick with countably many transcendentals, because we can have at most countably many different names for numbers. So if you have two real numbers that can't be named, how can you distinguish between them?

If you do not include 'or equal to' in your trick then i choose 2 DIFFERENT sets that can satisfy 'greater than pi property' 1st set... 'The set of all transcendentals greater than or equal to 2e' ...2nd set... 'The set of all transcendentals greater than or equal to 3e' ...these 2 sets are different yet they 'enjoy' the same mentioned property, namely they are both greater than pi. Many more can be picked, an uncountable number of them and they can all satisfy greater than pi. So this property ALONE is not enough. Do you see now why 'or equal to' is so brilliant? Greater than or equal to pi defines only ONE set...UNIQUE! Just like you said in your post...read it again, YOU SAID IT!

So you have found a property that makes an uncountable set of transcendentals (without omissions of course) UNIQUE

What do i mean by without omissions? It is not allowed to say...Take the set of all transcendental greater than or equal to pi TWICE. Remove 2e from the second set. Now you have 2 different sets sharing the same property and UNIQUENESS COLLAPSES.

What i mean by property is like 'greater than or equal to' personally, i don't find this property interesting, unless it turns out to be the only one possible in my search, then it will be VERY interesting to me. Finite length and not too complicated to state. Another property i find more interesting is 'not the root of any polynomial equation with integer coefficients' This is what i mean by property, nothing too fancy, although if you have a fancy complicated idea i would like to hear it. Thanx again man. Let me know if i got my point across.

 

[agentredlum]

I haven't got the hang of this yet, the 1st paragraph under the quote is also a quote from steveL27

 

[SteveL27]

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.

 

[agentredlum]

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?

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.

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!

 

[SteveL27]

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?

 

[agentredlum]

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 that's 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.

 

[agentredlum]

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

 

[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?

 

[SteveL27]

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.

 

[agentredlum]

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.

[STRIKE][/STRIKE]

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?

 

[SteveL27]

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.

 

[agentredlum]

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.

What about fractals? http://en.wikipedia.org/wiki/Fractal

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?

 

[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!

 

[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