Are the number of operations countable or uncountable?by agentredlum Tags: countable, number, operations, uncountable 

#1
Jun2511, 08:31 PM

P: 460

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




#2
Jun2511, 08:38 PM

Mentor
P: 16,698

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: [tex]\sqrt[n]{x}[/tex] and we can do this for any real (nonzero) number n. Thus there are uncountably many operations... 



#3
Jun2511, 09:17 PM

P: 460





#4
Jun2511, 10:11 PM

P: 460

Are the number of operations countable or uncountable?
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? 



#5
Jun2611, 07:09 AM

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




#6
Jun2611, 11:10 AM

P: 460

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. 



#7
Jun2611, 11:50 AM

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P: 805

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




#8
Jun2611, 01:11 PM

P: 460

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" 



#9
Jun2611, 01:16 PM

P: 799

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 finitelength 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. 



#10
Jun2611, 02:02 PM

P: 460

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. 



#11
Jun2711, 12:43 PM

P: 799

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. 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/ma...constructive/ 



#12
Jun2711, 01:29 PM

P: 460

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!!! 



#13
Jun2711, 01:37 PM

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P: 16,698

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... 



#14
Jun2711, 02:59 PM

P: 460





#15
Jun2711, 03:23 PM

P: 460

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) 



#16
Jun2711, 06:15 PM

P: 799

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 [itex]\lt[/itex] versus [itex]\leq[/itex]. 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? 



#17
Jun2711, 07:29 PM

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of them from others. But what, exactly, are you looking for? 



#18
Jun2711, 08:49 PM

P: 460

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. 


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