Real number set and countablity

• husseinshimal
In summary, the purpose of this post is to show that real numbers could be generated intensively and counted somehow by defining a special surjective or injective function. I think the mathematical constructure of this post needs to be fixed by an expert, so I need some help here.
husseinshimal
the purpose of this post is an attempt tp show that real numbers set could be generated intensively,it also could be counted somehow by defining aspecial surjective or injectice function.i think the mathematical constructure of this post need to be fixed by an expert,thats why i need some help here.

Consider we express the tow positive real numbers ,A&B as,

A=Σam[(10)^(n-m)] B=Σbm[(10)^(n-m)] Where,( n,m=0,1.2,……) am,bm,positive integer Now if, am+bm=pm+10,pm<10 pm,positive integer Then we define the relationship R, ARB={pm*(10)^(n)}+{(pm+1)*(10)^(n-1)+... Obviously, R; looks like adding backwards.e.g, 341R283=525 =(3+2)=5,(4+8)=12,(1+1+3)=5 Let's now pick up arbitrarily the infinite sequence

S0=Σn\(10)^(n),+Σn\(10)^(n+1) + Σn\(10)^(n+2)+...

Where n=1to9 ,10to99,100to999 ,...etc.respectively

i.e, S0=0.123456789101112131415161718192021222324... ,. In order to generate or count* the real numbers within the interval,e.g. (0,1),

We define the surjective function,F;

F:N→positive irrational numbers subset in(0,1)

Where ,
F(n)=SnR0.1,

Sn, the set of sequences,

S1=s0 R 0.1,

S2=s1 R 0.1,

Sn=Sn-1R0.1,

etc.

notice that (n-1) is suffix,

post(1)

There are an infinite sequences,s1,s2 that we can make S1RS2 Close enough to any real number.

post(2)
if the relation,R,has aseriouse mathematical use , can we solve equations of the form,
xRx=s,where,s=0.3,0.5,..etc ,or xR1=10? i mean can the relation,R,be generalized to involve such equations?or even negative numbers?

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The fundamental problem is that your whole concept is flawed. The operations you define can only produce a countable set of numbers and the set of all real numbers is not countable. Or do you refuse to accept that? You define
"the surjective function,F;

F:N→positive irrational numbers subset in(0,1)"
which is, of course, impossible since that would imply the set of irrational numbers between 0 and 1 was countable.

what is impossible in terms of mathematics?

HallsofIvy said:
The fundamental problem is that your whole concept is flawed. The operations you define can only produce a countable set of numbers and the set of all real numbers is not countable. Or do you refuse to accept that? You define
"the surjective function,F;

F:N→positive irrational numbers subset in(0,1)"
which is, of course, impossible since that would imply the set of irrational numbers between 0 and 1 was countable.
i don't get it,you said that The operations i defined can only produce a countable set of numbers and we know the set is countable if there exists acountable subset belong to the original set.

husseinshimal said:
i don't get it,you said that The operations i defined can only produce a countable set of numbers and we know the set is countable if there exists acountable subset belong to the original set.
The numbers that both HallsofIvy and I see here are only the rational numbers, but real numbers include the irrational numbers, so your set is incomplete. While you can make rational numbers close to irrational numbers they are still only the rational numbers and not the irrational numbers. Moreover for every rational number you say is close to an irrational number you can create an infinite number of irrational numbers still closer, so which of these irrational numbers does your counting number actually count?

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husseinshimal said:
we know the set is countable if there exists acountable subset belong to the original set.

No, that's non-sense. A set if countable it there exist a countable set containing it, not the other way around! Any set of real numbers, complex numbers, etc. contains the natural numbers- that does prove that any such set is countable!

another comment

HallsofIvy said:
No, that's non-sense. A set if countable it there exist a countable set containing it, not the other way around! Any set of real numbers, complex numbers, etc. contains the natural numbers- that does prove that any such set is countable!
. first let me express my great apreciation for your being patient with me thank you all guys.I am just confused here alittle bit.i am not saying that real number set is countable.all what iam trying to understand is this,if the arbitrary sequence,S0,that i gave represents one irrational number in(0,1) and if the realtion,R,that i defined is not mathematically flawed.and if the definition of counable set which is,(G, is countable, i.e. there exists an injective function ,F:G→N.
Either, G, is empty or there exists a surjective function,F:N→G) is right.then don't you think that what i was trying to do might be right? all what iam saying is ,F(1)=S0R0.1=0.223456789101112...f(2)=S0R0.2=S1R0.1=0.323456...,i.e,F,is asurjective function.but ,F,keeps all the irrational numbers within(0,1) as long as it just shifts the digits to the left as it defined in the relation,R,the question here guys,and iam sure that you are the experts,is , if there is no logical or mathematical objection about this then don't you think it is worth to study it?thank you

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husseinshimal said:
. first let me express my great apreciation for your being patient with me thank you all guys.I am just confused here alittle bit.i am not saying that real number set is countable.all what iam trying to understand is this,if the arbitrary sequence,S0,that i gave represents one irrational number in(0,1) and if the realtion,R,that i defined is not mathematically flawed.and if the definition of counable set which is,(G, is countable, i.e. there exists an injective function ,F:G→N.
Either, G, is empty or there exists a surjective function,F:N→G) is right.then don't you think that what i was trying to do might be right? all what iam saying is ,F(1)=S0R0.1=0.223456789101112...f(2)=S0R0.2=S1R0.1=0.323456...,i.e,F,is asurjective function.but ,F,keeps all the irrational numbers within(0,1) as long as it just shifts the digits to the left as it defined in the relation,R,the question here guys,and iam sure that you are the experts,is , if there is no logical or mathematical objection about this then don't you think it is worth to study it?thank you
If your function creates unique irrational numbers from a set of k "ordered" non-negative integers where k is a constant then I would say that your set is countable because all such sets of k integers can be sorted first by the sum of the k integers, then by ascending order, for instance from left to right of the k integers. Thus if k = 4, 1 = F(0,0,0,0), 2 =F(0,0,0,1), 3 = F(0,0,1,0), 4 = F(0,1,0,0), 5 = F(1,0,0,0), 6 = F(0,0,0,2), 7 = F(0,0,1,1), etc. When k itself is infinite this is not the case.

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1. What is the definition of the real number set?

The real number set is a set of numbers that includes all rational and irrational numbers. These numbers are represented on a continuous number line and have infinite decimal places.

2. How are real numbers different from natural numbers?

Unlike natural numbers, which include only positive whole numbers, real numbers include all rational and irrational numbers, including negative numbers and numbers with decimal places.

3. What is the significance of the term "countability" in relation to real numbers?

Countability refers to the ability to count or list all the elements in a set. In terms of real numbers, it signifies whether or not the numbers in a set can be counted or listed in a finite or infinite sequence.

4. Are all real numbers countable?

No, not all real numbers are countable. In fact, the majority of real numbers are uncountable, meaning they cannot be listed in a finite or infinite sequence. Examples of uncountable real numbers include irrational numbers such as pi or the square root of 2.

5. How does the concept of countability relate to infinity?

The concept of countability is closely related to infinity, as it determines whether or not a set of numbers can be listed in a finite or infinite sequence. If a set is countable, it has a finite or infinite number of elements. If a set is uncountable, it has an infinite number of elements that cannot be listed in a sequence.

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