## real number set and countablity

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 Lets 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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 Recognitions: Gold Member Science Advisor Staff Emeritus 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.

 Quote by HallsofIvy 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 dont 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.

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## real number set and countablity

 Quote by husseinshimal i dont 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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