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is that because they are uncountable?

well you will "see" two lines , y = 1 and y = 0

it doesnt matter there are more irrationals than rational, so what

Are there more irrational numbers than rational is really the question

or more generally, pick two points, between them is an irrational and a rational

pick 2 distinct points

so it is a scattering of individual points, neither line exists

very true, imagine we had a microscope ;)

right, its not a line ,. but at the macro level it "looks" like a line

on the number line. so ...

there cannot be two distinct irrationals without a rational between them, correct

so its kind of paradoxical

Do you have a good source where I could read more? Wiki gets too techie too quickly

yeah, thats true. well you can google cantor's infinity

but this question is not addressed specifically, its something that my math professor told me

there are many layman easy websites

Yeah, I want something between the two levels

yeah i know the feeling

Also, I think it's neat to see hebrew letters come into math for this question!

well college libraries have some gradual immersion books

yeah, thats cool

it can be shown that aleph null is the "least" infinity , as weird as that sounds

so we start with | N | , where N = { 1,2,3,...}

ok

where |N | means cardinality , that means the size of the set

right

now we know for any set , |P (S)| > |S|

P(S) is the power set of S

only for finite sets?

for any set

ok

it comes in handy for infinite sets

what a power set ?

2^S, right?

the number of permutations within the set

right, thats the shorthand for it

well, its the set of all subsets of the set

i dont know about permutations, hmmm

I thought infinity was defined as the set that has the same cardinality as any of its proper subsets

yes thats a definition of an infinite set

but im talking about power set, what is the power set operation

ok say you have { 1, 2 } , the power set is
{ {}, {1}, {2} , {1,2} }

S= { 1, 2 } , then P(S)= { {}, {1}, {2} , {1,2} }

right

now cantor proved a delightful theorem , that
| P ( S) | > | S| for any set S

so the powerset of integers is larger I mean has a higher cardinality than the set of integers

exactly

ok so

| P ( N ) | > | N | , correct ?

I sense a trick coming up...

but, yes

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

ok it turns out that |R| which we use shorthand c , it turns out
|R| = | P ( N ) | ,

now the question is, is there an infinity between |N| and | P ( N ) |

is the real numbers the smallest infinity after the natural numbers

Wouldn't rational and irrational be smaller than real?

we have a hierarchy of infinities,
|N| < |P(N)| < |P (P(N))| < ...

oh boy, rational is the same cardinality as natural numbers

the irrationals are uncountable

So cantor's work doesn't cover uncountable sets?

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