[Retric]

We agree that {A,B,C} has a lower cardinality than {A,B}.

Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.

[pcmonk]

{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false.

For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...

[Retric]

The distance between 0 and 1 is smaller than the distance between 0 and 2. The number of points between 0 and 1 is larger than the number of rational numbers.

[pcmonk]

> The distance between 0 and 1 is smaller than the distance between 0 and 2.

This is correct. However, the number of real-valued points between 0 and 1 is the same as the number of real-valued points between 0 and 2.

> The number of points between 0 and 1 is larger than the number of rational numbers.

This is also true because there are uncountably many real-valued points between 0 and 1 and countably many rational numbers.

[yongjik]

1. e + ei is not real.

2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise.

This function maps all real numbers to irrational numbers, 1-to-1.

[Retric]

e is real, e + ei is irrational.

[pcmonk]

I believe you mean "complex". You can form a bijection between the reals and the complex numbers by interleaving the digits, as any Google search can tell you.

[Retric]

For every point on your mapimg I define two points X + 1 and x + 1i. You can assign infinity to one of them but not both.

[yongjik]

I'm afraid you have some basic confusion with mathematical concepts, including "mapping", "irrational", "complex", and "infinity".

So, after you define x+1 and x+i, now what? Also please bear in mind that "infinity" is neither real nor complex: if you have a well-defined mapping into complex numbers, then by definition, it never maps to infinity.

(Yes, there are some "functions" like y = 1/x that "maps to infinity", but it's simply mathematicians being lazy and abusing notations because everybody around them understands what's going on.)

[Retric]

O, I get it.

There is a basic contradiction in set theory. Called Russell's paradox, there are two ways around it. First is ignoring it, aka everything builds from it's self nothing can become recursive. Or Zer's something or other that basically removed membership and equality and hides in the corner crying.

As such infinity is generally assumed not to exit in R. And causes all this all infinite set's map able to each other are equivalent size crap. It's also why real mathematicians laugh at the set guys.

But, sorry the way R was initially defined it included infinity and you only get to put it into 1 place on your mapping. Or as a math professor said, what angle is the highest number in R mapping to.

[jakub_h]

e is irrational, e+ei can't be irrational because it isn't real (all irrationals are reals).

[Retric]

Sorry, imaginary number composed of two irrational numbers.