[pcmonk]

Where in math is the subset partial ordering used to describe one set as larger than another?

Diagonalization isn't showing that a number in set A isn't in set B - that's obviously true for reals and integers, but it's also true for rationals and integers. It's showing that there does not exist a mapping from B to A where there's an element in B for each element in A.

We're obviously not using the same definition of "size". I generally think in terms of cardinality, what are you thinking of?

[Retric]

Cardinality is the number of elements in a set.

If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} < {A,B,C}

There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.

[pcmonk]

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers.

There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers.

Try reading this: https://en.wikipedia.org/wiki/Cardinality#Infinite_sets

[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.

[jakub_h]

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

[stouset]

The cardinality of the set of prime numbers and of rationals are both aleph-null[1]. Same with the list of all integers, the list of all positive integers, and the list of all even integers. They all have the same cardinality, and that cardinality is aleph-null.

But please, continue to argue with mathematical definitions that have been established for over a century.

[1]: https://en.wikipedia.org/wiki/Aleph_number#Aleph-naught