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?
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.
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.
{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.
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.
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.
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.)
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.
Where did you get all this idea? You know enough terms, yet somehow you have an incorrect understanding of pretty much everything.
If you are interested, please read an actual math textbook. (Yes, they can be a giant time sink, but at least you'll learn the correct meanings of sets and functions.)
I have taken plenty of high level math classes for fun and easy A's, I even had a department head yell at me for not getting a PHD. So, I can speak the lingo.
But, the absurd results are not generalizable outside of their assumptions sorry Axioms. Set theory being one of the most obvious cases.
It's sadly like a religion in many ways, follow enough false statements and you can prove anything. Yet, if you find a contradiction then don't actually accept at least one of your assumptions are false.
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.
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?