...screw it, I’ll reply to this one just to point out what you should be looking up. That is not Cantor’s proof, Cantor used (invented) the diagonal argument[0]. Nor is that a correct proof; if it were, it would prove that Q would have a larger cardinality than Z. You may remember this surprise? Q has the same cardinality as Z but R has strictly larger cardinality? If this doesn’t sound familiar to you, you need to relearn basic set theory. If this does sound familiar to you but you don’t see why it applies, you need to better develop your ability to analyze arguments, and relearn basic set theory (going by your previous statements).
EDIT: JoshuaZ points out a clearer counterexample to your argument in a brother comment.
That should do for a start, though a more organized textbook may be preferable. Now you have at least something to read and I will spend no more time addressing your arguments myself as the linked pages do so plenty well.
[0]Yes, I know this was not his original proof. That is not the point.
...screw it, I’ll reply to this one just to point out what you should be looking up. That is not Cantor’s proof, Cantor used (invented) the diagonal argument[0]. Nor is that a correct proof; if it were, it would prove that Q would have a larger cardinality than Z. You may remember this surprise? Q has the same cardinality as Z but R has strictly larger cardinality? If this doesn’t sound familiar to you, you need to relearn basic set theory. If this does sound familiar to you but you don’t see why it applies, you need to better develop your ability to analyze arguments, and relearn basic set theory (going by your previous statements).
EDIT: JoshuaZ points out a clearer counterexample to your argument in a brother comment.
Here. Here are some Wikipedia links to get you started.
http://en.wikipedia.org/wiki/Hume%27s_principle
http://en.wikipedia.org/wiki/Galileo%27s_paradox
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
http://en.wikipedia.org/wiki/Equinumerosity
http://en.wikipedia.org/wiki/Bijection
http://en.wikipedia.org/wiki/Dedekind-infinite_set
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
http://en.wikipedia.org/wiki/Cantor%27s_theorem
http://en.wikipedia.org/wiki/Cardinality
http://en.wikipedia.org/wiki/Cardinal_number
http://en.wikipedia.org/wiki/Injective_function
http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem
That should do for a start, though a more organized textbook may be preferable. Now you have at least something to read and I will spend no more time addressing your arguments myself as the linked pages do so plenty well.
[0]Yes, I know this was not his original proof. That is not the point.