Cantor/Hilbert Diagnolization Argument: There are countably many rational numbers. That is, there are as many fractions as there are whole numbers.
Cantor's Uncountability Theorem: There are uncountably many infinite sequences of 0's and 1's.
Corollary: There are uncountably many subsets of N. There are uncountably many real numbers. That is, there are more real numbers than whole numbers (but the same number of whole numbers as there are fractions).
Tarski's Undefinability of Truth Theorem: TRUTH, the set of numbers which encode true sentences of number theory, is not definable in number theory.
Godel's Completeness Theorem: First order logic is complete.
Godel's First Incompleteness Theorem: Any adequate axiomatizable theory is incomplete. In particular the sentence "This sentence is not provable" is true but not provable in the theory.
Godel's Second Incompleteness Theorem: In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system is not provable in the system.
Choice Is Consistent With Set Theory (Godel): The axiom of choice is consistent within ZFC set theory.
Not-Choice Is Consistent With Set Theory (Cohen): The negation of the axiom of choice (nonexistence of the choice function) is consistent with ZFC set theory.
Corollary: Choice is logically independent of set theory.
Continuum Hypothesis Independent Of Set Theory: It can not be disproved (Godel), it can not be proven (Cohen). Hence the Continuum Hypothesis is logically independent from ZFC set theory.
References:
http://www.math.hawaii.edu/~dale/godel/godel.html
http://en.wikipedia.org/wiki/Kurt_Gödel
http://en.wikipedia.org/wiki/Continuum_hypothesis
Buggies in a jar!
She is all I want :]
I was just a normal guy... until one day I met a girl who turned my life upside down (in a good way). Words can't describe how it feels to go to sleep with a deep feeling of satisfaction every single day and to wake up excited about that day every single day. It's amazing that a girl can complete your life like that. I am so lucky to have found her. I love you Jamie.