Axiom of choice
In set theory, the Axiom of Choice (a.k.a. AC) is a totally unwarranted assumption which blatantly disregards the fundamental right of sets and subsets to remain free from having their elements wantonly dismembered by cruel and unfeeling mathematicians.
The Axiom of Choice was accidentally discovered by the Irish mathematician James Choice in 1939. Its first formulation was deeply embedded in the undergraduate novel Finnegans Wedge, but since no one was able to decipher the symbolism of the book, the discovery went unnoticed until 1977.
In the distant future of nonstandard mathematics, AC will be replaced by MULTIVAC, a huge God-like espresso-powered analogic calculator. MULTIVAC, which first appeared as a hypothetical extrapolation in an Asimov et al paper on LAAThSFSLPhJ, will have the astounding property of separating milk from coffee by reversing entropy itself. (George M. Cohen successfully wrote a song in 1963 about how espresso may be chosen independently of milk and sugar.)
Without loss of generality, the AC has infinite applications to every finite field of mathematics, even though when it shows up most mathematicians frown and suddenly start mumbling unintelligibly about something else.
The Axiom of Choice has an uncountable number of equivalent formulations (the proof of which is left to the reader as an exercise). However, for the sake of ennumerability, we will select the following ones for no apparent reason:
- Banach-Tarski Paradox: an implementation of the Fundamental Principle of Wiki - "duplicate material excessively".
- Zorro's Lemma: no matter how bad the current situation is, Zorro will get out of prison and kick Sgt. Garcia's ass.
- The Existence of Bases: every baseball field has at least one base.
- Well-ordering Principle: you cannot order a Chinese meal without taking rice (to which the Chinese Remainder Theorem is a corollary).
- A=A^2, which in turn implies A=0 or A=.999999999 for every A
- Porn's Lemma: every non-empty porn site in which every chained girl has an upper bound contains at least one maximally annoying popup window
- The Resistance of Vector Bases: all your vector base are belong to us
- Lobotomy's Law: either I do not know this law, or you are too stupid to understand it, or both of them.
- The Existence of Marxist Ideals: every ideal actually belongs to a greater Communist Ideal.
- Tychonoff's Theorem: the product of compact discs is always a compact disc
- The Axiom of Choice for Dummies™: among an infinite number of pairs of nonchiral socks, one can select one sock out of every pair, even though nobody knows how to go about doing this. (Note that the above is obviously false for a finite number of nonchiral socks, as there is always a missing one.)