Axiom of choice
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The Axiom of Choice (a.k.a. AC) is the classiest, most desired axiom of set theory (hence its name). At some point, the other axioms rebelled against it in jealous rage, leaving mathemagicians with no longer any constructive way to use its high-horse results. It now lives independently from those other axioms (if you know what I mean), and prolonged consideration of its truth will probably result in Christians knocking down your door.
Formal Statement
For any set of disjoint pregnancies, the mother is able to choose one child from each pregnancy to abort or not to abort. Naturally, this was found highly offensive by Pro-Lifers and Pro-Deathers, but was finally upheld in Roe vs. Wade (the issue of twin or triplet abortions being addressed somewhat later in Roe vs. Wade vs. Paddle). In finalizing that decision, the Supreme Court had to consider, among other sets of possible pregnancies, sets that contained themselves or that contained an infinite nest of abortable babies-within-babies, making it a historic landmark decision.
History
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 (MULTIple Vacuous Axioms of Choice), which will allow us to choose, in turn, from different Axioms of Choices themselves. MULTIVAC, which first appeared as a hypothetical supercomputer in an Asimov et al paper on LAAThSFSLPhJ, will have the astounding property of separating milk from coffee by reversing entropy. (George M. Cohen successfully wrote a song in 1963 about how espresso may be chosen independently of milk and sugar.)
Equivalent Formulations
The Axiom of Choice has an uncountable number of equivalent formulations (of which a bullshit attribute can ironically be selected from each one). For the sake of ennumerability, we will list the following ones for no apparent reason:
- Tychonoff's Theorem: the product of compact discs is always a compact disc
- The Existence of Marxist Ideals: every ideal actually belongs to a greater Communist Ideal.
- The Existence of Bases: every baseball field has at least one base.
- Hausdorff Maximal Principle: Every partially-ordered meal contains a totally-ordered side dish.
- Well-ordering Principle: Any Chinese meal is well-ordered with rice (to which the Chinese Remainder Theorem is a corollary).
- 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.
- Tarski's Theorem: For every infinite A, we have A=A^2, which in turn implies A=0 or A=.999999999
- Zorro's Lemma: no matter how bad the current situation is, Zorro will get out of prison and kick Sgt. Garcia's ass.
- 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.)
Weak-Ass Formulations
These are not as cool as the Axiom of Choice, but rely on it for dear life and generally hang out in the same clique.
- Banach-Tarski Paradox: an implementation of the Fundamental Principle of Wiki - "duplicate material excessively".
- From Category Theory: Every small category has a skeleton in its closet.
- Ultrafilter Lemma: Every toilet filter able to block out your bullshit is, in fact, contained within a maximal filter: a plug.