Rejection letters, correspondence, and miscellanea from the otherwise empty annals of the Journal of Universal Rejection.

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Thursday, December 8, 2011

The Choice of Axioms

Dear D          ,

Thank you for submitting "An Empty Article About the Empty Set" to the Journal of Universal Rejection.

Rest assured your article does indeed fulfill the requirements for rejection at our prestigous Journal.

We have previously received 22 empty documents, but never before an empty (aside from title, author, affiliation, date, and the word 'Abstract') document that was explicitly about the empty set.

Before I delve fully into rejecting your document, allow me a brief aside about the empty set for the benefit of our readership at Reprobatio Certa where I will be posting this letter. 

The empty set is an object which exists in every set-theoretic mathematical model that I know of.  Let's see if we can deal with that.  We'll focus on the most common axioms used today in mathematics, the ZF (Zermelo-Fraenkel) axioms.  There are eight axioms in this model.  It is the third axiom--the Axiom of Restricted Comprehension--which can be used to guarantee the existence of an empty set, once the existence of a single set (call it w) is known.  This existence is guaranteed by another axiom--the Axiom of Infinity.  For example we can get the empty set by doing:

Here we see that the empty set is the set of all elements in w that both do and do not contain themselves.  Since (P and not P) is a contradiction, there are no such elements, and the empty set is, well, empty.

We can also denote the empty set as {}.  The braces are standard set-theory notation where the contents of the set are listed within them.  Here there is nothing in between them.  But typography alone does not guarantee the existence of this set.

So we've got the existence of the empty set squared away.  Or do we?  We've used the Axiom of Infinity and the Axiom of Restricted Comprehension.  Maybe you don't want to believe in an infinite set.  Well, we'd just need any set to exist.  So if we want to not believe in the empty set, we have to either
(A) postulate a model for set-theory in which there are no sets whatsoever, or
(B) postulate a model for set-theory in which we don't have the Axiom of Restricted Comprehension.

If we live in a universe of pure nothingness, and our existences, our love songs, addictions, brilliant colors, taste of pomegranates, our fashion sense, the electric touch of our electronic gadgets, cups of tea, soft evening breezes, all qualia, pure nonmoving vibrations of nothingness, then I will choose option A.

If we live in a universe of confusion, of dead ends, residues, tastes that we can't get out of our mouths no matter how much we brush our teeth, the electric touch of our electronic gadgets, addictions, songs of love stuck forever in our gullet, all things we will never understand, crackling and crunching into an everlasting cinder, then I will choose option B.

To me, the answer to this quandry is obvious.

Therefore your paper is rejected.

Best regards,

Caleb Emmons, PhD
Journal of Universal Rejection

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