The Number One, Part Two

My previous post could use some further explanation. The gigantic and unreadable definition for the number one which I posted is the result of giving the number one a precise, complete, logical definition without resorting to using any numbers except zero. Other numbers are avoided in the definition because they use the number one in their definitions; if we defined the number one using them, we would have a circular definition. In English, it's hard to define the word the word "is" without using the word "is". Similarly, it's difficult to define numbers without using numbers, but that's just what Russell, Whitehead, and later Quine, were trying to do. Hence the complexity.

Quine actually defines one and zero as:

where the funny symbols represent things commonly used in set theory (Quine uses the word "class" rather than "set") and symbolic logic, which aren't numbers or arithmetical operations. To translate into plain English, the Lamba means the null or empty set, which is the class with no members; the mathematical version of an empty plastic bag with all the air sucked out of it. Iota followed by some object means the class containing only that object; a plastic bag with one thing in it and all the air sucked out of it. So zero, according to the logicians, is a plastic bag containing another plastic bag that is empty. It's weird, but it works.

Furthermore, the "x" with the carat ("^") followed by a statement means the class of all objects x for which the statement (which usually mentions x) happens to be true. The backwards "E" followed by "y" means that there is something in existence called y and the statement following the "E" and the "y" (which usually describes y in some useful way) is true. The square dot means about the same as the word "and". y-epsilon-x means that y is a member of class x. The arc turned downward combines two classes into a single class made up of everything found in both of those classes (which might be nothing at all - the empty set). Finally, iota-y with the line over the top of it means "everything except y".

Most of these symbols can be defined in terms of even more basic symbols, many of which can be defined in terms of even more basic symbols, just as one is defined in terms of zero, and zero is defined in terms of the empty set and a set with just the empty set in it. The definition given in my previous post is an attempt at unraveling these definitions in such a way that one is defined in terms of the most basic logical and set theoretical concepts.

I couldn't translate the huge definition of one into plain English, but I can try to translate the definition given above. Here it goes... the number one is every class, collection or set we can define which has an object in it, and nothing other than that particular object in it. Interestingly, one, by this definition, is not just one thing, but the aggregate of everything that has the property of oneness e.g. God is one, I am one, the set of all wives which belong to me is one, etc.

What It Means To Be Number One

I just finished reading W.V.O. Quine's textbook, Mathematical Logic. It only took me about 3 years to finish it (literally), and it was by far the most difficult book I have ever read, but it was worth it. The book defines predicate logic, basic set operations, natural, rational, and real numbers, arithmetic, and some other things all in terms of set membership, universal quantification, and joint denial. Then it proves and explains a version of Godel's Incompleteness Theorem. It is fascinating stuff. I hope I'll understand it all someday.

To celebrate, I wrote in truly nerdy fashion a Perl script to generate the natural number one expressed in terms of class membership, class abstraction, universal quantification, and joint denial, according to Quine's definitions. Here it is.