Transfinity

Let me let you in on a secret: There are no infinitely large sets in mathematics. Or infinitely divisible lines or spaces. (According the philosophy of mathematics called intentionalism, that is.)

Now whether there are such things in nature, one may speculate about. But from the point of view of quantum theory, there is a smallest size (Planck length) and time interval (Planck time). And inflationary cosmology says our universe likely has a finite number of particles, albeit an incredibly large number of them.

How does the idea of infinity enter mathematics? It enters by the language of nesting quantifiers.

Take one of the axioms of arithmetic of natural numbers:

(forall (m) (if (natural m) (exists (n) (and (natural n) (succ m n)))))

Here I'm using a formulation of the axioms with S-expressions for the object language. (succ m n) says n is the successor of m.

(forall list-of-variables expression)
(exists list-of-variables expression)

are interpreted in the standard way as the universal and existential quantifiers.

But suppose a domain of quantification is added after each quantifier:

(forall P list-of-variables expression)
(exists Q list-of-variables expression)

where P and Q are the domains over which the variables range.

Then the above axiom becomes:

(forall P (m) (if (natural m) (exists Q (n) (and (natural n) (succ m n)))))

The interpretation is that when domains of quantification are nested, then the domain of the inner quantification is a superset of the domain that encloses it (Q ⊃ P) — i.e. the inner-more you go, the bigger the sets get. And not only that, but all domains of quantification are finite! (When the domains are the same as in standard logic one is "committed" to an infinite set of natural numbers, but this is not the case for nesting domains.)

If one has an entire mathematical theory (expressed as a collection of S-expressions in this language) then one is talking about variables ranging only over finite sets, but one gets bigger and bigger finite sets as needed in any practical application of the theory. A book by Shaughan Lavine (based on work by Jan Mycielski, discussed in previous posts January 25, 2009 and January 8, 2012) likens this to getting bigger and bigger bags of beans when needed. (I guess it would help to be a bean counter.)

So whatever mathematicians told you about infinity: Take with a grain of salt — or a bag of beans!



This post is the second in a series of seven for the 7 Day Blogging Challenge for Bloggers from +Jenson Taylor.