Wednesday, December 28, 2011

"Transfinite" vs "Infinite"

Ask a third grader how large the set of alllll the numbers are and he'll cry out "infinity!"
Ask a set theorist and he'll very likely say "transfinite". Transfinite? Really asshole? Come on...

So why are the cardinalities of infinite sets referred to as "transfinite cardinals"? It turns out that the reason is somewhat historical. Georg Cantor, the man responsible for placing set theory at the very foundation of all modern mathematics (and, therefore, science) was a deeply religious man who adhered to the idea that the one true god was the only "absolutely infinite" construct which could exist in the Universe; He is beyond any concept, whether it be a set or a world or a galaxy. This led Cantor to coin the term "transfinite" to describe the notion of a quantity which is not finite, but which is not the absolute infinity which he believed could only describe God - it is somewhere in between. It is the naturals; it is the reals; it is every subsequent power set of these sets, etc.

At first glance, the matter seems trivial at best and childish at worst, especially to those deeply skeptical of religion. But one must be careful to look at the perspective of the time: Cantor's Theorem, which directly implies that there are an infinite amount of successively larger infinities, was exotic. More than exotic. Ludicrous. Insane. Heretical to the omnipresence, omnipotence, and omniscience of God. A challenge - a provably correct challenge! - to the very fabric of how human beings understand the notion of abstract quantity, and, by extension, mathematics, science, philosophy, ontology, and the Universe as a whole. I find it reasonable that Cantor would draw a distinction between the hierarchy of infinities and the "absolute infinity" which he believed to be only attainable by God.

Even if the notion of God is removed, however, the question remains relevant. What is the "largest" object which can be constructed while remaining logically consistent with the rest of mathematics? Does such a construction exist? If so, should "less numerous" infinities, such as the cardinality of the natural numbers, still be titled "infinite"? Should an infinite set be one which always contains more elements, or should it be defined as a set which contains all elements? Can such a set exist? Can an infinite set exist at all in the physical world beyond the abstract universe of mathematical thought?

Answers are sought to many of these questions within the domain of large cardinal theory, and although they admittedly rarely come into practical consideration in any remotely applicable area of mathematics, it is nonetheless instructive to consider them, if only to feel a bit humble and to realize just how fucked up and confused the foundations of mathematics and science remain.