One of the standard axioms of Zermelo-Frankel set theory, the axiom of infinity, asserts the existence of an infinite set. "There is a nonempty set W such that x in W implies xu{x} in W." The other axioms of set theory guarantee an infinite number of sets but are rather agnostic about an infinite set. They neither prove nor disprove that an infinite set exists.

This axiom is now accepted dogma but was highly controversial a hundred years ago. Everyone agreed that there are infinitely many objects but the idea that there was or should be an infinite object was hard for some, particularly the intuitionists and constructivists (the mathematical atheists) such as Brouwer, Kronecker. Poincare and Weyl.

Kronecker and Weyl would object to my identifying them as atheist. In fact they regarded themselves as the orthodoxy and considered set theorists with their transcendental enties to be mystics. According to Kronecker numbers were the "work of God" but infinite sets were the suspect inventions of man and more mysticism than mathematics. He didn't even accept the existence of irrationals with their infinitude of decimal places. He once commented to Lindemann "of what use is your beautiful investigation regarding pi? Why study such problems, since irrational numbers are non-existent?"

Weyl waxed almost revivalist in denouncing these speculative entities: "classical logic was abstracted from the mathematics of finite sets and their subsets.... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory, for which it is justly punished by the antimones. It is not that such contradictions showed up that is surprising, but that they showed up at such a late stage of the game."

Despite their protestations of orthodoxy, in hindsight, they seem to be the ones of little faith. Everyone now accepts the controversial axiom of infinity as a article of the mathematical faith.

But why accept the existence of an object which cannot be constructed in any reasonable sense? Basically the answer boils down to a matter of simplicity.

The intuitionists labored mightly (and messily) to establish an alternative, constructive foundation for classical analysis. This culmunated in the publication of the now-out-of-print 1967 text "Constructive Analysis" by Errett Bishop.

While Bishop showed that mathematics could be done without the infinite, neither he nor anyone else has shown that it can be done with elegance. More than anything else, wading through convoluted intuitionist mathematics leaves one with a renewed appreciation for the beauty and simplicity of classical math.

One could argue that irrational numbers are unnessary in science since no measurements are ever made to more than a finite number of decimals places. But try to express the relation between a radius and a circumference without them. Or try to get by without the least upper bound theorem in analysis as Bishop does. One quickly discovers that mathematical life is hard without these unneeded and speculative objects.

Ordinarily those of us with skeptical inclinations invoke Occam's razor to dispatch entities of dubious existence, be they dragons or dieties. But not here. Mathematics is simpler, easier and more elegant in a set theory universe with these controversial infinities. With respect to the existence of infinity, we've all become believers.