New Axioms

Since Hilary does mention new axioms, I’ll take the opportunity, to say a little about the topic, although I suspect that Hilary will not disagree with much of what I say. Zermelo’s axiom of choice is the obvious example of a proposed axiom that has gained general acceptance although it excited considerable controversy in the early years of the twentieth century. There was a prominent group of French mathematicians who went to considerable lengths to avoid the axiom, so that, for example, they didn’t permit themselves to say without qualification: The union of a countable set of countable sets is countable. By the second half of the century, it had become an indispensable tool in various branches of mathematics.

The branch of mathematics called descriptive set theory, pioneered in Eastern Europe during the first half of the twentieth century, provides an interesting example of the power of a new axiom. We write R for the set of real numbers and for a set B e Rn+1, we write Proj(B) for the set

The hierarchy of projective sets is defined simultaneously in all Rn as follows:

Lusin proved the key hierarchy theorem: For non-negative integers m, ?m c ^+1 and nlm c nm+1. Also, for m > 0, ?lm - nlm = 0.

Souslin proved that the Borel sets are exactly those that are in both ?} and П1, and in 1917 Lusin showed that every set in ?j is Lebesgue measurable. It seemed plausible to researchers, noting that known proofs of the existence of non-measurable sets used the axiom of choice, that sets that had explicit definitions should be measurable. This leads to the conjecture that projective sets, evidently being explicitly definable, should all be Lebesgue measurable. But efforts to prove this failed. When Cohen developed his forcing method, it became clear why success was so elusive. The proposition that all projective sets are Lebesgue measurable turned out to be undecidable in ZFC.

A new axiom seemed to be called for to settle the question, and it turned out that the concept of determinacy provided the key: Associated with a set A of real numbers is an infinite game defined as follows: Players I and II alternately move by each specifying a binary digit 0 or 1. They thus specify the binary expansion of a real number x in the unit interval. If x is the fractional part of a member of A, then I wins; otherwise II wins. The set A is determined if either I or II has a winning strategy. The axiom of projective determinacy (PD), states that every projective set of real numbers is determined. And PD does yield the desired result that every projective set is Lebesgue measurable. In fact a number of other open questions about projective sets can be settled when PD is assumed. Tony Martin and John Steel were able to derive PD from a suitable large cardinal axiom, thus providing a satisfying conclusion to those who view large cardinals with equanimity.[1]

On the other hand, axioms asserting the existence of “large” cardinals are certainly not being widely accepted. Harvey Friedman has found a considerable number of propositions in combinatorial mathematics, some of them rather attractive, that can only be proved by assuming such axioms.

  • [1] The cardinal in question is in fact quite large: a countable infinity of Woodin cardinals with ameasurable cardinal above them.
 
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