Cardinal Arithmetic for Skeptics

When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with “consistency” rather than “truth” may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of 20 , cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic. To see the subject in this light it will be necessary to carry out a substantial shift in our point of view. To make a very rough analogy with another generalization of ordinary arithmetic, the natural response to the loss of unique factorization caused by moving from Z to other rings of algebraic integers is to compensate by changing the definitions, rescuing the theorems. Similarly, after shifting the emphasis in cardinal arithmetic from the usual notion of exponentiation to a somewhat more subtle variant, a substantial body of results is uncovered that leads to new theorems in cardinal arithmetic and has applications in other areas as well. The first shift is from cardinal exponentiation to the more general notion of an infinite product of infinite cardinals; the second shift is from cardinality to cofinality; and the final shift is from true cofinality to potential cofinality (pcf). The first shift is quite minor and will be explained in §1. The main shift in viewpoint will be presented in §4 after a review of basics in §1, a brief look at history in §2, and some personal history in §3. The main results on pcf are presented in §5. Applications to cardinal arithmetic are described in §6. The limitations on independence proofs are discussed in §7, and in §8 we discuss the status of two axioms that arise in the new setting. Applications to other areas are found in §9. The following result is a typical application of the theory.