Generic Large Cardinals as Axioms

Abstract

We argue against Foreman's proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.Shortly after proving that the set of all real numbers has a strictly larger cardinality than the set of integers, Cantor conjectured his Continuum Hypothesis (CH): that there is no set of a size strictly in between that of the integers and the real numbers [1]. A resolution of CH was the first problem on Hilbert's famous list presented in 1900 [19]. Gödel made a major advance by constructing a model of the Zermelo-Frankel (ZF) axioms for set theory in which the Axiom of Choice and CH both hold, starting from a model of ZF. This showed that the axiom system ZF, if consistent on its own, could not disprove Choice, and that ZF with Choice (ZFC), a system which suffices to formalize the methods of ordinary mathematics, could not disprove CH [16]. It remained unknown at the time whether models of ZFC could be found in which CH was false, but Gödel began to suspect that this was possible, and hence that CH could not be settled on the basis of the normal methods of mathematics. Gödel remained hopeful, however, that new mathematical axioms known as "large cardinals" might be able to give a definitive answer on CH [17].The independence of CH from ZFC was finally solved by Cohen's invention of the method of forcing [2]. Cohen's method showed that ZFC could not prove CH either, and in fact could not put any kind of bound on the possible number of cardinals between the sizes of the integers and the reals. Lévy and Solovay further developed the forcing machinery, and noticed that it also destroyed Gödel's hopes for large cardinals. Forcing allowed one to manipulate the cardinal value of the set of reals, passing from one model of ZFC to another giving a different answer on CH, without disturbing any large cardinals in the process [22].This was not the last word on CH from the community of set theorists. Several programs to develop acceptable axioms that settle CH have been put forward. Matthew Foreman, a leading set theorist, has suggested a solution to CH via axioms called "generic large cardinals." Our goal here is to critically examine Foreman's proposal. First, we describe the goals these axioms are supposed to meet and the kinds of considerations in their favor, highlighting the claim that the favorable considerations for traditional large cardinals transfer to the generic ones. Second, we discuss many technical difficulties in accommodating generic large cardinals in a single axiomatic framework, and present some new "mutual inconsistency results" that raise troubles for the program. Third, we examine the considerations in favor of traditional large cardinals and argue that they do not have the same import for the generic variety. Finally, we consider an alternative take on these kinds of axiomsThe author would like to thank Sean Walsh and Neil Barton for many helpful discussions.

Incompatibility of Generic Hugeness Principles

“…Regarding the history: Woodin proved, in unpublished work mentioned in [8, p. 1126], that it is inconsistent for to be minimally generically 3-huge while is minimally generically 1-huge. Subsequently, the author [3] improved this to show the inconsistency of a successor cardinal being minimally generically n -huge while is minimally generically almost-huge, where . The weakening of the hypothesis to being only generically 1-huge uses an idea from the author’s work with Cox [1].…”

Incompatibility of generic hugeness principles

“…Regarding the history: Woodin proved that it is inconsistent for ω 1 to be minimally generically 3-huge while ω 3 is minimally generically 1-huge. Subsequently, the author [3] improved this to show the inconsistency of a successor cardinal κ being minimally generically n-huge while κ +m is minimally generically almost-huge, where 0 < m < n. The weakening of the hypothesis to κ being only generically 1-huge uses an idea from the author's work with Cox [1].…”