Does mathematics need new axioms? (Q5890202)

These papers are revisions of lectures and responses presented by the four authors at the Association of Symbolic Logic annual meeting in Urbana-Champaign in June 2000. Solomon Feferman's views are the most radical. He believes that working mathematicians now feel no pressing need for new axioms and that there is no evidence that the open problems of interest to those mathematicians will require for their solution anything more than ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Further, he claims that the mathematics they do can be carried out in even weaker systems and that the scientifically applicable part of their work can be formalized in systems that are conservative over Peano arithmetic (as well as over much weaker theories). Even more radically, he regards the Continuum Hypothesis (CH) to be an inherently vague statement and thinks that the power set of the natural numbers is not a definite mathematical object. Of course, much of modern set theory also has to count as inherently vague. In line with these views, the question of the title of this paper is taken by Feferman to be an essentially philosophical problem, not susceptible to a generally accepted mathematical solution. Penelope Maddy, on the other hand, argues that mathematics has no need of philosophical justification and that the acceptance of any new axioms in a given domain is a matter to be decided by the leading mathematicians working in that area. The basis for their decision would be whether or not the axioms benefit their research goals. To support this approach, she cites the example of the axiom of choice (AC), which is now almost universally used but was once the subject of debate. Presumably her point is that the usefulness of AC trumped any doubt about its truth. Moreover, she casts scorn on any concern about whether or not the CH has a determinate truth value in some Platonic world of sets. She doesn't consider the objection that mathematicians need the image of some such Platonic world to guide their research (as Gödel claimed with respect to his own work). In addition, there are intrinsic arguments in favor of AC; many mathematicians implicitly used AC in their reasoning before AC was explicitly singled out by Zermelo. John Steel agrees, for the most part, with Maddys point of view and is himself one of the leading set theorists whose research should determine the acceptability of various axioms concerning large cardinals and determinacy principles. However, he believes that those axioms do have some sort of intrinsic validity in that they are expressions of an intrinsically plausible informal reflection principle. He discusses some of those axioms and contrasts them with the generally rejected set theory based on the axiom of constructibility, V = L. Although he is against tossing out good mathematics to save inherently vague philosophy, he does allow that philosophy may have a role to play in that a resolution of CH may need some accompanying analysis of what it is to be a solution to the Continuum Problem. Thus, his position seems to lie somewhere between those of Feferman and Maddy. Harvey Friedman believes that mathematicians working in the traditional areas have little interest or respect for foundational research except for its value in providing interpretations within ZFC of their basic notions and techniques. He attributes this attitude partly to excessive generality in the formulation of the problems which allowed for pathological cases, but he does not mention the traditional analyst's willingness to live with the pathological examples that arise in analysis itself. Friedman also expresses some skepticism about the direct intuition about projective sets of reals that present-day set theorists rely on to justify various large cardinal axioms and to reject V = L. Nevertheless, he thinks that large cardinal axioms will begin to be accepted as new axioms for mathematics with controversy because they will be consequences of certain quite concrete and useful assumptions that are embodied in his new Boolean relation theory (BRT). A very condensed sketch of that theory is given in footnotes on pp. 438-439. Friedman also provides a list of criteria that should govern the adoption of new axioms. (BRT) is designed to satisfy these criteria.