SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard

SOME SET THEORIES ARE MORE EQUAL 7use of the stronger axioms looks to them like an unnecessary logicians’finicking.There are other examples like Martin’s proof of Borel Determinacywhich initially was inferred from his proof of analytic determinacy usingmeasurable cardinals and the Wadge Borel determinacy , which lookslike any genuine statement studied by analysts, which was proved originallyfrom Borel determinacy, hence used ZFC in a strong way butwas later proved in second order arithmetic.In summary I think that it is very likely that the working mathematicianwill run into problems that the only way to settle them willbe by expanding the usual axiom system of Set Theory. And the mathematicalcommunity will accept such a solution as legitimate if the settheoretical principles used will be accepted as part of a canonical widelyaccepted system of axioms.We , as set theorists , face the problem of how to choose among themultitude of possible set theories those that will be mathematicallyfruitful,and acceptable in the long run. The next sections will chartsome of the considerations that can lead us in this process.4. The search for new axiomsWe are going to present several criteria or principles that shouldserve as guidance in the search for new axioms. These are rather informalprinciples which leave a large leeway in their interpretation in aparticular case.We see the search for new axioms as a ongoing process, not dissimilar to the process in other fields of science , by which a scientifictheory is crystalized by a sequence of trials and errors, where atany particular moment there may be several competing options. Thecriteria of testing a particular axiom or a set of axioms is very likelyvery different form those used in Science but there are similarities. Thefamous quote from Gödel [15] is very appropriate introduction to theprocess we envision.. . . Even disregarding the intrinsic necessity of some newaxiom. . . a probable decision about its proof is possiblealso . . . by studying its success. Success here meansfruitfulness in consequences. . . There might exist axiomsso abundant in their verifiable consequences, sheddingso much light upon a whole field and yielding such powerfulmethods for solving problems . . . that, no matterwhether or not they are intrinsically necessary, theywould have to be accepted at least in the same senseas any well-established physical theory