NewAxioms_12pt.6.5.00

NewAxioms_12pt.6.5.00 - 1 DOES MATHEMATICS NEED NEW AXIOMS...

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1 DOES MATHEMATICS NEED NEW AXIOMS? ASL Meeting, Urbana Panel discussion: June 5, 2000 Harvey M. Friedman The point of view of the set theory community is well represented here. I want to concentrate on the perspective of mathematicians outside set theory. 1. MATHEMATICIANS’ VIEWPOINT. New axioms are needed in order to settle various mathematic- ally natural questions. Yet no well known mathematician outside set theory is even considering adopting any new axioms for mathematics, even though they are aware of at least the existence of the independence results. The difference in perspective of set theorists versus mathe- maticians who are not set theorists, is enormous. Recall that mathematics goes back, say, 2,500 years - whereas set theory in the relevant sense dates back to the turn of the 20th century. For 2,500 years, mathematici-ans have been concerned with matters of counting and geometry and physical notions. These main themes gave rise to arithmetic, algebra, geometry, and analysis. The interest in and value of mathematics is judged by mathematicians in terms of its relevance and impact on the main themes of mathematics. It is generally recognized by most mathematicians that set theory is the most convenient vehicle for achieving rigor in mathematics. For this purpose, there has evolved a more or less standard set theoretic interpretation of mathematics, with ZFC generally accepted as the current gold standard for rigor. It is simply false that a number theorist is interested in and respects set theory just as they are interested in and respect group theory, topology, differential geometry, real and complex analysis, operators on Hilbert space, etcetera.
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2 The reason for this attitude is quite fundamental and ex- tremely important. A number theorist is of course interested in complex analysis because he uses it so much. But not so with operators on Hilbert space. Yet there is still a distant respect for this because of a web of substantive and varied interconnections that chain back to number theory. Set theory does not have comparable interconnections. For the skeptical, the degree of extreme isolation can be subjected to various tests including citation references - broken down even into their nature and quality. Using a critical notion from statistics, set theory is an extreme outlier. Nor is set theory regarded as intrinsically interesting to mathematicians, independent of its lack of impressive inter- connections. Why? For the mathematician, set theory is regarded as a con- venient way to provide an interpretation of mathematics that supports rigor. A natural number is obviously not a set, an ordered pair is obviously not a set, a function is obviously not a set of ordered pairs, and a real number is obviously not a set of rationals.
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