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histincphen_5_21_97 - 1 SOME HISTORICAL PERSPECTIVES ON...

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1 SOME HISTORICAL PERSPECTIVES ON CERTAIN INCOMPLETENESS PHENOMENA Harvey M. Friedman Department of Mathematics Ohio State University [email protected] www.math.ohio-state.edu/~friedman/ May 21, 1997 We have been particularly interested in the demonstrable unremovability of machinery, which is a theme that can be pursued systematically starting at the most elementary level - the use of binary notation to represent integers; the use of rational numbers to solve linear equations; the use of real and complex numbers to solve polynomial equations; and the use of transcendental functions to solve differential equations. Practical situations arise such as the use of complex variables in number theory, or group theory in topology. Here there has been no demonstrable unremovability. But it appears that when the machinery is removed, clarity and power is lost. This kind of unremovability is extremely difficult to get at rigorously. Over the years, a growing collection of cases of demonstrable unremovability of increasing interest have been developed. Here is a brief synopsis of some of the highlights. Around 1967, Tony Martin solved a crucial problem in infinite game theory involving Borel sets, using a massive amount of machinery, going well beyond the usual axioms for mathematics. Around 1968, we proved that a small part of the machinery was unremovable - uncountably many uncountable cardinals. In 1974, after an extended effort, Martin reduced the amount of machinery he used to uncountably many uncountable cardinals. We later gave the following reinterpretation of Martin's theorem: every Borel set in the plane that is symmetric about the line y = x contains or is disjoint
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2 from the graph of a Borel function, and showed the unremovability of uncountably many uncountable cardinals for this statement.
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histincphen_5_21_97 - 1 SOME HISTORICAL PERSPECTIVES ON...

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