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Unformatted text preview: 1 TRANSFER PRINCIPLES IN SET THEORY by Harvey M. Friedman Department of Mathematics Ohio State University May 21, 1997 [email protected] www.math.ohiostate.edu/~friedman/ TABLE OF CONTENTS PART A. HIGHLIGHTS. Introduction. A1. Two basic examples of transfer principles. A2. Some formal conjectures. A3. Sketch of some proofs. A4. Ramsey Cardinals. A5. Towards a new view of set theory. PART B. FULL LIST OF CLAIMS. (Based on 5/1996 abstract) 1. Transfer principles from N to On. A. Mahlo cardinals. B. Weakly compact cardinals. C. Ineffable cardinals. D. Ramsey cardinals. E. Ineffably Ramsey cardinals. F. Subtle cardinals. G. From N to <On. H. Converses. 2. Transfer principles for general functions. A. Equivalence with Mahloness. B. Equivalence with weak compactness. C. Equivalence with ineffability. D. Equivalence with Ramseyness. E. Equivalence with ineffable Ramseyness. F. From N to <On. G. Converses. H. Some necessary conditions. 3. Transfer principles with arbitrary alternations of quantifiers. 4. Decidability of statements on N. 5. Decidability of statements on <On and On. NOTE: Talks are based on Part A only 2 PART A. HIGHLIGHTS INTRODUCTION The results presented here establish unexpected formal relationships between the functions on N and the functions on On. (Here N is the set of all natural numbers and On is the class of all ordinal numbers). These results provide a reinterpretation of certain large cardinals axioms as extensions of known facts about functions on N to functions on On. More specifically, the transfer principles assert that any assertion of a certain logical form that holds of all functions on N holds of all functions on On. These transfer principles are proved using certain large cardinal axioms. In fact, we show that these transfer principles are equivalent to certain large cardinal axioms. A1. TWO BASIC EXAMPLES OF TRANSFER PRINCIPLES Let N = {0,1,...} and On be the class of all ordinals. We begin by considering the sentences *) ( 2200 f 1 ...f p :N k → N)( 2200 x 1 ...x q )( 5 y 1 ...y r ) A(x 1 ...x q ,y 1 ...y r )), where A is a Boolean combination of inequalities between (possibly nested) terms involving the f’s, x’s, and y’s. Constants for elements of N are allowed. The x’s and y’s range over N. And consider the corresponding sentence **) ( 2200 f 1 ...f p :On k → On)( 2200 x 1 ...x q )( 5 y 1 ...y r ) (A(x 1 ...x q ,y 1 ...y r )). The x’s and y’s range over On. Note that **) is a sentence in class theory. Now consider this transfer principle: 3 T ) for all suitable k,p,q,r,A, * → **. Unfortunately, it is easy to refute this transfer principle, even for k = 1 and no constants allowed. We say that f:N k → N is weakly regressive iff for all x ∈ N k , f(x) ≤ min(x). Here min(x) is the least coordinate of x....
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
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