TransPrin12pt050797

TransPrin12pt050797 - 1 TRANSFER PRINCIPLES IN SET THEORY...

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1 TRANSFER PRINCIPLES IN SET THEORY by Harvey M. Friedman Department of Mathematics Ohio State University May 7, 1997 friedman@math.ohio-state.edu http://www.math.ohio-state.edu/~friedman/ 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. TWO BASIC EXAMPLES OF TRANSFER PRINCIPLES Let N = {0,1,. ..} and On be the class of all ordinals. We begin by considering the sentences *) ( " f 1 ...f p :N k Æ N)( " x 1 ...x q ) ( $ y 1 ...y r )(A(x 1 ...x q ,y 1 ...y r )), where A is a Boolean combin-ation of inequalities between (possibly nested) terms in-volving 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 **) ( " f 1 ...f p :On k On) ( " x 1 ...x q )( $ y 1 ...y r )(A(x 1 ...x q ,
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2 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: T 0 ) 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. Consider the following sentences. *’) ( " wr f 1 ...f p :N k N) ( " x 1 ...x q )( $ y 1 ...y r ) (A(x 1 ...x q ,y 1 ...y r )) **’) ( " wr f 1 ...f p :N k On) ( " x 1 ...x q )( $ y 1 ..y r ) (A(x 1 ...x q ,y 1 ,...y r )) Again, the x’s and y’s in the first form range over N, and the x’s and y’s in the second form range over On. And the transfer principle: T 1 ) for all suitable k,p,q,r,A, *’ **’. Our first interesting trans-fer principle T 1 is equivalent to a large cardinal principle. Here we use VB + AxC as the base theory. We can even weaken this transfer principle to T 1 ’) for all suitable k,p,q,r,A, * **’ and obtain the same results. We now introduce another modification of T 0 involving quantification over all functions on N. Fix E = {2 n : n N}, and E^ = {2 a :a On}.
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3 *^ ( " f 1 ...f p :N k N) ( " x 1 ...x q )( $ y 1 ...y r E) (A(x 1 ...x q ,y 1 ...y r )) **^ ( " f 1 ...f p :On k On) ( " x 1 ...x q )( $ y 1 ...y r E^) (A(x 1 ...x q ,y 1 ...y r )) We were deliberately vague as to what kind of exponentia- tiation is used in the definition of E^. We can take it to be either ordinal exponentiation or cardinal exponentiation. The results are the same. T 2 ) for all suitable k,p,q,r,A, *^ **^. This second transfer prin-ciple is equivalent to a class theoretic large cardinal axiom.
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TransPrin12pt050797 - 1 TRANSFER PRINCIPLES IN SET THEORY...

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