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Constructing Cardinals from below
"... this paper are all formulated in terms of a formula #(X), with only X free. For now, the formula is one in the language of basic set theory and X is a secondorder variable. The corresponding condition is that #(A) is true in R(#) for some A and, for no # # is #(A R(#)) true in R(#). The fo ..."
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this paper are all formulated in terms of a formula #(X), with only X free. For now, the formula is one in the language of basic set theory and X is a secondorder variable. The corresponding condition is that #(A) is true in R(#) for some A and, for no # # is #(A R(#)) true in R(#). The formal expression that this condition is an existence condition is the axiom #X[#(X) # ### R(#))] (1) (X) is the result of restricting the first and secondorder bound variables in #(X) to R(#) and R(# + 1), respectively. Axioms of this form have been called reflection principles, because they express the fact that R(#)'s possession of a certain property is reflected by R(#)'s possession of it for some # #