Abstract: First, we review the man ideas of modal-structural (MS) interpretations of mathematical theories, especially in second-order formulations, e.g. Dedekind-Peano Arithmetic, and Real Analysis (with 2d-order LUB axiom), and set theories like Zermelo, ZFC, and extensions with (small) large cardinal axioms. Next we turn to the problem of introducing higher-order reflection principles into the MS versions of Z- (Zermelo less Infinity). (This choice will be explained). Even the usual motivation of such reflection has to be adjusted, since MS recognizes no “absolutely infinite” totalities such as “the universe of all sets" or "all the ordinals." Then we consider two attempts, one based on a translation scheme due to Putnam, but which runs into inconsistency; and a second which we claim is consistent relative to standard ZFC + e.g. an Erdös cardinal (borrowing from recent work of W.W.Tait and Peter Koellner). We close with discussion of significance of such results.
Professor Helman is from the University of Minnesota
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