By Françoise Delon, Ulrich Kohlenbach, Penelope Maddy, Frank Stephan

The once a year ecu assembly of the organization for Symbolic common sense, often referred to as the good judgment Colloquium, is likely one of the so much prestigious annual conferences within the box. the present quantity, common sense Colloquium 2007, with contributions from plenary audio system and chosen exact consultation audio system, includes either expository and examine papers via the superior logicians on the planet. This quantity covers many parts of up to date good judgment: version conception, evidence concept, set idea, and laptop technology, in addition to philosophical common sense, together with tutorials on cardinal mathematics, on Pillay's conjecture, and on automated constructions. This quantity should be important for specialists in addition to these drawn to an summary of crucial modern subject matters in mathematical common sense.

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10] A. , Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. [11] A. Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, Higher Set Theory (Proc. , Math. , Oberwolfach, 1977), Lecture Notes in Mathematics, vol. 669, Springer, Berlin, 1978, pp. 99–275. [12] J. Ketonen, Strong compactness and other cardinal sins, Annals of Pure and Applied Logic, vol. 5 (1972/73), pp. 47–76. CARDINAL PRESERVING ELEMENTARY EMBEDDINGS 31 [13] P. Larson, The Stationary Tower: Notes on a Course by W.

For each term t[x ] of type with a distinguished variable x of type , we can construct a term q of type → whose variables are those of t except for x such that, for every term s of type and atomic formula A with a distinguished variable w , HA0 A[t[s/x]/w] ↔ A[qs/w]. For instance, if t is of type 0, then qs = t[s/x]. t[x])s = t[s/x] ( reduction). t[x])s in any formula.

There is no formal definition of what a large cardinal is, but a few features can be distinguished. Typically, they are regular cardinals κ such that Vκ is itself a model of ZFC and, more importantly for present purposes, one can associate to κ a family of elementary embeddings j : V → M where M is a transitive class. The association is usually (as in the case of measurable, strong or supercompact cardinals) that κ is the critical point cp(j) of j, the first ordinal α such that j(α) > α, but it can take other shapes, as is the case with Woodin cardinals.

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