# Repository | Book | Chapter

(2018) *The hyperuniverse project and maximality*, Dordrecht, Springer.

Let *M* be a transitive model of ZFC. We say that a transitive model of ZFC, *N*, is an outer model of *M* if *M* ⊆ *N* and ORD ∩ *M* = ORD ∩ *N*. The outer model theory of *M* is the collection of all formulas with parameters from *M* which hold in all outer models of *M* (which exist in a universe in which *">M* is countable; this is independent of the choice of such a universe). Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal *κ*, we show that it is consistent to have a transitive model *M* of ZFC of size *κ* in which the outer model theory is lightface definable, and moreover *M* satisfies *V* = *HOD*. The proof combines the infinitary logic *L*_{∞,ω }, Barwise's results on admissible sets, and a new forcing iteration of length strictly less than *κ*^{+} which manipulates the continuum function on certain regular cardinals below *κ*. In the Appendix, we review some unpublished results of Mack Stanley which are directly related to our topic.

Publication details

DOI: 10.1007/978-3-319-62935-3_7

Full citation:

Honzik, R. , Friedman, S. (2018)., Definability of satisfaction in outer models, in C. Antos, R. Honzik, C. Ternullo & S. D. Friedman (eds.), * The hyperuniverse project and maximality*, Dordrecht, Springer, pp. 135-160.

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