Applications of Quantifier Elimination in Banach Spaces

Stefano Baratella & Siu-Ah Ng

Following an approach to quantifier elimination pursued in an abstract setting by Keisler [K], in [BN] we introduced a notion of quantifier elimination (briefly: QE) for structures based on Banach spaces. Such a notion can be proved equivalent to Henson’s quantifier elimination in sufficiently saturated nonstandard hulls. QE refers to a class of infinitary expressions, called by Keisler neocompact formulas, extending the class of positive bounded formulas introduced in [H].

First we present some properties equivalent to QE, then we discuss consequences of QE (or weaker properties like density or back-and-forth) on the geometry of a Banach space.

Eventually, we point out relationship between the abovementioned properties and fixed point theorems in reflexive nonstandard hulls of Banach spaces.

[BN] S. Baratella and S.-A. Ng, Neocompact quantifier elimination in structures based on Banach spaces, to appear in Annals of Pure and Applied Logic.

[H] C.W. Henson, Nonstandard hulls of Banach spaces, Israel J. Math. 25 (1976), pp. 108-144.

[K] H.J. Keisler, Quantifier elimination for neocompact sets, Journal of Symbolic Logic 63, no. 4 (1998), pp. 1442-1472.

Addresses Stefano Baratella: Dipartimento di Matematica, via Sommarive 14, 38050 Povo TN - Italia

E-mail: baratell@science.unitn.it

Siu-Ah Ng: University of Natal, Pietermaritzburg, 3209 South Africa

E-mail: ngs@math.unp.ac.za