In paper [5] it was shown that a great part of model theory of logic with the generalized quantifier Q x = there exist uncountably many x is reducible to the model theory of first order logic with an extra binary relation symbol. In this paper we consider when the quantifier Q x can be syntactically defined in a first order theory T. That problem was raised by Kosta Doen when he asked if the quantifier Q x can be eliminated (...) in Peano arithmetic. We answer that question fully in this paper. (shrink)

A topological class logic is an infinitary logic formed by combining a first-order logic with the quantifier symbols O and C. The meaning of a formula closed by quantifier O is that the set defined by the formula is open. Similarly, a formula closed by quantifier C means that the set is closed. The corresponding models are a topological class spaces introduced by Ćirić and Mijajlović (Math Bakanica 1990). The completeness theorem is proved.