The Hanf number of stationary logic
The Hanf number of a logic is the least cardinal \(\kappa\) such that every sentence with a model of power \(\geq \kappa\) in fact has arbitrarily large models. For \(L(Q_ 1)\) where \(Q_ 1=''there\) exist uncountably many'', this number is known to be \(\beth_{\omega}\). For the logic L(aa) or ''stationary logic'', we show that the Hanf number \(\kappa\) is much larger than \(\beth_{\omega}\). In fact, \(\omega <\kappa =\beth_{\kappa}\) and \(\kappa\) is at least as large as the Hanf number for logic with quantification over countable sets. In the universe L we show that the Hanf numbers of L(aa) and of second-order logic are the same. The result also holds in L(D) where D is a normal ultrafilter on some cardinal. We also relate the Hanf number of L(aa) to large cardinals, by giving the consistency (relative to the existence of certain large cardinals) of the assertion that it exceeds many measurable cardinals. The main tool for most of these results is Theorem 3.3, which says that \(\aleph_ 1\) is an anomaly in the following sense: while L(aa) behaves nicely on models of power \(\aleph_ 1\), i.e., there is compactness and completeness, there is nevertheless a sentence \(\psi\) of L(aa) with arbitrarily large models, such that \(\psi\) describes models of set theory in which every countable subset of the model is an element of the model, as long as the model has power greater than \(\aleph_ 1\).
scientific article; zbMATH DE number 3993560