If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or stationary logic [6]. Our first set of results show that both [Formula: see text] and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between [Formula: see text] and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals [Formula: see text] of [Formula: see text] are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals, the Continuum Hypothesis is true in the relativized model.
Assume ℵ 0 , ℵ 1 → λ, λ + . Assume M is a model of a first order theory T of cardinality at most λ + in a vocabulary L(T ) of cardinality ≤ λ. Let N be a model with the same vocabulary. Let ∆ be a set of first order formulas in L(T ) and let D be a regular filter on λ. Then M is ∆-embeddable into the reduced power N λ /D, provided that every ∆-existential formula true in M is true also in N . We obtain the following corollary: for M as above and D a regular ultrafilter over λ, M λ /D is λ ++ -universal. Our second result is as follows: For i < µ let M i and N i be elementarily equivalent models of a vocabulary which has has cardinality ≤ λ. Suppose D is a regular filter on µ and ℵ 0 , ℵ 1 → λ, λ + holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length λ + on i M i /D and i N i /D. This yields the following corollary: Assume GCH and λ regular (or just ℵ 0 , ℵ 1 → λ, λ + and 2 λ = λ + ). For L, M i and N i be as above, if D is a regular filter on λ, then i M i /D ∼ =
We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle fin λ,D introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which M λ /D is not λ ++ -universal and elementarily equivalent models M and N of size λ for which M λ /D and N λ /D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1].
This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
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