It is by now well known that Gödel first advocated the philosophy of Leibniz and then, since 1959, that of Husserl. This raises three questions: 1. How is this turn to Husserl to be interpreted? Is it a dismissal of the Leibnizian philosophy, or a different way to achieve similar goals? 2. Why did Gödel turn specifically to the later Husserl's transcendental idealism? 3. Is there any detectable influence from Husserl on Gödel's writings? Regarding the first question, Wang [96, p.165] reports that Gödel ‘[saw] in Husserl's work a method of refining and consolidating Leibniz' monadology’. But what does this mean? In what for Gödel relevant sense is Husserl's work a refinement and consolidation of Leibniz' monadology? The second question is particularly pressing, given that Gödel was, by his own admission, a realist in mathematics since 1925. Wouldn't the uncompromising realism of the early Husserl's Logical investigations have been a more obvious choice for a Platonist like Gödel? The third question can only be approached when an answer to the second has been given, and we want to suggest that the answer to the first question follows from the answer to the second. We begin, therefore, with a closer look at the actual turn towards phenomenology. Some 30 years before his serious study of Husserl began, Gödel was well aware of the existence of phenomenology. Apart from its likely appearance in the philosophy courses that Gödel took, it reached him from various directions.

Assume (ℵ₀, ℵ₁) → (λ, λ⁺). Assume M is a model of a first order theory T of cardinality at most λ⁺ in a language of cardinality ≤ λ. Let N be a model with the same language. Let Δ be a set of first order formulas in and let D be a regular filter on λ. Then M is Δ-embeddable into the reduced power N^λ/D, provided that every Δ-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 language which has cardinality ≤ λ. Suppose D is a regular filter on λ and (ℵ₀, ℵ₁) → (λ, λ⁺) holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ⁺ on Π_iM_i/D and Π_iN_i/D. This yields the following corollary: Assume GCH and λ regular (or just (ℵ₀, ℵ₁) → (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 Π_iMⁱ/D ≅ Π_iNⁱ/D.

Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.

We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle 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].