Strong Reflection Principles and Large Cardinal Axioms

We are concerned with the distinction between intrinsic and extrinsic justifications for large-cardinal axioms, as outlined in for example Section 3 of the 1964 version of Gödel's essay on the continuum problem. William Tait and Peter Koellner have examined the question of which reflection principles can be said to be intrinsically justified, in the sense of merely unfolding the content of the iterative conception of set. We formulate a new reflection principle which subsumes all of the reflection principles which which were considered by Tait and Koellner and are also known to be consistent, and which is itself consistent relative to an ω-Erd˝ os cardinal (because equivalent to the existence of a remarkable cardinal). We suggest that this principle can be said to be intrinsically justified in this sense. Initially we say that a cardinal κ is extremely reflective if V κ satisfies the form of reflection in question, however it turns out that κ is extremely reflective if and only if κ is remarkable.

Studia Logica, 2019

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement ϕ(a) in the first-order language of set theory is true in the set-theoretic universe V , then it is also true in a proper inner model W V. A stronger principle, the ground-model reflection principle, asserts that any such ϕ(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π 2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.

Philosophia Mathematica, 2018

In recent work of McCallum a new large-cardinal axiom was formulated of strength intermediate between a totally indescribable cardinal and an ω-Erd˝ os cardinal, positing the existence of an " extremely reflective cardinal " , and it was demonstrated that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and an argument was made that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an α-reflective cardinal was formulated. Welch and Roberts have recently put forward a family of reflection principles, Welch's principle implying the existence of a proper class of Shelah cardinals and provably consistent relative to a superstrong cardinal, and Roberts' principle implying the existence of a proper class of 1-extendible cardinals and provably consistent relative to a 2-extendible cardinal. Roberts tentatively argued that his principle should be seen as intrinsically justified (at least on the assumption that a weaker form of reflection involving reflection of second-order formulas with a second-order parameter should be seen as intrinsically justified). This work overlapped with previous work of Victoria Marshall's on reflection principles. We analyze the relationship between reflection principles equivalent to those studied by McCallum and stronger but similar reflection principles which are natural extensions of those of Welch and Roberts. We also show how a natural strengthening of Roberts' reflection principle yields the existence of supercompact cardinals, and in the process solve a question which Marshall left open, of whether her theory B 0 (V 0) is strong enough to imply the existence of supercompact cardinals. We also manage to resolve negatively her question of whether her theory B 1 (V 0) implies the existence of a huge cardinal.

A pervasive thought in contemporary Philosophy of Mathematics is that in order to justify reflection principles, one must hold Universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting Universism with a Zermelian form of Multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the Universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, the status of reflection principles as axioms is left open for the Zermelian.

Pure and Applied Mathematics Journal, 2014

Annals of Mathematical Logic, 1978

British Journal of Mathematics & Computer Science, 2015

Boccuni, F. - Sereni, A. (eds.) Objectivity, Realism and Proof (FilMat Studies in the Philosophy of Mathematics), Springer, Berlin, 2016

The Hyperuniverse Programme, introduced in Arrigoni-Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, that is the `maximal iterative concept', and the programme identifies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements in countable transitive models, the collection of which constitutes the `hyperuniverse', has remarkable consequences, which should be viewed as new intrinsically motivated set-theoretic axioms (H-axioms).