A consistency proof for some restrictions of Tait's reflection principles

Abstract

In [5] Tait identifies a set of reflection principles which he calls Γ(2) n -reflection principles which Peter Koellner has shown to be consistent relative to κ(ω), the first ω-Erdös cardinal, in [2]. Tait also goes on in the same work to define a set of reflection principles which he calls Γ (m) n -reflection principles; however Koellner has shown that these are inconsistent when m > 2 in [3], but identifies restricted versions of them which he proves consistent relative to κ(ω). In this paper we introduce a new large-cardinal property with an ordinal parameter α > 0, calling those cardinals which satisfy it α-reflective cardinals. Its definition is motivated by the remarks Tait makes in [5] about why reflection principles must be restricted when parameters of third or higher order are introduced. We prove that if κ is κ+α+1 -supercompact and 0 < α < κ then κ is α-reflective. Furthermore we show that αreflective cardinals relativize to L, and that if κ(ω) exists then the set of cardinals λ < κ(ω), such that λ is α-refective for all α such that 0 < α < λ, is a stationary subset of κ(ω). We show that an ω-reflective cardinal satisfies some restricted versions of Γ (m) n -reflection, as well as all the reflection properties which Koellner proves consistent in [3].Part of this paper was written while I was a Research Intensive Academic at the Australian Catholic University. I made further revisions to it while I was an Adjunct Lecturer at the University of New South Wales, and while I held a post-doctoral position at the University of Münster. I am grateful to these institutions for their support. I am also thankful to Peter Koellner for giving me helpful feedback on an early draft.

Which axioms of set theory are intrinsically justified?

“…If we modify the reflection principle so as only to apply to formulas in which unbounded existential quantifiers for first-order variables do not appear within the scope of a higher-order quantifier (but existential quantifiers bound by a firstorder variable are allowed), then the resulting reflection principle becomes consistent relative to an ω-Erdős cardinal. In fact a level V κ satisfies this form of reflection if and only if κ is ω-refelctive in the sense defined in [6], and when this is so then V κ satisfies Γ…”

“…In Section 2 I will describe the ideas about intrinsic justification which originally led me to think that the existence of remarkable cardinals was intrinsically justified. Then in Section 3 I will analyze the relationship between the reflection principles of [6] and [7] with those of [1], [2] and [14], and consider the philosophical question whether there can be any principled grounds for accepting the former but not the latter. I will also discuss how a natural extension of the reflection principle put forward by Roberts is equivalent to the existence of a supercompact cardinal.…”

All large-cardinal axioms not known to be inconsistent with ZFC are justified

“…Formulating a notion of an α-reflective cardinal for ordinals α > 0 and seeking to motivate this along lines inspired by remarks in the work of Tait, I showed in [7] that if κ is ω-reflective then V κ satisfies RP m,n for all m, n with some restrictions on φ slightly more restrictive than Tait's ones. And I showed that it is consistent relative to an ω-Erdős cardinal that there is a proper class of α-reflective cardinals for each α > 0.…”

Large Cardinals as Principles of Structural Reflection

“…But Koellner [16], building on some results of Tait, shows that no V κ can reflect the class of formulas of the form ∀X ∃Yϕ(X, Y, Z), where X is of third-order, Y is of any finite order, Z is of fourth order, and ϕ has only first-order quantifiers and its only negated atomic subformulas are either of first order or of the form x ∈ X , where x is of first order and X is of second order. Other kinds of restrictions on the class of sentences to be reflected are possible (see [26] for the consistency of some forms of reflection slightly stronger than Tait's Γ (2) ), but Koellner [16] convincingly shows that the existence of a cardinal κ such that V κ reflects any reasonable expansion of the class of sentences Γ (2) , with parameters of order greater than 2, either follows from the existence of κ( ) or is outright inconsistent. These results seem to put an end to the program of providing an intrinsic 3 justification of large-cardinal axioms, even for axioms as strong as the existence of κ( ), by showing that their existence follows from strong higherorder reflection properties holding at some V κ .…”