Theory of g-Tg-Derived and g-Tg-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks

In a generalized topological space Tg = (Ω, Tg) (Tg-space), the g-topology Tg : P (Ω) −→ P (Ω) can be characterized in the generalized sense by specifying the generalized open, generalized closed sets (g-Tg-open, g-Tg-closed sets), generalized interior, generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω) (g-Tg-interior, g-Tg-closure operators), or generalized derived, generalized coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω) (g-Tg-derived, g-Tg-coderived operators), respectively. For very many Tg-spaces, the δth-iterates g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω) of g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, defined by transfinite recursion on the class of successor ordinals are also themselves g-Tg-derived, g-Tg-coderived operators for new g-topologies in the generalized sense on Ω. Thus, the use of novel definitions of g-Tg-derived, g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, based on a very clever construction, together with their δth-iterates g-Tg-operators g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω), defined by transfinite recursion on the class of successor ordinals, will give rise to novel generalized g-topologies on Ω. The present authors have been actively engaged in the study of g-Tg-operators in Tg-spaces. The study of the essential properties and the commutativity of novel definitions of g-Tg-interior and g-Tg-closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in Tg has formed the first part, and the study of the essential properties and sets of consistent, independent axioms of novel definitions of g-Tg-exterior and g-Tg-frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, has formed the second part. In this work, which forms the last part on the theory of g-Tg-operators in Tg-spaces, the present authors propose to present novel definitions and the study of the essential properties of g-Tg-derived and g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, and their δth-iterates, and the notions of g-Tg-open and g-Tg-closed sets of ranks δ in Tg-spaces.

PREDICATIVE COLLAPSING PRINCIPLES

We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal α there exists an ordinal β such that $1 + \beta \cdot \left( {\beta + \alpha } \right)$ (ordinal arithmetic) admits an almost order preserving collapse into β. Arithmetical comprehension is equivalent to a statement of the same form, with $\beta \cdot \alpha$ at the place of $\beta \cdot \left( {\beta + \alpha } \right)$. We will also characterize the principles that any set is contained in a countable coded ω-model of arithmetical transfinite recursion and arithmetical comprehension, respectively.

MINIMUM MODELS OF SECOND-ORDER SET THEORIES

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR.

NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM

In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using transfinite induction up to ε0, which was considered a finitistically acceptable procedure by both Gentzen and Paul Bernays. Gentzen’s method of arithmetising ordinals and thus avoiding the Platonistic metaphysics of set theory traces back to the 1920s, when Bernays and David Hilbert used the method for an attempted proof of the Continuum Hypothesis. The idea that recursion on higher types could be used to simulate the limit-building in transfinite recursion seems to originate from Bernays. The main difficulty, which was already discovered in Gabriel Sudan’s nearly forgotten paper of 1927, was that measuring transfinite ordinals requires stronger methods than representing them. This paper presents a historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction.

Fraïssé’s conjecture in Π11-comprehension

We prove Fraïssé’s conjecture within the system of [Formula: see text]-comprehension. Furthermore, we prove that Fraïssé’s conjecture follows from the [Formula: see text]-bqo-ness of 3 over the system of Arithmetic Transfinite Recursion, and that the [Formula: see text]-bqo-ness of 3 is a [Formula: see text]-statement strictly weaker than [Formula: see text]-comprehension.

PREDICATIVITY THROUGH TRANSFINITE REFLECTION

Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.For a set of formulas Γ, define predicative oracle reflection for T over Γ (Pred–O–RFNΓ(T)) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then$$\forall \,\lambda < {\rm{\Lambda }}\,([\lambda |X]_T^{\rm{\Lambda }}\varphi \to \varphi ).$$In particular, define predicative oracle consistency (Pred–O–Cons(T)) as Pred–O–RFN{0=1}(T).Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left( {{\rm{RCA}}_0^{\rm{*}} } \right) \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left( {{\rm{ACA}}} \right).$$We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic.Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles.

THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König’s Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.