A Generic Model in Which the Russell-Nontypical Sets Satisfy ZFC Strictly between HOD and the Universe

Vladimir Kanovei,Vassily Lyubetsky

doi:10.3390/math10030491

Vladimir Kanovei, Vassily Lyubetsky

Open Access

https://doi.org/10.3390/math10030491

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Journal: Mathematics	Publication Date: Feb 3, 2022
Citations: 2	License type: CC BY 4.0

Affiliation: Institute for Information Transmission Problems

Abstract

The notion of ordinal definability and the related notions of ordinal definable sets (class OD) and hereditarily ordinal definable sets (class HOD) belong to the key concepts of modern set theory. Recent studies have discovered more general types of sets, still based on the notion of ordinal definability, but in a more blurry way. In particular, Tzouvaras has recently introduced the notion of sets nontypical in the Russell sense, so that a set x is nontypical if it belongs to a countable ordinal definable set. Tzouvaras demonstrated that the class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and satisfies HOD⊆HNT. In view of this, Tzouvaras proposed a problem—to find out whether the class HNT can be separated from HOD by the strict inclusion HOD⫋HNT, and whether it can also be separated from the universe V of all sets by the strict inclusion HNT⫋V, in suitable set theoretic models. Solving this problem, a generic extension L[a,x] of the Gödel-constructible universe L, by two reals a,x, is presented in this paper, in which the relation L=HOD⫋L[a]=HNT⫋L[a,x]=V is fulfilled, so that HNT is a model of ZFC strictly between HOD and the universe. Our result proves that the class HNT is really a new rich class of sets, which does not necessarily coincide with either the well-known class HOD or the whole universe V. This opens new possibilities in the ongoing study of the consistency and independence problems in modern set theory.

Highlights

We recall that a set X is ordinal definable if X can be defined by a formula with ordinals as parameters in the universe of all sets
Tzouvaras [8,9] connected these notions with some philosophical and mathematical ideas of Bertrand Russell and works of van Lambalgen [10] et al on the concept of randomness. They contribute to the ongoing study of important classes of sets in the set theoretic universe V which themselves satisfy the axioms of set theory, to Gödel’s class L and the class HOD
Tzouvaras [9] established the non-strict inclusion HOD ⊆ HNT, and proposed a problem: to find out whether the class HNT can be separated from HOD by the strict inclusion HOD HNT, and can be separated from the universe V of all sets by the strict inclusion HNT V, in suitable set theoretic models

Summary

Introduction

See more on these fundamental notions of modern set theory in [1] (Chapter 13) or [2] (Section II.8), or [3] as the original reference. Tzouvaras [8,9] connected these notions with some philosophical and mathematical ideas of Bertrand Russell and works of van Lambalgen [10] et al on the concept of randomness They contribute to the ongoing study of important classes of sets in the set theoretic universe V which themselves satisfy the axioms of set theory, to Gödel’s class L and the class HOD. (We recall that the equivalence relation E0 is defined on 2ω so that x E0 y iff x(k) = y(k) for all but finite k.) This property of P is responsible for a P-generic real a to belong to HNT, and to L[a] ⊆ HNT, in L[a, x] The reader envisaged is assumed to have some knowledge of the pointset topology of the Baire space ωω (we give [15] and [1] [Chapter 11] as references) along with some basic knowledge of forcing and Gödel’s constructibility (we give [1,2,16] as references)

Silver Trees

Reduction of Borel Maps to Continuous Ones

Normalization of Borel Maps

The Forcing Notion for Theorem 1

Proof of the Extension Lemma

10. Conclusions and Discussion