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It was established by Jensen in 1970 that there is a generic extension L [ a ] of the constructible universe L by a non-constructible real a ∉ L , minimal over L , such that a is Δ 3 1 in L [ a ] . Our first main theorem generalizes Jensen's result by constructing, for each n ≥ 2 , a generic extension L [ a ] by a non-constructible real a ∉ L , still minimal over L , such that a is Δ n + 1 1 in L [ a ] but all Σ n 1 reals are constructible in L [ a ] . Jensen's forcing construction has found a number of applications in modern set theory. A problem was recently discussed as to whether Jensen's construction can be reproduced entirely by means of second-order Peano arithmetic PA 2 , or, equivalently, ZFC − (minus the power set axiom). The obstacle is that the proof of the key CCC property (whether by Jensen's original argument or the later proof using the diamond technique) essentially involves countable elementary submodels of L ω 2 , which is way beyond ZFC − . We demonstrate how to circumvent this difficulty by means of killing only definable antichains in the course of a Jensen-like transfinite construction of the forcing notion, and then use this modification to define a model with a minimal Δ n + 1 1 real as required as a class-forcing extension of a model of ZFC − plus V = L . [ABSTRACT FROM AUTHOR]

A model of set theory ZFC is defined in our recent research, in which, for a given n ≥ 3 , (A n) there exists a good lightface Δ n 1 well-ordering of the reals, but (B n) no well-orderings of the reals (not necessarily good) exist in the previous class Δ n − 1 1 . Therefore, the conjunction (A n) ∧ (B n) is consistent, modulo the consistency of ZFC itself. In this paper, we significantly clarify and strengthen this result. We prove the consistency of the conjunction (A n) ∧ (B n) for any given n ≥ 3 on the basis of the consistency of PA 2 , second-order Peano arithmetic, which is a much weaker assumption than the consistency of ZFC used in the earlier result. This is a new result that may lead to further progress in studies of the projective hierarchy. [ABSTRACT FROM AUTHOR]

We make use of generalized iterations of the Sacks forcing to define cardinal-preserving generic extensions of the constructible universe L in which the axioms of ZF hold and in addition either (1) the parameter-free countable axiom of choice AC ω * fails, or (2) AC ω * holds but the full countable axiom of choice AC ω fails in the domain of reals. In another generic extension of L, we define a set X ⊆ P (ω) , which is a model of the parameter-free part PA 2 * of the 2nd order Peano arithmetic PA 2 , in which CA (Σ 2 1) (Comprehension for Σ 2 1 formulas with parameters) holds, yet an instance of Comprehension CA for a more complex formula fails. Treating the iterated Sacks forcing as a class forcing over L ω 1 , we infer the following consistency results as corollaries. If the 2nd order Peano arithmetic PA 2 is formally consistent then so are the theories: (1) PA 2 + ¬ AC ω * , (2) PA 2 + AC ω * + ¬ AC ω , (3) PA 2 * + CA (Σ 2 1) + ¬ CA . [ABSTRACT FROM AUTHOR]

The problem of the existence of analytically definable well-orderings at a given level of the projective hierarchy is considered. This problem is important as a part of the general problem of the study of the projective hierarchy in the ongoing development of descriptive set theory. We make use of a finite support product of the Jensen-type forcing notions to define a model of set theory ZFC in which, for a given n > 2 , there exists a good Δ n 1 well-ordering of the reals but there are no such well-orderings in the class Δ n − 1 1 . Therefore the existence of a well-ordering of the reals at a certain level n > 2 of the projective hierarchy does not imply the existence of such a well-ordering at the previous level n − 1 . This is a new result in such a generality (with n > 2 arbitrary), and it may lead to further progress in studies of the projective hierarchy. [ABSTRACT FROM AUTHOR]

Using a nonLaver modification of Uri Abraham's minimal Δ31 collapse function, we define a generic extension L[a] by a real a , in which, for a given n ≥ 3 , { a } is a lightface Πn1 singleton, a effectively codes a cofinal map ω → ω1L minimal over L , while every Σn1 set X ⊆ ω is still constructible. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]