arXiv subjects

Sy D. Friedman

Publications and source records attributed to Sy D. Friedman.

15 recordsLinked to original sources

arXiv2017

Coherent systems of finite support iterations

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń's diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a $Δ^1_3$ well-order of the reals.

math.LO↗

arXiv1997

David's trick

We put into a general setting a technique of Rene' David (see "A Very Absolute Pi^1_2 Singleton, Annals of Pure and Applied Logic, 1982) to show that for S a Sigma^1_1 statement quantifying over subclasses of V of a special form, there is a stronger Sigma^1_3 statement quantifying over reals which can be forced over any model of S. Then I mention some application to Pi^1_2 Singletons and to Sigma^1_3 absoluteness.

math.LO↗

arXiv1997

New Sigma^1_3 facts

We use ``iterated square sequences'' to show: There is an L-definable partition n: L-singulars --> omega such that if M is an inner model without 0#: (a) For some n, M satisfies that {alpha | n(alpha)=n} is stationary. (b) For each n there is a generic extension of M in which 0# does not exist and {alpha | n(alpha)<n} is non-stationary. The above result is then applied to show that if M is an inner model without 0# then some Sigma^1_3 sentence not true in M can be forced over M.

math.LO↗

arXiv1996

Generic Saturation

Assuming that ORD is $ω+ω$-Erdös we show that if a class forcing amenable to $L$ (an $L$-forcing) has a generic then it has one definable in a set-generic extension of $L[O^\#]$. In fact we may choose such a generic to be {\it periodic} in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be {\it almost codable} in the sense that it is definable from a real which is generic for an $L$-forcing (and which belongs to a set-generic extension of $L[O^\#]$).

math.LO↗

arXiv1996

Ordinal Recursion Theory

In this article, intended for the Handbook of Recursion Theory, we survey recursion theory on the ordinal numbers, with sections devoted to $α$-recursion theory, $β$-recursion theory and the study of the admissibility spectrum.

math.LO↗

arXiv1995

Strict Genericity

We show that an inner model of a class-generic extension of L need not itself be such an extension. Our example is of the form L[R], where R is a real belonging to a class-generic extension of L and constructible from 0#.

math.LO↗

arXiv1995

Coding Without Fine Structure

We present a proof of Jensen's Coding Theorem (assumong -0#) which quotes the covering lemma, but otherwise makes no appeal to fine structure theory. The key idea is to use a modified definition of the coding at limit cardinals, using "coding delays".

math.LO↗

arXiv1995

Delta-1-2 Without Sharps

We show that the supremum of the lengths of boldface Delta-1-2 prewellorderings of the reals can be Aleph-2, with Aleph-1 inac- cessible to reals, assuming only the consistency of an inaccessible.

math.LO↗

arXiv1993

Iterated Class Forcing

In this paper we isolate the notion of Stratified class forcing and show that Stratification implies cofinality-preservation and is preserved by iterations with the appropriate support. Many familiar class forcings are stratified and therefore can be simultaneously iterated without changing cofinalities, provided the proper support is used. Easton forcing, Backward Easton forcings and some modifications of Jensen coding are stratified. Jensen coding is not stratified but instead obeys a related property, Delta-Stratification, which is also preservedby iteration with an appropriate larger support.

math.LO↗

arXiv1993

Provable Pi-1-2 Singletons

In this note I show that a pi-1-2 singleton R of L-degree strictly between 0 and 0# can be obtained so as to be the unique solution to a pi-1-2 formula which provably has at most one solution, in the theory ZFC+(*) where (*) has the approximate strength of an ineffable cardinal.

math.LO↗

arXiv1992

Jensen's Σ^* theory and the combinatorial content of V=L

The purpose of this article is to indicate how a reformulation of Jensen's $Σ^*$ theory (developed for the study of core models) can be used to provide a more satisfactory treatment of uniformization, hulls and Skolem functions for the $J_α$'s. Then we use this approach to fine structure to formulate a principle intended to capture the combinatorial content of the axiom $V=L.$

math.LO↗

arXiv1992

A large Pi-1-2 set absolute for set forcing

Let k be a definable L-cardinal. Then there is a set of reals X, class-generic over L, such that L(X) and L have the same cardinals, X has size k in L(X) and some pi-1-2 formula defines X in all set-generic extensions of L(X). Two corollaries, both assuming the consistency of an inaccessible: It is consistent for the Perfect Set Property to hold for boldface sigma-1-2 sets, yet fail for some lightface pi-1-2 set. It is consistent that the Perfect Set Property holds for boldface sigma-1-2 sets yet some lightface pi-1-2 wellordering of some set of reals has length aleph-1000.

math.LO↗

arXiv1992

The Genericity Conjecture

In this paper we produce a real r such that 0<r<0# in L-degree, yet R is NOT generic over L (for a forcing amenable to L). This answers a question of Beller-Jensen-Welch.

math.LO↗

arXiv1992

A simpler proof of Jensen's coding theorem

We present a simplification of Jensen's proof of his Coding Theorem (even in the case where 0# exists). The proof avoids Jensen's split into cases according to whether or not 0# exists. In addition, the paper contains self-contained proofs of the necessary forms of Square and Diamond, based on an approach to fine structure using Jensen's $Σ^*$ theory.

math.LO↗

arXiv1992

Minimal universes

An inner model M is MINIMAL if there is a class A such that is amenable yet has no transitive proper elementary submodel. We study minimal universes in the context of 0#. For example we prove: If 0# exists then there is an inner model which is minimal and locally generic over L(i.e., every set in the inner model is set-generic over L). This answers a question of Mack Stanley.

math.LO↗