A strengthening of Jensen's □ principles

Beller, Aaron J.; Litman, Ami

doi:10.2307/2273186

Cited by 38 publications

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“…PROOF. The strengthening of Jensen's O principle proved by Beller and Litman [1] yields the following consequence which we will need. There is a stationary E C X and a sequence (C a : a < X) such that the following properties hold for each a < X:…”

Section: ) T N / \ W ( C J R ) Iff ® T = / \^( C > F R )mentioning

confidence: 89%

“…Having constructed such a sequence, we then let JV = (J{J" Q : a < «}. Property (1) implies that JV is A-like. Clearly, JV = JVQ = Jl.…”

Section: ) T N / \ W ( C J R ) Iff ® T = / \^( C > F R )mentioning

confidence: 99%

“…C v are such that if t e 5", then rank(/) e R and if ? G 5 1 and rank(?) > a £ R, then there is i € S such that s <a t and rank(^) = a.…”

Section: ) T N / \ W ( C J R ) Iff ® T = / \^( C > F R )mentioning

confidence: 99%

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A reflection principle and its applications to nonstandard models

Schmerl

1995

J. symb. log.

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Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) > ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

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Section: ) T N / \ W ( C J R ) Iff ® T = / \^( C > F R )mentioning

confidence: 89%

“…Having constructed such a sequence, we then let JV = (J{J" Q : a < «}. Property (1) implies that JV is A-like. Clearly, JV = JVQ = Jl.…”

Section: ) T N / \ W ( C J R ) Iff ® T = / \^( C > F R )mentioning

confidence: 99%

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A reflection principle and its applications to nonstandard models

Schmerl

1995

J. symb. log.

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“…in a, type(£>") < a, and if /3 is a limit point of D a then /3 is singular and Dp = D a f| /3. The existence of such a sequence is proved by Beller and Litman in [1]. Since K is not Mahlo we can also choose a c.u.b.…”

Section: (K = L) If K Is Not Mahlo Then B(k) Is Truementioning

confidence: 96%

On a generalization of Jensen's □_κ, and strategic closure of partial orders

Velleman

1983

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Link to this article: http://journals.cambridge.org/abstract_S0022481200037415How to cite this article: Dan Velleman (1983). On a generalization of Jensen's □ κ , and strategic closure of partial orders . Introduction.It is well known that many statements provable from combinatorial principles true in the constructible universe L can also be shown to be consistent with ZFC by forcing. Recent work by Shelah and Stanley [4] and the author [5] has clarified the relationship between the axiom of constructibility and forcing by providing Martin's Axiom-type forcing axioms equivalent to 0 and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to • , . The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of • * .In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = is a partial order. For each ordinal a we will consider a game G" played by two players, GOOD and BAD. 'The players choose, in order, the terms in a descending sequence of conditions (Pp\ ]3 < «>• GOOD chooses all terms/^ for limit /3, and BAD chooses all the others. BAD wins if for some limit fi has no lower bound. Otherwise, GOOD wins. Of course, we will be rooting for GOOD.A strategy for GOOD is a function S a such that for any limit ordinal /3 < a and any descending sequence p = (p r \f < j3>, S a (p) is a (possibly empty) set of lower bounds for p-in other words, a set of possible next moves for GOOD. If p = (p r \ y < |3> is a partially completed game of G£, we say that GOOD has played by S a if for each limit ordinal d < /3, Ps e S a (p [ d). S a is a winning strategy for GOOD if for each limit ordinal /3 < a and partially completed game p =

i n w h i c h G O O D h a s played by S a , S a (p) # 0.Recall that n , is the statement that there is a sequence < C a | a < K + , a a limit ordinal) such that C a is c.u.b. in a, if /3 is a limit point of C a then C^ = C a |~| j3, and if cf(a) < K then type(C 0 ) < K. Here "c.u.b." abbreviates "closed and unbounded", and "type(C")" is the order type of C a . Our generalization of • is defined as follows: DEFINITION 1. For any infinite cardinal K, B(K) is the statement: There is a sequence « C a , O | a < K, a a limit ordinal) such that for all limit ordinals a < K:(1) C" is c.u.b. in a.

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“…Silver machines were indeed invented to avoid the Jensen hierarchy of J α 's for this very purpose (see [4] and [15]). Both the Silver machines, and the Hyperfine Structure of Friedman and Koepke ([17]) however make use of a Finiteness Property that works against the ITTM formalism (as indeed does the Σ 2 -nature of the limit rules.…”

Section: Ordinal Length Tapesmentioning

confidence: 99%

Transfinite machine models

Welch¹

2014

Turing's Legacy

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A strengthening of Jensen's □ principles

Abstract: The aim of this paper is to prove strengthenings of three theorems appearing in Jensen [1].

Cited by 38 publications

References 2 publications

A reflection principle and its applications to nonstandard models

A reflection principle and its applications to nonstandard models

On a generalization of Jensen's □_κ, and strategic closure of partial orders

Transfinite machine models

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A strengthening of Jensen's □ principles

Abstract: The aim of this paper is to prove strengthenings of three theorems appearing in Jensen [1].

Cited by 38 publications

References 2 publications

A reflection principle and its applications to nonstandard models

A reflection principle and its applications to nonstandard models

On a generalization of Jensen's □κ, and strategic closure of partial orders

Transfinite machine models

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Product

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On a generalization of Jensen's □_κ, and strategic closure of partial orders