Killing them softly: degrees of inaccessible and Mahlo cardinals

2017 ◽  
Vol 63 (3-4) ◽  
pp. 256-264
Author(s):  
Erin Kathryn Carmody
Keyword(s):  
Mahlo Cardinals

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

1983 ◽  
Vol 48 (4) ◽  
pp. 1046-1052 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Recent Work ◽  
Partial Orders ◽  
Closure Condition ◽  
Forcing Axioms ◽  
Combinatorial Principles ◽  
The Relationship ◽  
Mahlo Cardinals ◽  
Forcing Axiom

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 ◊ and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to □κ. The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of □κ.In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = 〈P, ≤〉 is a partial order. For each ordinal α we will consider a game played by two players, Good and Bad. The players choose, in order, the terms in a descending sequence of conditions 〈pβ∣β < α〉 Good chooses all terms pβ for limit β, and Bad chooses all the others. Bad wins if for some limit β<α, Good is unable to move at stage β because 〈pγ∣γ < β〉 has no lower bound. Otherwise, Good wins. Of course, we will be rooting for Good.

Regressive partitions and Borel diagonalization

1989 ◽  
Vol 54 (2) ◽  
pp. 540-552 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Natural Generalization ◽  
Large Cardinal ◽  
Unit Interval ◽  
Combinatorial Proof ◽  
Consistency Strength ◽  
Several Variables ◽  
Borel Measurable ◽  
Mahlo Cardinals ◽  
Level Analysis

Several rather concrete propositions about Borel measurable functions of several variables on the Hilbert cube (countable sequences of reals in the unit interval) were formulated by Harvey Friedman [F1] and correlated with strong set-theoretic hypotheses. Most notably, he established that a “Borel diagonalization” proposition P is equivalent to: for any a ⊆ co and n ⊆ ω there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. In later work (see the expository Stanley [St] and Friedman [F2]), Friedman was to carry his investigations further into propositions about spaces of groups and the like, and finite propositions. He discovered and analyzed mathematical propositions which turned out to have remarkably strong consistency strength in terms of large cardinal hypotheses in set theory.In this paper, we refine and extend Friedman's work on the Borel diagonalization proposition P. First, we provide more combinatorics about regressive partitions and n-Mahlo cardinals and extend the approach to the context of the Erdös cardinals In passing, a combinatorial proof of a well-known result of Silver about these cardinals is given. Incorporating this work and sharpening Friedman's proof, we then show that there is a level-by-level analysis of P which provides for each n ⊆ ω a proposition almost equivalent to: for any a ⊆ co there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. Finally, we use the combinatorics to bracket a natural generalization Sω of P between two large cardinal hypotheses.

0# and inner models

2002 ◽  
Vol 67 (3) ◽  
pp. 924-932 ◽  
Author(s):  
SY D. Friedman
Keyword(s):  
High Order ◽  
Covering Theorem ◽  
Inner Models ◽  
Inner Model ◽  
Mahlo Cardinals ◽  
Reverse Easton Iteration

In this paper we examine the cardinal structure of inner models that satisfy GCH but do not contain 0#. We show, assuming that 0# exists, that such models necessarily contain Mahlo cardinals of high order, but without further assumptions need not contain a cardinal κ which is κ-Mahlo. The principal tools are the Covering Theorem for L and the technique of reverse Easton iteration.Let I denote the class of Silver indiscernibles for L and 〈iα ∣ α ϵ ORD〉 its increasing enumeration. Also fix an inner model M of GCH not containing 0# and let ωα denote the ωα of the model M[0#], the least inner model containing M as a submodel and 0# as an element.

Positive ∑ operations on ordinals and normal filters on greatly mahlo cardinals

1989 ◽  
Vol 54 (1) ◽  
pp. 226-233
Author(s):  
Thomas Jech
Keyword(s):  
Normal Filter ◽  
Uncountable Cardinal ◽  
Mahlo Cardinals ◽  
Ordinal Function

If ℱ is a normal filter on a regular uncountable cardinal κ, let ║f║ be the ℱ-norm of an ordinal function f. We introduce the class of positive ordinal operations and prove that if F is a positive operation then ║F(f)║ ≥ F(║f║). For each η < κ+ let fη be the canonical ηth function. We show that if F is a ∑ operation then F(fη) = fF(η).As an application we show that if κ is greatly Mahlo then there are normal filters on κ of order greater than κ+.

Solovay models and forcing extensions

2004 ◽  
Vol 69 (3) ◽  
pp. 742-766 ◽  
Author(s):  
Joan Bagaria ◽  
Roger Bosch
Keyword(s):  
Generic Absoluteness ◽  
Solovay Model ◽  
Mahlo Cardinals

Abstract.We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly- absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions.

An ideal characterization of Mahlo cardinals

1989 ◽  
Vol 54 (2) ◽  
pp. 467-473 ◽  
Author(s):  
Qi Feng
Keyword(s):  
Regular Cardinal ◽  
Normal Ideal ◽  
Mahlo Cardinal ◽  
Mahlo Cardinals

AbstractWe show that a cardinal κ is a (strongly) Mahlo cardinal if and only if there exists a nontrivial κ-complete κ-normal ideal on κ. Also we show that if κ is Mahlo and λ ≧ κ and λ<κ = λ then there is a nontrivial κ-complete κ-normal fine ideal on Pκ(λ). If κ is the successor of a cardinal, we consider weak κ-normality and prove that if κ = μ+ and μ is a regular cardinal then (1) μ< μ = μ if and only if there is a nontrivial κ-complete weakly κ-normal ideal on κ, and (2) if μ< μ = μ < λ<μ = λ then there is a nontrivial κ-complete weakly κ-normal fine ideal on Pκ(λ).

L-Mahlo Cardinals

1974 ◽  
Vol 20 (13-18) ◽  
pp. 229-231
Author(s):  
Paul E. Cohen
Keyword(s):  
Mahlo Cardinals
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