Models of ZF with the same sets of sets of ordinals

Monro, G. P.

doi:10.4064/fm-80-2-105-110

Cited by 14 publications

(26 citation statements)

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“…We note that Col inj (X, Y) is isomorphic to Col inj (Y, X). In this section, we will focus on Col inj (A, κ) when κ is an infinite ordinal, 4 and since A is Dedekind-finite, the conditions are finite. It will be easier to work with Col inj (κ, A) instead, and as it is isomorphic to Col inj (A, κ), we can do that without a problem.…”

Section: Injective Collapsementioning

confidence: 99%

How to have more things by forgetting how to count them

Karagila

Schlicht

2020

Proc. R. Soc. A.

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Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ . In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2 A is extremally disconnected, or [ A ] < ω is Dedekind-finite.

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Section: Injective Collapsementioning

confidence: 99%

How to have more things by forgetting how to count them

Karagila

Schlicht

2020

Proc. R. Soc. A.

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“…We are indebted to THOMAS JECH for Remark 1 and for a preprint of MONRO'S R e m a r k 1. The boundability of -KW follows immediately from the equivalent "For every family 9 of infinite sets there is a function F on 8 such that paper [15].…”

Section: Imentioning

confidence: 99%

On the Independence of the Kinna Wagner Principle

Pincus¹

1974

Mathematical Logic Qtrly

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in Seattle, Washington (U.S.A.) § 1. Introduction KINSA and WAGNER [6] studied the following statement of ZERMELO-FRAENKEL set theory ZF: (KIT') arly ordered.) it is clear that Every set i s similar to a subset of some P (a), a E On .l)If AC is the axiom of choice and 0 is the ordering principle (Every set can be line-Nether of these implications can be reversed. HALPERN and LEVY [5] proved the irreversability of the first arrow and FELGNER [l] proved the irreversability of the second.Two other statements stronger than 0 have been studied extensively. They are the order extension principle OE and the prime ideal theorem for Booman algebras PI. These statements obey the implications AC -+ PI +IOE -+ 0 . Again none of the implications are reversable. HALPERN and LEVY [5], prove the irreversability of the first arrow, FELGNER [ 2 ] , the second, and MATHIAS [8], the third. The models used in the 3 above papers all satisfy KW. Thus from [S] -(KW + PI).It is the purpose of this paper to prove a theorem on the transfer of FRAENHEL-MOSTOWSKI consistencies to ZF set theory. This theorem has as a consequence the nonimplication N (PI -+ KW). FRAEKKEL-MOSTOWSKI models suggest themselves since' the above nonimplication holds in the ordered model of MOSTOWSKI [9]. HALPERN [3] proves that PI holds there. On the other hand the power set of an ordinal is well orderable in every FRSENKEL-MOSTOWSKI model so KW ts AC holds in every FRAENKEL-MOSTOWSKI model. Unfortunately we are not done a t this point since FRAENKEL-MOSTOWSKI models do not satisfy ZF set theory but only the weaker theory ZFA.The model used by FELGNER in [l] is closely related to MOSTOWSKI'S ordered model. I n fact it is obtained from MOSTOWSKI'S model by a general technique for extending FRA4ENXEL-~fOSTOwSKI models to models of ZF set theory. This technique is set forth in [12] and is used in [13] to obtain the following metatheorem. l) S o t a t i o n . Sets are similar if they can be put in 1 : 1 correspondence. P ( z ) denotes the power set of .c. On denotes the class of ordinal numbers.

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“…In this paper we develop a technique for iterating symmetric extensions. We can see precursors to this idea in the works of Gershon Sageev [11,12] and Gordon Monro [9]. Sageev constructs two models using iterations and symmetries.…”

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confidence: 99%

“…Section 8 will be devoted to a variant of the general method where some added restrictions will allow us to access filters which are not necessarily generic for the iteration. We finish the paper with an example of a symmetric iteration in which we subsume the work of Monro in [9] into this new framework, we discuss Kinna-Wagner Principles and construct a sequence of models, each an extension of the previous, where slowly more and more Kinna-Wagner Principles fail.…”

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confidence: 99%

Iterating Symmetric Extensions

Karagila¹

2019

J. symb. log.

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The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.

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Models of ZF with the same sets of sets of ordinals

Cited by 14 publications

References 0 publications

How to have more things by forgetting how to count them

How to have more things by forgetting how to count them

On the Independence of the Kinna Wagner Principle

Iterating Symmetric Extensions

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