The Complexity of Classification Problems for Models of Arithmetic

Coskey, Samuel; Kossak, Roman

doi:10.2178/bsl/1286284557

Cited by 5 publications

(11 citation statements)

References 14 publications

(34 reference statements)

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“…The proof of this fact will make use of the close analogy between models of

sans-serifZFC

and models of

sans-serifPA

, together with the fact that the analogous result has already been established for countable models of

sans-serifPA

in [2]. In that article, the authors used a construction due to Gaifman called a “canonical I ‐model” to establish that for any completion T of

sans-serifPA

, the classification of countable models of T is Borel complete.…”

Section: Introductionmentioning

confidence: 98%

“…In that article, the authors used a construction due to Gaifman called a “canonical I ‐model” to establish that for any completion T of

sans-serifPA

, the classification of countable models of T is Borel complete. In the present article, we shall show how Gaifman's construction may be used to build models of

sans-serifZFC

, and how the argument of [2] thus gives the desired conclusion for models of

sans-serifZFC

.…”

Section: Introductionmentioning

confidence: 99%

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The classification of countable models of set theory

Clemens

Coskey

Dworetzky³

2020

Mathematical Logic Qtrly

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“…The proof of this fact will make use of the close analogy between models of

sans-serifZFC

and models of

sans-serifPA

, together with the fact that the analogous result has already been established for countable models of

sans-serifPA

in [2]. In that article, the authors used a construction due to Gaifman called a “canonical I ‐model” to establish that for any completion T of

sans-serifPA

, the classification of countable models of T is Borel complete.…”

Section: Introductionmentioning

confidence: 98%

“…In that article, the authors used a construction due to Gaifman called a “canonical I ‐model” to establish that for any completion T of

sans-serifPA

, the classification of countable models of T is Borel complete. In the present article, we shall show how Gaifman's construction may be used to build models of

sans-serifZFC

, and how the argument of [2] thus gives the desired conclusion for models of

sans-serifZFC

.…”

Section: Introductionmentioning

confidence: 99%

The classification of countable models of set theory

Clemens

Coskey

Dworetzky³

2020

Mathematical Logic Qtrly

Self Cite

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“…In Chapter 2, we will examine the argument used in [1] showing that models of PA are Borel complete. The idea is that we can find a Borel reduction from the isomorphism relation on the class of countable linear orders to the isomorphism relation on the class of countable models of PA.…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…This chapter will establish the following result due to Coskey-Kossak: the class of countable models of PA is Borel complete [1]. We know that the class of countable linear orders is Borel complete.…”

Section: Pamentioning

confidence: 89%

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The Classification Problem for Models of ZFC

Dworetzky¹

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Models of ZFC are ubiquitous in modern day set theoretic research. There are many different constructions that produce countable models of ZFC via techniques such as forcing, ultraproducts, and compactness. The models that these techniques produce have many different characteristics; thus it is natural to ask whether or not models of ZFC are classifiable. We will answer this question by showing that models of ZFC are unclassifiable and have maximal complexity. The notions of complexity used in this thesis will be phrased in the language of Borel complexity theory.In particular, we will show that the class of countable models of ZFC is Borel complete. Most of the models in the construction as it turns out are ill-founded. Thus, we also investigate the sub problem of identifying the complexity for well-founded models. We give partial results for the well-founded case by identifying lower bounds on the complexity for these models in the Borel complexity hierarchy.iv

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The Structural Complexity of Models of Arithmetic

Montalbán¹,

Rossegger²

2023

J. symb. log.

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We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $ . Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $ -jump of models of an arbitrary completion T of $\mathrm {PA}$ we show that every countable ordinal $\alpha>\omega $ is realized as the Scott rank of a model of T.

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The Complexity of Classification Problems for Models of Arithmetic

Cited by 5 publications

References 14 publications

The classification of countable models of set theory

The classification of countable models of set theory

The Classification Problem for Models of ZFC

The Structural Complexity of Models of Arithmetic

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