A Candidate for the Generalised Real Line

Galeotti, Lorenzo

doi:10.1007/978-3-319-40189-8_28

Cited by 6 publications

(18 citation statements)

References 12 publications

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“…We should like to mention that Galeotti developed the basic theory of an generalised analogue

R_{κ}

of the real numbers in on the basis of Conway's surreal numbers . The space

R_{κ}

allows us to define appropriate notions of κ‐metric and κ‐Polish spaces and gives hope for a generalisation of measure theory that might shed some light on the question of the generalisation of random forcing.…”

mentioning

confidence: 99%

Questions on generalised Baire spaces

Khomskii

Laguzzi

Löwe

et al. 2016

Mathematical Logic Qtrly

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“…We should like to mention that Galeotti developed the basic theory of an generalised analogue

R_{κ}

of the real numbers in on the basis of Conway's surreal numbers . The space

R_{κ}

mentioning

confidence: 99%

Questions on generalised Baire spaces

Khomskii

Laguzzi

Löwe

et al. 2016

Mathematical Logic Qtrly

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show abstract

“…Then we use this framework to induce a notion of computability over the generalised real line R κ , showing that, as in the classical case, by using suitable representations, the field operations are computable. Finally we will generalise Weihrauch reducibility to spaces of cardinality 2 κ and extend a classical result by showing that the generalised version of the IVT introduced in [10] is Weihrauch equivalent to a generalised version of the boundedness principle B I . Throughout this paper κ will be a fixed uncountable cardinal, as usual assumed to satisfy κ <κ = κ, which in particular implies that κ is a regular cardinal.…”

Section: Introductionmentioning

confidence: 84%

“…We therefore need to replace Dedekind completeness with a weaker property. This was done in [9,10], and we repeat the central definitions here.…”

Section: The Generalised Real Linementioning

confidence: 99%

“…Theorem 4 (Galeotti [10]). The field R κ is the unique Cauchy-complete real closed field extension of R which is an η κ -set of cardinality 2 κ , degree κ, and in which No <κ can be densely embedded.…”

Section: The Generalised Real Linementioning

confidence: 99%

“…Recently, the study of the descriptive set theory of the generalised Baire spaces κ κ and Cantor spaces 2 κ for cardinals κ > ω has been of great interest to set theorists. In [10] the second author provided the foundational basis for the study of generalised computable analysis, namely the generalisation of computable analysis to generalised Baire and Cantor spaces. In particular, in [10] the second author introduced R κ , a generalised version of the real line, and proved a version of the intermediate value theorem (IVT) for that space.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Towards Computable Analysis on the Generalised Real Line

Galeotti

Nobrega

2017

Unveiling Dynamics and Complexity

Self Cite

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Abstract. In this paper we use infinitary Turing machines with tapes of length κ and which run for time κ as presented, e.g., by Koepke & Seyfferth, to generalise the notion of type two computability to 2 κ , where κ is an uncountable cardinal with κ <κ = κ. Then we start the study of the computational properties of Rκ, a real closed field extension of R of cardinality 2 κ , defined by the first author using surreal numbers and proposed as the candidate for generalising real analysis. In particular we introduce representations of Rκ under which the field operations are computable. Finally we show that this framework is suitable for generalising the classical Weihrauch hierarchy. In particular we start the study of the computational strength of the generalised version of the Intermediate Value Theorem.

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The Theory of the Generalised Real Numbers and Other Topics in Logic

Galeotti¹

2019

Bull. symb. log

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The real line R and Baire space are two of the most fundamental objects in descriptive set theory and have been studied extensively. In recent years, set theorists have become increasingly interested in generalised Baire spaces κ κ , i.e., the sets of functions from κ to κ for an uncountable cardinal κ. In the thesis, two κ-analogues of R are discussed, one at https://www.cambridge.org/core/terms.

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A Candidate for the Generalised Real Line

Cited by 6 publications

References 12 publications

Questions on generalised Baire spaces

Questions on generalised Baire spaces

Towards Computable Analysis on the Generalised Real Line

The Theory of the Generalised Real Numbers and Other Topics in Logic

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