Abstract
Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power and meta-theoretic properties when comparing first-order and second-order logic.
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Acknowledgements
Order of authors is alphabetical. We would like to thank two anonymous reviewers for very helpful comments on the paper. The first author is grateful to the VolkswagenStiftung for their support through the project Forcing: Conceptual Change in the Foundations of Mathematics. The second author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement ERC-2014-CoG 648276 (AUTAR).
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Barton, N., Müller, M. & Prunescu, M. On Representations of Intended Structures in Foundational Theories. J Philos Logic 51, 283–296 (2022). https://doi.org/10.1007/s10992-021-09628-2
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DOI: https://doi.org/10.1007/s10992-021-09628-2