On Representations of Intended Structures in Foundational Theories

Journal of Philosophical Logic 51 (2):283-296 (2022)


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.


Added to PP

305 (#36,090)

6 months
126 (#4,703)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Neil Barton
Universität Konstanz

References found in this work

Undecidable Theories.Alfred Tarski - 1953 - Amsterdam: North-Holland Pub. Co..
Second-Order Logic and Foundations of Mathematics.Jouko Väänänen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.

View all 21 references / Add more references

Citations of this work

No citations found.

Add more citations

Similar books and articles

Structural Relativity and Informal Rigour.Neil Barton - forthcoming - In Objects, Structures, and Logics, FilMat Studies in the Philosophy of Mathematics.
Higher-Order Logic or Set Theory: A False Dilemma.S. Shapiro - 2012 - Philosophia Mathematica 20 (3):305-323.
The HOROR Theory of Phenomenal Consciousness.Richard Brown - 2015 - Philosophical Studies 172 (7):1783-1794.
Theories, Models, and Representations.Mauricio Suárez - 1999 - In L. Magnani, N. J. Nersessian & P. Thagard (eds.), Model-Based Reasoning in Scientific Discovery. Kluwer/Plenum. pp. 75--83.
Category Theory: The Language of Mathematics.Elaine Landry - 1999 - Philosophy of Science 66 (3):27.
A Higher-Order, Dispositional Theory of Qualia.John O'dea - 2007 - Annals of the Japan Association for Philosophy of Science 15 (2):81-93.
Decidable Fragments of First-Order Temporal Logics.Ian Hodkinson, Frank Wolter & Michael Zakharyaschev - 2000 - Annals of Pure and Applied Logic 106 (1-3):85-134.
Representing the World with Inconsistent Mathematics.Colin McCullough-Benner - 2020 - British Journal for the Philosophy of Science 71 (4):1331-1358.
Set Theory and Structures.Neil Barton & Sy-David Friedman - 2019 - In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Springer Verlag. pp. 223-253.
Logic and Mathematics.Jan Wolénski - 1995 - Vienna Circle Institute Yearbook 3:197-210.
The Collapse Argument.Joseph Gottlieb - 2019 - Philosophical Studies 176 (1):1-20.