Abstract
After a very brief comment on Yuri Matiyasevich’s contribution, I discuss at greater length proposals to use modal logic to clarify foundational issues in set theory. Finally, I very sadly bid farewell to my friend and collaborator Hilary Putnam.
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Notes
- 1.
The cardinal in question is in fact quite large: a countable infinity of Woodin cardinals with a measurable cardinal above them.
- 2.
Recent work by Joel Friedman on modalism [1] should also be mentioned.
- 3.
“It is difficult to see that the word if acquires when written \(\supset \), a virtue it did not possess when written if.” [3], p. 156.
References
Friedman, J. (2005). Modal platonism: An easy way to avoid ontological commitment to abstract entities. Journal of Philosophical Logic, 34, 227–273.
Hellman, G. (1989). Mathematics without numbers: Toward a modal-structural interpretation. Oxford.
Poincaré, H. (2012). Science and method, translated from French by Francis Maitland, Thomas Nelson and Sons, London 1914. Facsimile Reprint: Forgotten Books. http://www.forgottenbooks.org.
Weyl, H. (1950). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50, 612–654.
Zermelo, E. (1996). Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16, pp. 29–47. English Translation: On Boundary numbers and domains of sets: New investigations in the foundations of set theory. In W. B. Ewald (Ed.), From Kant to Hilbert: A source book in the foundations of mathematics (pp. 1219–1233). Oxford University Press.
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Davis, M. (2016). Concluding Comments by Martin. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_15
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