Abstract
We address the problem of the choice of new axioms for set theory. After discussing some classical views about the notion of axiom in mathematics, we present the most currently debated candidates for a new axiomatisation of set theory, including Large Cardinal axioms, Forcing Axioms and Projective Determinacy and we illustrate some of the main arguments presented in favour or against such principles.
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- 1.
“There are also extrinsic reasons for rejecting V = L, most prominently that it implies the existence of a \(\Delta _2^1\) well-ordering of the reals, and hence that there is a \(\Delta _2^1\) set which is not Lebesgue measurable.” (Maddy 1988)
- 2.
Given a tree of height ω whose levels are finite, if every finite subtree has a branch of the same length as the height subtree, then the whole tree also has a branch of the same length as the height of the tree.
- 3.
A function j: V →M is an elementary embedding if for every formula φ and parameters a 1, …, a n one has V ⊧φ(a 1, …, a n) if and only if M⊧φ(j(a 1), …, j(a n)).
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Acknowledgements
I would like to thank the anonymous reviewer for his careful reading and for his constructive comments. I am also indebted to Juliette Kennedy, Menachem Magidor, Neil Barton and Claudio Ternullo who gave me many useful suggestions that helped improve the quality of this paper.
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Fontanella, L. (2019). How to Choose New Axioms for Set Theory?. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_2
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