Abstract
I discuss three potential sources of evidence for truth in set theory, coming from set theory’s roles as a branch of mathematics and as a foundation for mathematics as well as from the intrinsic maximality feature of the set concept. I predict that new non first-order axioms will be discovered for which there is evidence of all three types, and that these axioms will have significant first-order consequences which will be regarded as true statements of set theory. The bulk of the paper is concerned with the Hyperuniverse Programme, whose aim is to discover an optimal mathematical principle for expressing the maximality of the set-theoretic universe in height and width.
Originally published in S. Friedman, Evidence for set-theoretic truth and the Hyperuniverse Programme. IfCoLog J. Log. Appl. “Proof, Truth, Computation” 3(4), 517–555 (2016).
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Notes
- 1.
For a discussion of this notion of derivability see the final Sect. 4.13.
- 2.
Woodin has in fact proposed such an axiom which he calls Ultimate L.
- 3.
For the experts, to get PFA one must allow non-transitive models of size ℵ 1.
- 4.
As a specific example, let \(\mathfrak a\) denote the least size of an infinite almost disjoint family of subsets of ω, and \(\mathfrak b\) (\(\mathfrak d\)) the least size of an unbounded (dominating) family of functions from ω to ω ordered by eventual domination. Then \({\mathfrak {b}} < {\mathfrak {a}} < {\mathfrak {d}}\) is consistent; shouldn’t it in fact be true?
- 5.
This is derivable once we add maximality to the iterative conception, but is convenient to assume already as part of the iterative conception.
- 6.
The set of ctm’s is called the Hyperuniverse; hence we arrive at the Hyperuniverse Programme.
- 7.
Height actualism with just GB (Gödel-Bernays) appears inadequate for a fruitful analysis of maximality. A referee has informed us about agnostic Platonism, the view that there is a well-determined universe V of all sets but without taking a position on whether ZFC holds in it. But as this perspective allows for the possibility of height actualism with just GB, it is problematic for the HP.
- 8.
These comments were made during a lively e-mail exchange among numerous set-theorists and philosophers of set theory from August until November 2014, triggered by my response to Sol Feferman’s preprint The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem. Some of this discussion is documented at <http://logic.harvard.edu/blog/?cat=2>, but regrettably Hellman’s comments do not appear there.
- 9.
But I am not 100% sure that there could not be such an analogous iteration process, perhaps provided by a wildly successful theory of inner models for large cardinals.
- 10.
We thank one of the referees for pointing out that an earlier version of cardinal-maximality with a weaker parameter-absoluteness assumption is inconsistent. A similar phenomenon with weakly absolute parameters occurs in Theorem 10 of [18].
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Friedman, SD. (2018). Evidence for Set-Theoretic Truth and the Hyperuniverse Programme. In: Antos, C., Friedman, SD., Honzik, R., Ternullo, C. (eds) The Hyperuniverse Project and Maximality. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-62935-3_4
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