Abstract
Consistency, interpretability and probability are three key instruments in the mathematical philosopher’s kit when it comes to questions of foundational theory comparison. This paper aims to bring these tools together with a focus on theories capable of providing foundations for mathematics with a particular emphasis on set theory. A number of counterintuitive results emerge which are then addressed by offering a novel framework based on what we call pointwise interpretability. We then investigate a plausible, existing instance of this framework, the generic multiverse, and demonstrate that it can be naturally situated within our pointwise interpretability framework.
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Notes
For a broader investigation into the relationship between probability and mathematics that delves deeper into practical probability problems see (Walsh, 2014) Our main focus here is on fundamental problems of coherence, although these problems are quite closely related to Walsh’s discussion of indiscernibility and language invariance.
These puzzles are relatively easy to construct and have the flavor of folk results. I’m not aware of them being recorded elsewhere, but the more experienced reader may which to gloss this section and more more quickly into Sect. 2.
This section has independent interest and some readers may wish to skip ahead to get to this content. Such a reader won’t miss to much if they start at Sect. 4 while glancing back for some notational conventions.
In fact, this theorem can be demonstrated in a much weaker theory like PRA. For a detailed discussion, see Section IV.8 and VII.9 in (Kunen, 2006).
There may be other options here worth investigating. For example, one might want to work in a theory which allows for a probability measure that is total on \(\mathcal {P}(\Omega )\), perhaps by sacrificing the axiom of choice and replacing it with determinacy. We leave this question open, but flag it for future consideration.
Later in the paper, we will consider stronger logics, however, they will always possess the Downward Löwenheim Skolem property; i.e., any suitable model will have a countable elementary submodel which is also suitable for that logic. We should also note that there are situations where results around interpretability can be affected by restricting to countable models although we do not believe that such cases are pertinent here.
Recall that \(\mathbb{H}\mathbb{C}\) is the set of hereditarily countable sets. It is often more convenient to use this space to represent models since no coding is required.
Working in the more general setting will only make the notation more tedious for our current purposes.
I’m using \(\sqcup \) to denote disjoint union.
See Chapter 13 of Jech (2003) for a proper explanation.
The converse does not hold. For example, ZFC and GBN are equiconsistent over ZFC, but ZFC cannot interpret GBN.
Note the unorthodox notation below is intended to indicate that there are interpretations t and s going in both directions.
We assume the usual gloss of terms like \(\emptyset \) into \(\mathcal {L}_{\in }\).
Note that there is some abuse of notation here: \(mod(\mathcal {L}_{0})\) is intended to denote the set of \(\mathcal {L}_{0}\)-models of the empty theory.
To illustrate this note that while we always have
$$\begin{aligned}{}[\varphi _{\mathcal {L}}]\cap [\lnot \varphi _{\mathcal {L}}]=\emptyset \end{aligned}$$it is often and perhaps counterintuitively the case that
$$\begin{aligned} cl_{DE}``[\varphi _{\mathcal {L}}]\cap cl_{DE}``[\lnot \varphi _{\mathcal {L}}]\ne \emptyset . \end{aligned}$$Thus, we might say that intersection of events and conjunction of sentences do not line up in \(\Omega _{DE}\) as nicely as they do over \(\Omega \). To see why this is the case, let \(\mathcal {L}=\{P\}\) be the language where P is a one-place relation symbol. Let \(\mathcal {M}\) and \(\mathcal {N}\) be models with domains containing just one object. Let \(P^{\mathcal {M}}=\emptyset \) and let \(P^{\mathcal {N}}=N\). By a similar argument to one we have used above, it can be seen that \(\mathcal {M}\sim _{DE}\mathcal {N}\). Now let \(\varphi \) be the sentence \(\exists xPx\). It should be clear that \([\varphi _{\mathcal {L}}]\cap [\lnot \varphi _{\mathcal {L}}]\) is empty. However, since \(\mathcal {M}\in cl_{DE}``[\varphi _{\mathcal {L}}]\), \(\mathcal {N}\in cl_{DE}``[\lnot \varphi _{\mathcal {L}}]\) and \(\mathcal {M}\sim _{DE}\mathcal {N}\) we see that
$$\begin{aligned} cl(\mathcal {M})\in cl_{DE}``[\varphi _{\mathcal {L}}]\cap cl_{DE}``[\lnot \varphi _{\mathcal {L}}]. \end{aligned}$$Very soon, a much improved framework will be offered in Sect. 3.2.
We’ll discuss the import of this theorem further in Sect. 4.
This obviously works for theories articulated in finite vocabularies, but with a little care with regard to which relation symbols are used, there is no problem with using infinitely many relation symbols of a particular arity; e.g., we might use \(R_{2n}^{i}\) for all \(n\in \omega \).
Strictly, the second clause is covered by the last, but a little redundancy hopefully adds a little clarity here.
See Steel (2014) for a particular good overview of this position.
These are point-wise versions of relations discussed extensively by Visser (2004).
This is closer to the original statement in Ali (2016).
See Theorem 19.14 in Jech (2003).
A detailed proof of this can be found in Reitz (2007).
The classic example here is ZFC and GBN. They are obviously equiconsistent, but ZFC cannot interpret GBN as can be seen by a reflection argument noting that GBN is finitely axiomatizable.
For ease of discussion, we avoid parameters here.
It’s worth noting that, depending on one’s metaphysical outlook, this could push us to say that there is no fact of the matter whether CH is true or not, since there will always be worlds where CH is true and worlds with the same information where it is false. But there is also a way to push back on this. Rather than taking what is essentially supervaluationist approach to the equivalence classes, we might be epistemicists and take it that just one world (up to, say, elementary equivalence) in each equivalence class is the correct one and we just can’t know which one it is Williamson (1994). Nothing hangs on this issue below.
More precisely, a \(\Sigma _{2}\) sentence \(\varphi \) is of the form \(\exists x\forall y\psi (x,y,z)\) where \(\psi \) is \(\Delta _{0}\) in the Lévy hierarchy (see Chapter 12 of Jech (2003) for more details). It can then be seen that \(\varphi \) is true if and only if there is some \(\alpha \) such that \(V_{\alpha }\models \varphi \wedge \chi \), where \(\chi \) says that \(\alpha \) is a \(\beth \)-fixed point.
See Lemma 3.3 of Koellner (2010).
See Theorem 2.5.10 of Larson (2004)
See Maddy and Meadows (2020) for a proof of a related claim from which this proposition easily follows.
Recall that a filter is upward closed and downward directed. Note that we are considering this to be a filter on a poset not on a field of sets.
References
Enayat, A. (2016). Variations on a Visserian theme. In J. van Eijck, R. Iemhoff, & J. J. Joosten (Eds.), Liber Amicorum Alberti: A tribute to Albert Visser (pp. 99–110). College.
Feferman, S., Friedman, H. M., Maddy, P., & Steel, J. R. (2000). Does mathematics need new axioms? The Bulletin of Symbolic Logic, 6, 401–446.
Gitman, V., & Hamkins, J. David. (2010). A natural model of the multiverse axioms. Notre Dame Journal of Formal Logic, 51, 475–484.
Hamkins, J. D. (2012). The set-theoretic multiverse. The Review of Symbolic Logic, 5, 416–449.
Hamkins, J. D., & Seabold, D. E. (2012). Well-founded Boolean ultrapowers as large cardinal embeddings. ArXiv e-prints, June
Hugh Woodin, W. (2004). Set theory after Russell; the journey back to Eden. In G. Link (Ed.), 100 years of Russell’s Paradox. De Gruyter.
Hugh Woodin, W. (2011). The realm of the infinite. In M. Heller & W. Hugh Woodin (Eds.), Infinity: New research frontiers. Cambridge University Press.
Hugh Woodin, W. (2012). The continuum hypothesis, the generic multiverse of sets, and the\(\Omega \)conjecture. Cambridge University Press.
Jech, T. (2003). Set Theory. Springer.
Koellner, P. (2009). Truth in mathematics: The question of pluralism, pp. 80–116. Palgrave Macmillan UK
Koellner, P. (2010). Strong logics of first and second order. Bulletin of Symbolic Logic, 16(1), 1–36.
Koellner, P. (2012). Woodin on ‘the realm of the infinite’. Unpublished manuscript.
Kunen, K. (1970). Some applications of iterated ultrapowers in set theory. Annals of Mathematical Logic, 1(2), 179–227.
Kunen, K. (2006). Set Theory: an introduction to independence proofs. Elsevier.
Larson, P. B. (2004). The stationary tower: Notes on a course by W. Hugh Woodin. University lecture seriesAmerican Mathematical Socity.
Maddy, P. (2016). Set-theoretic foundations. American Mathematical Society.
Maddy, P. (2019). What do we want a foundation to do? In S. Centrone, D. Kant, & D. Sarikaya (Eds.), Reflections on the foundations of mathematics: Univalent foundations, set theory and general thoughts (pp. 293–311). Springer.
Maddy, P. E., & Meadows, T. (2020). A reconstruction of steel’s multiverse project. Bulletin of Symbolic Logic, 26(2), 118–169.
Meadows, T. (2021). Two arguments against the generic multiverse. Review of Symbolic Logic, 14(2), 347–379.
Reitz, J. (2007). The ground axiom. Journal of Symbolic Logic, 72, 1299–1317.
Steel, J. R. (2014). Gödel’s program. In J. Kennedy (Ed.), Interpreting Gödel: Critical Essays. Cambridge University Press.
Visser, A. (2004). Categories of theories and interpretations. Utrecht Logic Group Preprint Series, 228.
Walsh, S. (2014). Empiricism, probability, and knowledge of arithmetic. Journal of Applied Logic, 12(3), 319–348.
Williamson, T. (1994). Vagueness (Vol. 81). Routledge.
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I’d like to thank Juliette Kennedy and Jouko Väänänen for inviting me to present this material to the Helsinki Logic Group in 2021. I’d also like to thank Benedict Eastaugh, Rohan French and Daniel Hermann for helpful conversations about this project.
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Meadows, T. Risk and theoretical equivalence in mathematical foundations. Synthese 202, 165 (2023). https://doi.org/10.1007/s11229-023-04374-1
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DOI: https://doi.org/10.1007/s11229-023-04374-1