Abstract
Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view—Countablism—according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, initially proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not.
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Notes
Meadows (2015) “expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. [...] elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort to make sense of Cantor’s troubling theorem” (191). Scambler (2021) develops the modal or ‘indefinite extensibility’ approach to CT that we favor, “generalizing [Linnebo’s] theory L, which offers a modal, indefinite extensibility solution to Russell’s Paradox, to a theory I call M [after Meadows], which offers an analogous ‘solution’ to Cantor’s theorem” (10), such that “in the modal setting Cantor’s theorem can be reconciled with the existence of only one infinite cardinality” (2); and which provides “a way of reconciling mathematics after Cantor with the idea there is one size of infinity” (21); we will be helping ourselves to Scambler’s formal results down the line. Pruss (2020) also develops a modal approach to CT, arguing that “there is an epistemic possibility that all infinite sets have the same size as the natural numbers” (604).
Scambler moreover suggests that the moral to draw is broadly anti-objectivist: “I think the results are best understood in the context of a kind of anti-objectivism about the question of whether there are different sizes of infinity: the guiding idea being that, when set-theoretic practice is formalized one way, one will find one verdict, and when it is formalized differently one will find quite a different one, with nothing objective to tell between them” (1099).
Relatedly, we aim to push back against the sort of anti-objectivist line of thought registered in the previous footnote.
It is more common to encounter RP as the claim that there is no set of all sets not containing themselves, but in the context of the axiom of foundation, the plurality of all sets not containing themselves is the same as the plurality of all sets simpliciter.
Here we intend V to be the set of all pure sets (sets whose transitive closure only includes sets), so that every \(x \in V\) is a set whose elements are all included in V, implying \(V \subset P(V)\).
This set will exist by the standard axiom schema of restricted comprehension.
We’ll discuss certain historical and contemporary exceptions to this rule shortly.
We should note that our position is not that there is anything inherently problematic about diagonalization arguments. After all, several of the most important proofs in logic appeal to some kind of diagonalization procedure, such as Gödel’s Incompleteness Theorems and the undecidability of the Halting problem. Relatedly, we are not questioning that CT and RP (and other diagonalization proofs) are perfectly valid formal results. We will only be arguing that CT does not have the philosophical import that it’s generally taken to have. Moreover, our reasons for thinking this do not generalize to every diagonalization argument, and our general skepticism about the uncountable typically will not apply to these other arguments (since, e.g., neither Gödel’s Incompleteness Theorems nor the undecidability of the Halting problem require any uncountable mathematics). See Meadows (2015: 206–7) for further discussion of why skepticism about the philosophical import of CT does not generalize to other diagonalization arguments.
Thanks to Øystein Linnebo for discussion here.
Note that this task may be accomplished even if Whittle is right that there is something like an intrinsic conception of size. Compare certain realist accounts of dispositions (as per, e.g., Martin (1996)) according to which these are intrinsic in that they might be had at a ‘lonely’ world, but which are necessarily such as to produce certain manifestations in certain circumstances.
Indeed, it is plausible that the notion of number itself arises from just such acts of counting—or tallying, as the simplest form of counting—and that the fact that this is so provides the basis for arithmetical relations between numbers being a priori and not subject to empirical disconfirmation; see Wilson (2000).
We think that even a Nominalist should be able to make sense of the possibility of there being such bijections. We discuss the relevant modality in more depth in Sect. 5.
One potential concern with this account of size is that it implicitly relies on the axiom of choice. In particular, someone who rejects the axiom of choice can consistently believe that some sets do not have a well-ordering. Consequently, such sets would not have any bijection from themselves to any ordinal number. Given our conception of size, this would in turn imply that the ‘same size as’ relation is not reflexive, since such sets would not have the same ‘number’ of elements as themselves (in fact, such sets could not be assigned any ‘number’ at all in the absence of such a well-ordering). We have three responses to this concern. First, as a dialectical matter, Whittle (2015) himself is perfectly content with the axiom of choice. Second, pursuing this objection to our account of size is mathematically revisionary, whereas Whittle’s initial case against Size \(\rightarrow\) Function was meant to be mathematically neutral. At the very least, endorsing a conception of size that is mathematically revisionary implies a significant cost. Lastly, the kind of denial of the axiom of choice that is needed for this objection to work is fairly radical. Our account of size only relies on the possibility of there being an appropriate bijection to an ordinal number. In order to pursue this kind of objection, one would have to maintain that there is a set such that it’s impossible for there to be a well-ordering of that set. A pluralist approach to set theory, according to which there are possible set-theoretic universes where the axiom of choice holds and possible set-theoretic universes where the axiom of choice does not hold, could still allow for this modal principle. Given the fact that ZFC is consistent so long as ZF is consistent, rejecting the mere possibility of there being suitable well-orderings strikes us as a fairly radical view. We will further discuss the modality at issue in our conception of size in Sect. 5.
We won’t be addressing every strategy for blocking RP. One notable strategy that we won’t discuss (beyond the following remarks) proceeds via dialetheism, accepting Naive Comprehension and the contradiction it generates, and altering the logic along paraconsistent lines (as in Priest (1995)). This approach could presumably be extended to CT, and moreover in a way that would fail to vindicate Countabilism (since moving towards a paraconsistent logic does not prevent there being, say, uncountably many spacetime points). It remains, in our view, that a dialethic approach to CT would be unsatisfactory, in facing the same main problem as the approach applied to RP: namely, the move to dialetheism and paraconsistent logic is too theoretically costly in light of the availability of viable and consistent approaches to the set-theoretic paradoxes.
We don’t intend to be arguing that Nominalism is false by making this point. In fact, one of us has defended Nominalism on independent grounds (see Builes (forthcomingb)). Rather, our point is simply that even a Nominalist should have a way of making sense of the practice of set theory.
We subsume the strategy of response to RP that involves appealing to a Ramified theory of types (see, e.g., Russell (1908)) under the present strategy, since Ramified type theory assumes a sort of predicativism/no circularity principle.
Poincaré’s critique targeted a version of CT aiming to show that arbitrary descriptions in a fixed language (corresponding to a countable set ordered by alphabetized sentences consisting of a finite number of words) could not be put in one-to-one correspondence with points of space; he characterizes Cantor’s purported result as “an illusion”, since “to classify these sentences and the corresponding points according to the letters which form the sentences [...] is to construct a classification which is not predicative” (61). See also Weyl’s (1918: 26–7) critique of Cantor’s theorem, which highlights that the result reflects the ‘purely mathematical’ creation of an impredicative condition. It is worth noting that these early critiques of Cantor’s theorem were less radical than Brouwer’s intuitionism: an entirely classical conception of natural numbers is retained, and classical logic is retained for reasoning about (predicatively acceptable) sets of natural numbers. Thanks to a referee here.
See Uzquiano (2015) for a more recent discussion on the relationship between CT and Predicativism. In particular, Uzquiano discusses a version of Bernays’ theorem, a result related to Cantor’s Theorem about classes, that can be proven with only predicative class comprehension. However, as Uzquiano notes, “while the proof of Bernays’ theorem does not require impredicative class comprehension, the link with the Cantorian lemma does presuppose it" (9), where the Cantorian lemma is the claim that every class has more subclasses than members.
Another (if approximate) precursor of the modal strategy, applied by Dummett (1993) to both the set-theoretic and semantic paradoxes, involves the claim that the notion of set is ‘indefinitely extensible’, where the notion of extensibility is not explicitly modal.
We address the potential worry that the non-existence of abstracta is ‘necessary’ in Sect. 5.1.
This follows from the fact that there is a possible set-theoretic structure witnessing the countability of both X and P(X).
Here we are following Scambler (2021) presentation of L, since Linnebo does not build in the axiom of choice in his presentation of the theory. Scambler includes the axiom of plural choice as part of the background plural logic, which, together with Modal Naive Comprehension, ensures the possible existence of a choice set for any set of disjoint non-empty sets.
See Dorr (2008, 2010) and Azzouni (2004) for related ontologically neutral approaches to mathematics. Just as these approaches often claim that they can vindicate the ‘nominalistic content’ of scientific theories that quantify over mathematical objects, such approaches can use similar means to extract the ‘nominalistic content’ of Countabilism, as we will further discuss in Sect. 7.
See Chalmers (2002) for discussion of the relevant notion of idealized conceivability.
There are also a number of historical challenges to the connection between conceivability and metaphysical possibility. For example, in her Essays Upon the Relation of Cause and Effect (1824), Mary Shepherd maintains that Hume errs in supposing that whatever he can conceive (e.g., that a new existent could occur without a cause, or that some similar cause might produce a different effect) is genuinely possible: “Mr. Hume makes also a great mistake in supposing because we can conceive in the fancy the existence of objects contrary to our experience, that therefore they may really exist in nature; for it by no means follows that things which are incongruous in nature, may not be contemplated by the imagination, and received as possible until reason shows the contrary” (83). For a prominent attempt to reforge an indirect link between conceivability and metaphysical possibility in terms of ‘epistemic two-dimensionalism’, see Chalmers (1996, 2002, 2006). For criticisms of Chalmers’s proposal, see Wilson (2006) and Melnyk (2008). For an alternative approach to implementing epistemic two-dimensionalism which appeals to abduction rather than conceiving, see Biggs and Wilson (2017, 2019).
However, in Sect. 6.3, we argue that width potentialism can be motivated on the grounds that there cannot be brute necessities. Insofar as one thinks that brute necessities are inconceivable, then this would motivate width potentialism as understood in terms of conceivability.
Thanks to a referee for raising this concern, and more generally for encouraging us to expand on what understanding(s) of modality is supposed to be at issue in Countabilism.
As Feferman (2005) observes, “the success of axiomatic set theory—as developed by Zermelo, Skolem and Fraenkel—[succeeded] in allaying fears about the paradoxes. Though not demonstrably consistent, intensive development of the subject without running into any difficulties gave comfort and confidence to its practitioners and gradually won the support of mathematicians at large” (11).
As a referee points out, one could perhaps introduce a stipulated notion of ‘possibility’ that makes Countabilism come out as trivially true, but we don’t think any of the three interpretations that we have discussed makes Countabilism come out as trivial. All three interpretations draw on modal notions whose coherence and importance have been defended on independent grounds by several philosophers. One could of course adopt a more radically skeptical stance that questions the intelligibility or objectivity of these modal notions (see, e.g., Clarke-Doane (2019, 2021), but arguing against this kind of view is beyond the scope of the current paper. At the end of the day, we think that the best reason for believing that Countabilism is both true and non-trivial is by looking at its many downstream consequences in mathematics, science, and philosophy. After all, Countabilism cannot be trivial or unintelligible if it has important non-trivial consequences about intelligible matters.
Another complicating factor about M is that it utilizes three different modal notions, a notion of ‘vertical’ possibility, a notion of ‘horizontal’ possibility, and a ‘combined’ notion of possibility that results from arbitrary iterations of vertical and horizontal possibility. The different interpretations of modality that we have so far been discussing are best viewed as interpretations of the more generalized combined notion of modality, whereas Scambler’s vertical and horizontal notions of possiblity should be seen as restrictions on this modality from which one can ‘generate’ the more general combined notion of modality. We also leave the important question of whether the relevant interpretation of the combined modal notion in M can be entirely ‘factored’ into restricted vertical and horizontal modalities for future work. Thanks to a referee for raising this concern.
This follows from a basic theorem of confirmation theory, according to which if some evidence E confirms H over \(\lnot\)H, then the evidence \(\lnot\)E confirms \(\lnot\)H over H.
We do not have any qualms with there being other conceptions of set besides the iterative conception, such as those that deny the axiom of foundation. We also don’t want to defend the claim that the concept of set at the heart of the iterative conception is in any way metaphysically superior than other conceptions of set. Our main interest lies in questions about the possible sizes of infinity, which we think has ramifications far outside mathematics. We are only focusing on the iterative conception of set here because such a conception provides an ideal setting for reasoning about possible sizes of infinity. If there are additional non-well founded sets that lie outside the cumulative hierachy, then in any case those sets will not influence any standard mathematics (see, e.g., Kunen (1980)). For an extended defense of the iterative conception of set and its advantages over other conceptions, see Incurvati (2020).
This follows from the fact that both theories are mutually interpretable (in the model-theoretic sense).
Of course, by Gödel’s theorem, there can be no truly complete axiomatization of \((H_{\aleph _0}, \in )\), but ZFC seems to determine the truth values of ‘natural’ mathematical statements about \((H_{\aleph _0}, \in )\). There are also less orthodox views on which the axioms of ZFC are simply false of their intended subject matter, which is (at least for our purposes) the iterative conception of set (see footnote 33). Although entering into the details of the controversies behind the particular axioms of ZFC is beyond the scope of the present paper, a few remarks are in order. With respect to the axiom of infinity, it seems to us that the only principled view which denies the possibilty of infinite sets is some version of Finitism, which we (pessimistically) discuss in Sect. 5.2. With respect to the axiom of choice, we are inclined to agree with Pollard (1988) that the axiom of choice for plurals (as opposed to sets) should be uncontroversial. With respect to plurals, the axiom of choice simply states that, for any disjoint non-empty sets there are some things which comprise exactly one element from each. Together with Modal Naive Comprehension (which we think is the most plausible substitute for Naive Comprehension), this plural version of the axiom of choice implies the standard axiom of choice for sets (this is how the modal set theory M derives the axiom of choice). With respect to the axiom schema of replacement, see Incurvati (2020: 90–100) for three different arguments that the iterative conception of set implies the axiom schema of replacement. For wider discussion of these controversies about ZFC, see Clarke-Doane (2013, 2020: Ch. 2).
This follows from the claim that \(\aleph _1\) is a regular cardinal.
The mathematical fact that ZFC without the powerset axiom is consistent with the statement ‘every set is countable’ supports this fact (the model \((H_{\aleph _1}, \in )\) of hereditarily countable sets satisfies all of the ZFC axioms, excluding the powerset axiom).
We should emphasize that Countabilism does not say that the pure mathematics that these scientific theories are based on (e.g. differential geometry) needs to be revised. It only says that the structure of the physical world is not adequately captured by the ‘uncountable’ mathematical objects studied in these mathematical theories.
While we focus here on gunky characterizations of spacetime, on which every spacetime region has a proper part, versions of Existence Monism, according to which the only concrete objecte object is the whole of spacetime, are also compatible with Countabilism. For more on Existence Monism, see Horgan and Potrc (2008), Cornell (2016), and Builes (2021).
See also Arntzenius and Hawthorne (2005) for a discussion of how a gunky conception of spacetime should be supplemented with an account of the variation of physical quantities across regions.
Thanks to a referee for this suggestion.
The following argument is generalized and defended at greater length in Builes (forthcominga).
One way to resist this argument (and the previous argument about mereology) is by rejecting the legitimacy of absolutely unrestricted quantification, which is the intended reading for the quantifiers in both Countabilism and Modal Naive Comprehension. For defenses of the legitimacy of absolutely unrestricted quantification, see Lewis (1991: 68), Sider (2001: xx-xxiv), van Inwagen (2002), and Williamson (2003, 2013). See Clarke-Doane (2019) and Rayo (2020) for independent reasons to think that metaphysical possibility (or ‘absolute’ possibility) might be open-ended.
Thanks to Tom Donaldson, Benj Hellie, Øystein Linnebo, Chris Menzel, Agustín Rayo, Chris Scambler, and Stephen Yablo for comments and conversation. Thanks also to the organizers of and participants at the 2019 Regress Arguments conference at Simon Fraser University and the 2021 Countabilism Workshop at Oslo University.
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Builes, D., Wilson, J.M. In defense of Countabilism. Philos Stud 179, 2199–2236 (2022). https://doi.org/10.1007/s11098-021-01760-8
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DOI: https://doi.org/10.1007/s11098-021-01760-8