Abstract
In this paper, I present and motivate a modal set theory consistent with the idea that there is only one size of infinity.
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Notes
In many applications, the generic multiverse is also taken to be closed under forcing grounds. But this is irrelevant here given the next assumption.
I emphasize of course we are restricted to set forcings; the point is the forcing theorem holds even for plural formulas.
See [7], Theorem 4.
I’m again indebted to Roberts for the central idea to this proof.
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Acknowledgements
This paper bears a huge intellectual debt to prior work by Øystein Linnebo and Sam Roberts, each of whome were also extremely helpful in its development; I owe a special debt of gratitude to Roberts in particular for the time he spent helping me to understand many of the more subtle logical points at issue. I also thank Neil Barton, Vera Flocke, and Martin Pleitz for useful comments on earlier drafts.
I’d also like to acknowledge the support of the ConceptLab project at the University of Oslo, led by Herman Wright Cappelen, Øystein Linnebo, and Camilla Serck-Hanssen, which provided me with generous funding to spend a summer discussing these and other issues with so many first rate scholars of logic and philosophy of mathematics it would be unpractical to list them here.
Finally, as this paper is the centerpiece of my Ph.D dissertation, undertaken at New York University, I’d also like to thank my committee (Hartry Field, Kit Fine, Graham Priest, and Crispin Wright) for their constant support and encouragement, and to express my gratitude to the NYU philosophy department as a whole for providing me with the perfect environment in which to pursue work of this kind.
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Appendix: Consistency of M
Appendix: Consistency of M
Let
be a ctm, and let \(M = V_{\mu }^{M_{0}} = L_{\mu }^{M_{0}}\).
Let Inac(α) abbreviate the claim that α is inaccessible.
Fact 1
For each formula φ, we have
Definition 1 (Generic Multiverse)
Let the generic multiverse \(\mathbb {M}\) on M be the closure of {M} under N↦N[G], G generic from some \(\mathbb {P}\) in N.Footnote 1
Remark 1
Since we chose M to be a model of V = L, we have that the generic multiverse on M consists of precisely those structures of the form M[G] for G generic over M by Usuba’s ‘Downward Directed Ground’ theorem [23].
Lemma 1
Every N in \(\mathbb {M}\) satisfies Fact 1.
Proof
Any forcing in M satisfies the λ chain condition for some λ < μ, and so does not destroy μ s Mahlo-ness in M0. Hence the lemma obtains in any model in \(\mathbb {M}\). □
Remark 2
We use Mahlo cardinals because we want reflection to inaccessibles in all models of the generic multiverse. A weaker analogue of this result can be had for inaccessibles (by simultaneously substituting ‘Mahlo’ for ‘inaccessible’ and ‘inaccessible’ for ‘worldly’ everywhere). But since (as Hamkins has shown) worldly cardinals are not downwards absolute in the generic multiverse, we would run into difficulties below, and a Mahlo cardinal will therefore prove much more convenient. I do expect, however, that equi-consistency with ZFC should be attainable.
Definition 2 (The model \(\mathbb {K}\))
Let \(\mathbb {K}\) be a Kripke model with the following components.
-
W, the set of possible worlds, is the set of all pairs 〈α,N〉, where \(N \in \mathbb {M}\) and α is inaccessible in N. Write w0, w1 for the first and second component of w ∈ W respectively.
-
D, the domain function, applies to a possible world and returns \(\langle V_{w_{0}}^{w_{1}}, V_{w_{0} + 1}^{w_{1}} \rangle \) as the singular and plural domain of the world respectively.
-
w is vertically accessible from u, written w ≤vu, when w1 = u1 and u0 ≤ w0.
-
w is horizontally accessible from u, written w ≤hu, when w0 = u0 and \(\exists \mathbb {P} \in V_{u_{0}}^{u_{1}} \exists G_{\mathbb {P}}[w_{1} = u_{1}[G_{\mathbb {P}}]]\).
-
w is generally accessible from u, written w ≤gu, when u0 ≤ w0 and \(\exists \mathbb {P} \in V_{w_{0}}^{w_{1}} \exists G_{\mathbb {P}}[w_{1} = u_{1}[G_{\mathbb {P}}]]\).
Here, the notation \(G_{\mathbb {P}}\) means that G is generic for \(\mathbb {P}\) (relative to the obvious ground model).
Given a world w and (singular and plural) variable assignment v we define truth at w in the standard Kripke-Tarskian way.
I will use \(w \vDash \varphi \) to mean that the world w satisfies φ in a Kripke model, and use the very same symbol \(\vDash \) for things like \(w_{1} \vDash \varphi \) to mean that a classical first order model satisfies φ. Context will always make it clear what is meant.
Note that since the plural domain of a world 〈α,N〉 is always the rank \(V_{\alpha +1}^{N}\), we immediately get that \(\mathbb {K}\) satisfies impredicative comprehension for formulas in non-modal (so plural and first order) fragment of the language. This in turn means first order truth is definable and that the forcing theorem holds for formulas in the plural language. See e.g. Krapf [11, p30].Footnote 2
We need a few lemmas on the interactions between 〈v〉 and 〈h〉. The first says that w is generally accessible to v iff it is a forcing extension of a height extension of v.
Lemma 2
w ≤gv iff there is a u with u ≤vv and w ≤hu.
Proof
The right to left is immediate. So suppose w ≤gv. The world u = 〈w0,v1〉 is vertically accessible from v, and w is horizontally accessible from u. □
Remark 3
We used the downward absoluteness of inaccessibility in the previous proof; the argument would fail were ‘inaccessibility’ replaced by ‘worldly’ in the definition of \(\mathbb {K}\).
Corollary 1
\(\mathbb {K} \vDash \Diamond \varphi \leftrightarrow \langle v \rangle \langle h \rangle \varphi \).
Proof
Immediate from the previous lemma. □
The second says that if something can be forced to hold in all vertical extensions, then in all vertical extensions it can always be forced.
Lemma 3
\(\mathbb {K} \vDash \langle h \rangle [ v ] \varphi \rightarrow [ v ] \langle h \rangle \varphi \).
Proof
If there’s a forcing extension such that all its height extensions φ, then this forcing will be available at all vertically accessible worlds. □
Lemma 4
For every expression ψ of \({\mathscr{L}}_{\Diamond }\) there is an expression t(ψ)(α) of the pure language of set theory (with free variable α) such that \(w_{1} \vDash t(\psi )(w_{0})\) if and only if \(w \vDash \psi \).
Proof
We define t(ψ) recursively. Here, ρ represents the standard rank function on sets.
-
t(x ∈ y) = x ∈ y
-
t(x ≺ xx) = x ≺ xx
-
t(φ ∧ 𝜃) = t(φ) ∧ t(𝜃)
-
t(¬φ) = ¬t(φ)
-
\(t(\forall x \varphi ) = (\forall x t(\varphi ))^{V_{\alpha }}\)
-
t(〈v〉φ) = ∃β > α[Inac(β) ∧ t(φ)(β)]
-
\(t(\langle h \rangle \varphi (x , xx)) = \exists \mathbb {P}, p \in \mathbb {P}[\rho (\mathbb {P}) < \alpha \wedge p \Vdash t(\varphi (\check {x}, \check {xx}))]\)
-
t(♢φ) = t(〈v〉〈h〉φ).
Here, plural check names are defined just as standard second-order check names are; see [11] here again for more details.
We now prove the claim by induction, assuming as inductive hypothesis that for all worlds u, \(u \vDash \varphi \) iff \(u_{1} \vDash t(\varphi )(u_{0})\) for φ of lower complexity than ψ.
Only the modal operators are significant. So suppose first that ψ := 〈v〉φ and \(w \vDash \psi \). Then there is u ≤vw with \(V^{u_{1}}_{u_{0}} \vDash \varphi \). By induction, \(u_{1} \vDash t(\varphi )(u_{0})\) and hence by definition \(u_{1} = w_{1} \vDash \exists \beta \geq w_{0} [Inac(\beta ) \wedge t(\varphi )(\beta ) ]\), as required. In the other direction, suppose
Then there is a vertically accessible world from w that satisfies t(φ)(β), and the result follows by induction.
Turning to 〈h〉, suppose first that \(w \vDash \langle h \rangle \varphi (x)\). Then there is u ≤hw with \(u \vDash \varphi \). By induction, \(u_{1} \vDash t(\varphi )(u_{0})\) which is precisely to say \(u_{1} \vDash t(\varphi )(w_{0})\). By the forcing theorem, there is a condition p for the relevant order \(\mathbb {P} \in V_{w_{0}}^{w_{1}}\) such that \(w_{1}\vDash p \Vdash t(\varphi (\check {x}))(\check {w_{0}})\). Thus
as required. The converse is similar to the previous.
The ♢ modality follows immediately using Corollary 2. □
Lemma 5
The modal logic of ≤v is S4.2.
Proof
S4 is immediate from the definitions. For .2, if two distinct worlds are v-accessible from a common one, whichever has the greater ordinal rank is accessible from the other. □
Lemma 6
If \(N \in \mathbb {M}\) has \(N \vDash p \Vdash \Box \varphi \) (some \(\mathbb {P} \in N\)), and N[G] is any forcing extension of N, then \(N[G] \vDash p \Vdash \Box \varphi \).
Proof
Assume that \(N \vDash p \Vdash \Box \varphi \) and \(N[G] \vDash \neg (p \Vdash \Box \varphi )\). Then there is \(q, \mathbb {Q} \in N\) such that \(N \vDash q \Vdash \text {``}\exists r \leq p[r \Vdash \Diamond \neg \varphi ]\text {''}\). But now a product forcing in the order \(\mathbb {P} \times \mathbb {Q}\) on a generic containing the condition 〈r,q〉 yields a contradiction. □
Lemma 7
The modal logic of ≤h is S4.2.
Proof
S4 is immediate from the definitions. For .2, we cannot make use of convergence of ≤h since it is not convergent.Footnote 3 Hence instead we make use of the t-translation. Suppose that \(w \vDash \langle h \rangle [ h ] \varphi \). Then
Consider any u ≤hw with u1 = w1[G], some generic G. We must show \(u \vDash \langle h \rangle \varphi \). Well, since \(\mathbb {P}\) and p are in the domain of u, we can construct a u1-generic filter H for \(\mathbb {P}\) containing p. By Lemma 12, we still have \(p \Vdash [ h ] t(\varphi )(\check {w_{0}})\) in u1, and so the world 〈w0,u1[H]〉 is an accessible world from u satisfying φ as required. □
Lemma 8
≤g satisfies S4.2.
Proof
Suppose \(\Diamond \Box \varphi \) holds at w. Then 〈v〉〈h〉[v][h]φ holds at w, and by Lemma 9 〈v〉[v]〈h〉[h]φ holds at w. Whence [v]〈v〉[h]〈h〉φ holds at w, and so finally we conclude \(\Box \Diamond \varphi \) using Lemma 9 again (contraposed). □
Lemma 9
For each formula φ(x,y,yy) in the language \({\mathscr{L}}_{\Diamond }\), any world w of \(\mathbb {K}\) satsifies
and similarly for any other number of parameters.
Proof
We already observed that every world satisfies impredicative comprehension for the pure second order fragment. So, let w be any world containing y, yy. Then the set of x with t(φ(x,y,yy)) is a member of the plural domain of any w (relative to any valuation v on w). The validity of the t-translation then implies the result. □
Lemma 10
\(\mathbb {K} \vDash \) VEv.
Proof
Any xx that don’t form a set at w form a set in the world given by the next inaccessible after w0 in w1. □
Lemma 11
\(\mathbb {K} \vDash \)HE.
Proof
Given a partial order and some dense sets in it in a world w, we know we can find a generic G that intersects all of them. The structure \(V_{w_{0}}^{w_{1}[G]}\) is then a horizontally accessible world with the required features. □
Lemma 12
\(\mathbb {K} \vDash \)Comp(\(\subseteq \))v
Proof
Given any x at a possible world, the set of all its subsets exists at the same world (since all of them are inaccessible ranks); and vertical expansion never adds further subsets. □
Lemma 13
Every instance of the 〈v〉 translation of replacement holds in \(\mathbb {K}\).
Proof
For the purposes of this proof, let ♢ stand for 〈v〉. Then suppose:
For each x ∈ a, we can therefore find a world w(x) with
by reflection to inaccessibles, we can find an inaccessible rank κ beyond all the w(x)0 such that Vκ of w1 reflects the situation for t(φ♢(x,y)) and hence for φ♢(x,y) in w1. The world u = 〈κ,w1〉 will then have
so, by replacement in u and Lemma 6
as required. □
Lemma 14
Every instance of the 〈h〉-translation of the replacement scheme holds in \(\mathbb {K}\).
Proof
For the purposes of this proof, let ♢ stand for 〈h〉; and let the clause for 〈h〉 in the t-translation temporarily be replaced by
-
\(t(\langle h \rangle \varphi (x , xx)) = \exists \mathbb {P}, p \in \mathbb {P}[ p \Vdash t(\varphi (\check {x}, \check {xx}))]\)
so now there is no need for reference to an ordinal parameter and in fact we have
since the only relevant modality is ♢ = 〈h〉.
Suppose
Then
For each x ∈ a let \(\mathbb {P}_{x}, p_{x}\) be a witnessing order and condition for which this holds. By replacement in w, we may form the products \(\mathbb {Q} = \varPi _{x \in a} \mathbb {P}_{x}\) and q =π x∈apx. Let G be generic for \(\mathbb {Q}\) containing q. Then if u = 〈w0,w1[G]〉 it is easy to see that \(u \vDash t(\forall x \in a\exists y \varphi (x,y))\). By replacement in u,
Using Lemma 6, we get
and we’re done.Footnote 4□
Lemma 15
Every instance of the ♢-translation of the replacement scheme holds in \(\mathbb {K}\).
Proof
Suppose
Then for every x ∈ a there is w(x) with
By Corollary 2, w(x) is a forcing extension of a height extension of w; and so in particular we can find a corresponding u(x) ≤vw with:
We then take κ an inaccessible rank above u(x)0 which reflects the situation for \(\exists p, \mathbb {P}[p \Vdash \exists y \varphi ^{\Diamond }(\check {x},y)]\). By product forcing, there is a horizontally accessible world v = 〈κ,w1[G]〉 with
and the rest is straightforward. □
Theorem 1
M is consistent relative to the existence of a Mahlo cardinal.
Proof
The only difficult parts are Lemmas 11, 13–21. The rest is routine. □
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Scambler, C. Can All Things Be Counted?. J Philos Logic 50, 1079–1106 (2021). https://doi.org/10.1007/s10992-021-09593-w
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DOI: https://doi.org/10.1007/s10992-021-09593-w