2.1. Reduction

To investigate the space of logics and consequence relations and for infinite cardinals and , membership in and will first be reduced to the so-called two-cardinal problem in model theory. As the two-cardinal problem has been investigated extensively, this relatively simple reduction immediately gives rise to a number of corollaries which would be extremely difficult to prove directly. Such corollaries will then be used to lay out the basic features of the space of logics and consequence relations determined by pairs of infinite cardinals, including results on compactness and recursive enumerability.

Let be the language of (non-modal) first-order logic with identity based on and V. Let a model be a tuple where D is a non-empty set and e is a function which maps every n-ary relation symbol R of to a set e(R) of n-tuples on D; informally, D and e will be called the set of ‘individuals’ and the ‘interpretation function, ’ respectively. Truth of an -formula in is defined as usual relative to an assignment function , written . From this, truth of a sentence in is derived as for all assignments a, and written . An -theory is a set of sentences of ; an -sequent is a tuple consisting of an -theory and an -sentence . A model of an -theory is a model in which all members of are true.

To state the two-cardinal problem, fix a unary relation symbol U of , and say that an -theory admits , for cardinals and , if there is a model of such that and . The two-cardinal problem is to determine under which conditions an -theory admitting a given pair of infinite cardinals admits another given pair of infinite cardinals. One might think of this as the problem of extending the standard Löwenheim–Skolem theorem to the case of two cardinals, one for the domain and one for the interpretation of U; this was in fact how Robert Vaught introduced the problem (Morley and Vaught Reference Morley and Vaught1962, Section 6). Let be the set of -theories admitting , and .

From now on, let , , , , etc. always be infinite cardinals. The following two elementary facts for any -theory will be required:

• Fact (i). If admits and , then admits .

• Fact (ii). If admits then admits .

They can be derived from the Löwenheim–Skolem theorem for first-order logic; proofs can be found in Chang and Keisler (Reference Chang and Jerome Keisler1990, Proposition 3.2.7 (i) and (ii)).

Membership in and is straightforwardly reduced to satisfiability on :

This lemma makes the connection to extensions of the Löwenheim–Skolem theorem precise. It should be noted that the particular class of Kripke models used here matters. E.g. if an accessibility relation and variable individual domains were introduced, it would be straightforward to prove that the same -theories can be satisfied on all pairs of infinite cardinals, since one could always add an unconnected world with an arbitrary number of new individuals. Of course, there are also more interesting questions one could pose in such a setting; see, e.g. Bowen (Reference Bowen1979, Chapter 5). Such variant classes of models won’t be considered in the following. Similarly, it is also essential to the results established below exactly which resources are available in . E.g. adding further modal operators, such as explicit quantifiers over worlds, would substantially change the dependence of and on and .

The main results to be established in this section are Theorems 3 and 6. The first reduces an -theory being satisfiable on to a corresponding -theory being a member of , and the second reduces an -theory being a member of to a corresponding -theory being satisfiable on . With various two-cardinal results from model theory, these two theorems will entail many useful corollaries, which will be derived in the two subsequent sections. To prove Theorems 3 and 6, observe first that since in any Kripke model, adding any number of copies of one of its worlds doesn’t change which -sentences are true in it, increasing the number of worlds in a frame doesn’t destroy satisfiability:

The syntactic mapping from -theories to -theories used in Theorem 3 is a version of the so-called standard translation, which translates necessity as explicit universal quantification (informally, over worlds). To do so, the arity of each relation symbol is increased by one, and each atomic predication is translated by adding a special variable to the arguments (which can informally be understood as recording the world of evaluation). Necessity is translated using a universal quantifier binding , restricted to a new relation symbol U (informally, the worlds). Quantifiers are translated by restricting them to the negation of U (informally, the individuals). Using this recursive translation, which will be called , the present version of the standard translation, called , is derived in a second step using a new unary relation symbol A (which informally can be understood as identifying the actual world): is the sentence which says that there is a unique U which is A, and letting be this element, is true.

To define this procedure more precisely, fix a unary relation symbol A of which is distinct from U, and . Let be an injection which maps every n-ary relation symbol of to an -ary relation symbol of , and every element of V to an element of V, such that none of U, A, and are in its image. Extend to a mapping from -formulas to -formulas using the following recursion clauses:

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For any -sentence , define

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and extend in the obvious way to any -theory :

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To prove Theorem 3, two model transformations corresponding to the syntactic mapping will be used, transforming a Kripke model into a corresponding model, and vice versa:

For Theorem 6, a complementary mapping from -sentences to -sentences will be used. The idea behind this translation is to add two constraints: on the one hand, for there to be at least as many worlds as Us, and on the other hand, for there to be infinitely many Us. The first can be achieved by stating that each U is possibly the unique U. To achieve the second, a new relation symbol R will be used, stating that it is a serial, transitive, and asymmetric on U.

More precisely, let be an injection which maps every n-ary relation symbol of to an n-ary relation symbol of distinct from R, and which maps U to itself. Extend to a function from -formulas to -formulas which only replaces a relation symbol F by . For any -sentence , define

where and are the following -sentences:

As with , extend in the obvious way to any -theory :

To prove Theorem 6, two auxiliary lemmas are established:

2.2. Logics and consequence relations

A helpful way of mapping out the logics and consequence relations determined by pairs of infinite cardinals, which also lends itself to reformulations in terms of analogs of the Löwenheim–Skolem theorem, is to establish conditions under which one is contained in another. Using Theorems 3 and 6, this can be reduced to the two-cardinal problem:

As a first corollary, it follows immediately that a frame in which there are at least as many worlds as individuals determines the unique weakest logic and consequence relation determined by any pair of infinite cardinals:

A second corollary follows from the so-called infinite gap two-cardinal theorem, due to Vaught (Reference Vaught, Addison, Henkin and Tarski1965a) (see also Chang and Keisler (Reference Chang and Jerome Keisler1990, Theorem 7.2.6)), which entails that if a theory admits which are sufficiently far apart, it admits . For this and the next few corollaries, two precise notions of distance among cardinals are required; one is in terms of successor cardinals, and one in terms of the powerset operation. So define, for any ordinal , and as follows: ; and ; and, if is a limit ordinal, and . The relevant consequence of the infinite gap two-cardinal theorem can now formulated as saying that an -theory admitting also admits (for ). It follows that any frame in which there are sufficiently more individuals than worlds determines the unique strongest logic and consequence relation determined by any pair of infinite cardinals:

An immediate consequence of these two corollaries is that if , then , and if , then , and analogously for the corresponding logics. To have non-arbitrary ways of referring to these, call them / and / , respectively.

What the first two corollaries leave open are and where ; call these ‘intermediate.’ The following two corollaries show that in this intermediate region, the comparative cardinalities matter; furthermore, in case the generalized continuum hypothesis (abbreviated , the claim that for all ordinals ) fails, they show that both the comparison in terms of successor cardinals and the comparison in terms of the powerset operation matter.

While (Chang and Keisler Reference Chang and Jerome Keisler1990, Proposition 3.2.7 (iv), attributed to Michael Morley, unpublished) record the existence of an -theory which admits, for every , but not , the construction of such a theory is left as an exercise to the reader. To ensure that the corresponding analog to Corollary 10 holds for logics, not just consequence relations, the following lemma shows that there is a finite such theory.

In this proof, witnesses . What does this sentence look like? Apart from relabeling R in , conjoins with two -sentences and which can informally be understood as requiring there to be infinitely many Us and there to be a world corresponding to each U. The most straightforward way of understanding how imposes a constraint on the relation between cardinalities of individuals and worlds is in two steps, one imposing a constraint on the relation between cardinalities of individuals and Us and one imposing a constraint on the relation between cardinalities of Us and worlds. This line of thought applies as well to the witnesses used in the proof of Corollary 10 above and the proofs of several further corollaries below.

Especially without the , the last two corollaries show that the intermediate region exhibits interesting structure. The following two corollaries impose two basic containment properties on the whole space, and so in particular on this intermediate region. The first simply applies the earlier observation that satisfiability can’t be destroyed by increasing the number of worlds; the second notes that it can also not be destroyed by reducing the number of individuals:

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The following diagram summarizes the positive corollaries obtained so far for logics and consequence relations determined by a pair of cardinals in the -series:

The two curves in the diagram divide the space of logics and consequence relations it represents into the region of , which includes those on the diagonal line, the shaded intermediate region, and the region of , which includes those on the vertical lines. marks the containment properties along both axes established in the last two corollaries. It should be noted that the diagram might be misleading in two respects: First, it only records pairs of cardinals in the -series, so unless the is true, it does not represent all pairs of infinite cardinals. Second, although the shaded area and lines suggest continuity, the represented space is obviously discrete. Not represented in the diagram is Corollary 10, which entails that any two represented consequence relations in the intermediate region are distinct if they, as one might say, differ in how far they are from the diagonal.

Assuming the , every intermediate consequence relations is identical to for some and . One might conjecture that in this case, at least the choice of does not matter. But it turns out that this conjecture is unlikely to be settled by , even assuming the : On the one hand, the conjecture follows from , the axiom of constructibility, which entails the . On the other hand, given the consistency of a very mild large cardinal hypothesis, is consistent with the negation of the conjecture.

For the first of these observation, the conjecture will be derived from the so-called gap n conjecture, which says that every -theory admitting also admits . According to Chang and Keisler (Reference Chang and Jerome Keisler1990, 524), that this holds for all was shown to follow from in unpublished work by Ronald Jensen. (Details of the gap 1 case can be found in Björn Jensen (Reference Jensen1972); details of the gap 2 case and a sketch of the gap n case can be found in Devlin (Reference Devlin1984, Chapter VIII).)

Assuming , the logics and consequence relations therefore form -chains, linearly ordered by the proper subset relation, and determined by the following sequence of pairs of cardinals:

The second observation is based on a consistency proof of the failure of the gap 2 conjecture due to Jack Silver. This proof uses an -sentence k which admits if and only if a certain claim holds. ( states the existence of a so-called Kurepa tree satisfying certain properties in terms of . A sketch of the construction of k can be found in Vaught (Reference Vaught and Bar-Hillel1965b), see also the discussion in Chang and Keisler (Reference Chang and Jerome Keisler1990, 527).) Using and (as in Lemma 11), this can be turned into a sentence which is a member of if and only if fails:

On the assumption that is consistent with the existence of two uncountable inaccessible cardinals, Silver (Reference Silver and Scott1971) has proven that is consistent with as well as consistent with . Thus on this assumption, neither settles whether is a member of , nor whether it is a member of , nor whether these logics are the same:

In the same way, an analogous negative result for and can be derived, and the independence of from (rather than ) can be obtained from the slightly weaker assumption that is consistent with the existence of one uncountable inaccessible cardinal. (See Chang and Keisler (Reference Chang and Jerome Keisler1990, 527–528), who attribute the relevant results to Silver and Robert Solovay.)

Some particular positive results have been obtained in the case of the gap 1 conjecture, which lead to some particular positive corollaries for logics and consequence relations determined by pairs of infinite cardinals , although negative results are also available for such pairs. The first positive corollary is based on what is known as Vaught's Two-Cardinal Theorem:

Using what is known as Chang's Two-Cardinal Theorem, this first positive corollary leads to a second one:

The relevant negative results depend on large cardinal assumptions which are stronger than those used in the gap 2 case. Similar to the use of Kurepa trees in the gap 2 case, they use so-called special -Aronszajn trees, whose existence will be written as . Again, there is an -sentence a which admits if and only if (a construction is given in Chang and Keisler (Reference Chang and Jerome Keisler1990, 7.2.11); the observation that such a sentence exists is attributed to Frederick Rowbottom and Silver, independently and unpublished).

While holds, the consistency of with the existence of a Mahlo cardinal entails the consistency of with , and the consistency of with the existence of a supercompact cardinal entails the consistency of with (see Chang and Keisler (Reference Chang and Jerome Keisler1990, 7.2.10. (i), 7.2.12. (i) & (iii)), who attribute these results to Nachman Aronszajn, Mitchell (Reference Mitchell1972) and Shelah (Reference Shelah, Boffa, van Dalen and McAloon1979)). Thus:

2.3. Axiomatizations

If it could be argued that is given by a particular pair , it would be interesting to specify it using a standard axiomatic calculus. This requires (i) to be compact, (ii) to satisfy the deduction/detachment property, and (iii) to be recursively enumerable. (ii) is easily seen to hold in general:

This raises the question which pairs of infinite cardinals determine compact consequence relations, and which determine recursively enumerable logics. For those pairs that satisfy both conditions, it is also interesting to consider whether there are any natural axiomatizations of the logics they determine. The next result shows that all of this is easily done in the case of . For concreteness, the construction will be based on the axiomatization described in Hughes and Cresswell (Reference Hughes and Cresswell1996, Chapter 13), which will be called . Its theorems are the same as those of the system which was presented in a more compressed manner in Kripke (Reference Kripke1959), and there proven to be sound and complete with respect to the class of all frames.

In case , the issues are more subtle. Theorems 3 and 6 allow us to derive a number of results on compactness and recursive enumerability. Starting with compactness, they yield the following theorem, which reduces the compactness of to a corresponding property for :

For and , results are available from which compactness follows:

No corresponding compactness results seem easily available for the remaining intermediate consequence relations. Indeed, Shelah (Reference Shelah, Baaz, Friedman and Krajíček2005) contains, for each , a consistency result for the existence of an -theory which does not admit although all of its finite subsets admit it. With Theorem 24, a corresponding consistency result for the incompactness of , for , follows.

The question of recursive enumerability can also be reduced to one which has been explored in model theory, using the standard notion of consistency in first-order non-modal logic:

As the preceding corollaries indicate, and stand out as compact consequence relations whose corresponding logics are easily shown to be recursively enumerable. It would be interesting to develop axiomatizations for them as well. Indeed, these two stand out also in that any of the other logics and consequence relations are likely to be extremely difficult to axiomatize; recall that previous corollaries exhibited various potential failures of (or even ) to decide membership in logics such as , and .

is an infinitary axiomatization of in the sense that it is obtained from by adding an axiom schema with infinitely many instances. It turns out that this is an essential feature, in the sense that there is no finite set of axioms which added to yields an axiomatization of (even if a rule of substitution is added to , which Hughes and Cresswell (Reference Hughes and Cresswell1996, Chapter 13) do not include). Moreover, this is the case for any logic ; in this sense, none of the logics under consideration here is finitely axiomatizable.

Without setting up the relevant variant of in detail, here is a sketch of the proof: Consider a model based on which interprets every n-ary relation symbol in as the function mapping every element of to , the set of all n-tuples on . For any , define to be such a model based on , i.e. one which interprets every n-ary relation symbol in as the function mapping every element of to . Using a back and forth system, it can be shown that any -sentence of quantifier rank up to i is true in if and only if it is true in . (The relevant definitions can be adapted from Fritz (Reference Fritz2013).) Considering any finite set , let i be the maximum of the quantifier ranks of its members. Since verifies all members of , so does . All axioms are true in (under any assignment function), and the -formulas true in (under any assignment function) are closed under the rules of . But , a sentence saying that there are at least individuals, is not true in . Thus , which is a member of , is not derivable in the axiomatization resulting from adding the members of to .

3.1. Williamson's argument

Let be the extension of by an infinite set of relational variables for each arity, as well as quantifiers binding them; let be the corresponding extension of . For any -formula , let be the result of uniformly replacing each relation symbol in by a distinct relational variable of appropriate arity, and prefixing the resulting formula by a string of universal quantifiers binding all of these variables; the result is an -formula. Unless explicitly noted otherwise, first- and second-order quantifiers will be interpreted unrestrictedly, and as expressing metaphysical necessity, as above. Thus, for any -sentence , is a fully interpreted -sentence. With this, the definition of can be summarized as follows:

Recall that this truth predicate is assumed to be transparent, as third- and in general higher order quantifiers are assumed to be available in the metalanguage.

As in Section 2.1, let be an injection which maps every n-ary relation symbol of to an -ary relation symbol of , and every element of V to an element of V, such that neither U nor are in its image; this was extended to a version of the standard translation from -formulas to -formulas. Instead of this extension of , the following will use a variant extension which differs from the above extension of only in that first-order quantifiers are not restricted to the negation of U. Thus, the most important clauses are:

Extend the notation introduced above by writing for the result of the same procedure, exempting U from being replaced by a variable and bound by a universal quantifier. Thus for any -sentence , is an -formula in which all relation symbols apart from U are interpreted, and no variable apart from is free.

Note that in the version of the standard translation used in this premise, the necessity operator is translated as a universal quantifier restricted to U, and first-order quantifiers are unchanged. In neither case is the relevant quantifier relativized to a condition formulated in terms of the variable representing the world of evaluation, so Premise 1 is where the assumptions of the necessity of what there is and what is necessary comes into the argument.

To state the second premise of the argument, say that an -theory is satisfiable on the intended interpretation if it is true on some interpretation of the non-logical constants, keeping the unrestricted interpretation of first- and second-order quantifiers. Say that such a such a theory is satisfiable on a set if it has a model in the standard set-theoretic semantics, according to which first-order quantifiers are interpreted as ranging over the members of a given set A, n-ary second-order quantifiers are interpreted as ranging over sets of n-tuples on A, and non-logical constants are interpreted accordingly.

• Premise 2: Every -theory satisfiable on the intended interpretation is satisfiable on a set.

Using these two premises, the argument runs as follows: By definition of , all members of are true, so by Premise 1, all members of the -theory are true, interpreting U as being a world and as the actual world. Thus is satisfiable on the intended interpretation, and so by Premise 2 satisfiable on a set. In particular, there is a model and assignment function a such that D is some cardinal , e(U) is a cardinal , , and each member of is true relative to and a. To establish , it now suffices to prove the following claim, which is routine: for any -sentence , iff is valid on .

It should be noted that Williamson's argument only applies to . To turn it into an argument for the claim that for some cardinals and , one could strengthen Premise 2 to apply to sentences of an extension of which allows for suitably infinitary conjunctions. Although the following focuses on Williamson's original argument concerning , much of it can be adapted to the case of .

3.2. Premise 2: Kreisel's principle

A version of Premise 2 is discussed in Kreisel (Reference Kreisel and Lakatos1967), and investigated formally in Shapiro (Reference Shapiro1987), who calls it Kreisel's principle. Shapiro formalizes Kreisel's principle in the context of second-order . Williamson notes that Shapiro proves it to have a standard model satisfying second-order , assuming that there are inaccessible cardinals. However, such a semantic consistency proof does not establish the truth of the premise. Williamson also notes that Shapiro derives in second-order that Kreisel's principle is equivalent to a certain reflection principle in second-order set theory. Such reflection principles have played an important role in modern set theory, and it has been suggested that adopting them can be justified on the basis of the iterative conception of set. However, some variants of such reflection principles have turned out to be inconsistent, and consequently, the status of the (as far as we know) consistent reflection principles is disputed, see, e.g. Koellner (Reference Koellner2009) for a recent overview.

Furthermore, it is not even clear that Shapiro's results apply to Premise 2 as used in Williamson's argument, since Shapiro (Reference Shapiro1987, 311) interprets all quantifiers to be restricted to pure sets, whereas Williamson appeals to an unrestricted reading of quantifiers. It is conceivable that Kreisel's principle holds on pure sets without holding in general, i.e. that every second-order theory satisfiable on the intended interpretation restricted to pure sets is satisfiable on a pure set even though there is some second-order theory satisfiable on the unrestricted intended interpretation which is not satisfiable on any pure (or even impure) set. Further assumptions of (impure) set theory are needed to show that this is not the case. Alternatively, one might directly endorse the relevant unrestricted version of the Kreisel Principle, which might turn out to be equivalent to a suitable reflection principle in impure set theory. Although reflection principles are usually discussed in the context of pure set theory, Burgess (Reference Burgess2004) introduces an impure set theory containing such a principle as one of its axioms.

3.3. Premise 1: worlds in higher order logic

Premise 1 captures the assumption that on a suitable understanding of possible worlds, necessity is equivalent to being true in all possible worlds. This won’t be questioned in the following. But Premise 1 also contains the assumption that possible worlds can be taken to be individuals, since replaces by a restricted universal first-order quantifier. Maybe they can be so understood according to the modal realism of Lewis (Reference Lewis1986), but Williamson (Reference Williamson2013, 17) explicitly rejects this view.

In line with a view which is perhaps more popular than modal realism (see, e.g. Stalnaker (Reference Stalnaker1976) and Fine (Reference Fine and Prior1977)), Williamson's treatment of possible world semantics in Williamson (Reference Williamson2013, Section 3.3) identifies possible worlds with possible propositions which are maximal in the sense of strictly entailing each proposition or its negation. Furthermore, in Williamson (Reference Williamson2013, Chapter 5), propositional quantification is treated as nullary second-order quantification, i.e. quantification into sentence position. It would therefore be natural for Williamson to treat quantification over worlds as restricted quantification into sentence position. He does not explicitly adopt this position in Williamson (Reference Williamson2013, Section 5.7), where he gives a homophonic interpretation of higher order modal logic. Rather, he leaves the understanding of worlds open by introducing a primitive type for worlds.

In the following, an argument will be sketched which shows that no matter how worlds are understood, Premise 1 can be adapted without loss of plausibility. For generality, will be extended by a type w for worlds, the interpretation of which will be left open. E.g. quantifiers binding variables of type w might be interpreted as propositional quantifiers restricted to propositions which are maximal in the sense described above. To define the syntax of this extended language , let the type of a relational symbol be given by a finite sequence of es and ws, indicating that the symbol combines with a corresponding string of individual and world variables to form an atomic predication. E.g. is the type of a relation symbol R which can be combined with two first-order variables x and y and a world variable to form an atomic predication . Based on a countably infinite supply of individual variables, world variables, as well as variables and constants of each relational type, which can be combined in the way just illustrated to form atomic predications, recursively define the formulas of using Boolean operations as well as quantifiers binding first-order, world, and relational variables. However worlds are understood, worlds and are taken to be identical if they are indistinguishable in the sense that , where X is a variable of type .

To adapt Premise 1 to , let be a version of the standard translation which differs from in using a world variable in place of the first-order variable , and translating using an unrestricted universal world-quantifier binding :

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Of course, each n-ary relation symbol must now also be translated using an -ary relation symbol of type . The required premise is then:

• Premise 1a: For any -sentence , is true iff is true, interpreting as the actual world.

Grant the truth of this premise for the sake of the argument – how to substantiate it will depend on the understanding of worlds. is in general a sentence of rather than , so the original argument using Premise 2 (Kreisel's principle) must be modified. The new argument distinguishes two cases: there being at least as many individuals as worlds, and there being more worlds than individuals.

Since set theory need not be applicable to worlds, how should this talk of ‘at least as many’ and ‘more’ be understood? There being at least as many individuals as worlds should be understood as there being an injective functional relation between worlds and individuals, and there being more words than individuals should be understood as there being no such relation. Here, quantification over relations between worlds and individuals should be understood as quantification binding variables of type , i.e. variables which combine with an expression of type w (denoting a world) and an expression of type e (denoting an individual) to form a sentence. For readability, the argument is written in English using phrases such as ‘relation between worlds and individuals, ’ rather than a formal higher order language, in which it would be expressed more properly.

• Case 1: There are at least as many individuals as worlds; i.e. there is an injection i from worlds to individuals. For this case, the argument relies on the following premise:

• Premise 1b: If there is an injection i from worlds to individuals, then for any -sentence , is true, interpreting as the actual world, iff is true, interpreting U as the image of i and as the image of the actual world under i.

This premise is motivated by the kind of unrestricted comprehension principle endorsed in Williamson (Reference Williamson2013, 227): with it, any sequence of relations witnessing the falsity of can be turned, via i into a counterexample of , and vice versa. Premises 1a and 1b entail:

• For any -sentence , is true iff is true, interpreting U as the image of i and as the image of the actual world under i.

This claim can take the place of Premise 1 in the original argument.

• Case 2: There are more worlds than individuals. (The following would also apply to the case in which there are as many worlds as individuals, but this is already covered by Case 1.) In this case, it will be shown that , first showing and then .

Assume first that . Then by Theorem 23, is derivable in . With Premise 1c it follows that , as required:

• Premise 1c: For any -sentence , if is derivable in , then is true.

A natural way of substantiating this premise is by an inductive argument over length of derivation in (a suitably modified version of) along the lines of the argument in Williamson (Reference Williamson2013, Section 3.3) for the propositional case.

For the other direction, it is helpful to establish an auxiliary claim for any given -sentence . Assume that is true, interpreting U as being an individual (and as some particular individual – it does not matter which). On this interpretation, is true, where (interpreting as identity). By Premise 2, is therefore satisfiable on a cardinal , interpreting U as a cardinal . Thus is valid on , but is not. Since is not valid on , , and therefore .

Contraposing this line of reasoning, if , then is false, interpreting U as being an individual and as some individual. With Premise 1d it follows that is false, interpreting as the actual world, and so, via Premise 1a, that , as required:

• Premise 1d: If there are more worlds than individuals, then for any -sentence , if is false, interpreting U as being an individual and as some individual, then is false, interpreting as the actual world.

This premise can be motivated by adapting the proof idea of Lemma 2: Since there are more worlds than individuals, there is a surjection s from worlds to individuals, mapping the actual world to the individual used to interpret . As in the argument for Premise 1b, given a sequence of relations which witnesses the falsity of , the unrestricted comprehension principle allows us to construct, via s, a sequence of relations witnessing the falsity of , interpreting as the actual world.