Every Countable Model of Arithmetic or Set Theory has a Pointwise-Definable End Extension

Main Theorem 1.

1. Every countable model of PA has a pointwise-definable end extension satisfying PA.

2. Indeed, for any particular m, there is a Σ _m -elementary such pointwise-definable end extension.

Main Theorem 2.

1. Every countable model of ZF has a pointwise-definable end extension satisfying ZFC + V = L.

2. More generally, every countable model of ZF with an inner model of a c.e. theory ZFC ‾ that includes ZFC + V = HOD has a pointwise-definable end extension satisfying ZFC ‾ .

3. Indeed, for any particular m, every countable model of ZFC + V = HOD admits a pointwise-definable Σ _m -elementary end extension. When m ≥ 1, these are top extensions.

Theorem 3.

(Blanck and Enayat 2017; Hamkins 2017, 2018; Woodin 2011). For any consistent c.e. theory PA⁺ extending PA, there is a Turing machine program e for computably enumerating a sequence with the following properties.

1. PA proves that the sequence enumerated by e is finite. And it is empty when computed in the standard model N .

2. For any model M ⊧ PA⁺, if s is the sequence enumerated by e in M and t is any finite extension of s in M, then there is an end extension of M to a model N ⊧ PA⁺ in which e enumerates exactly t.

Theorem 4.

For any consistent c.e. theory PA⁺ extending PA and any standard-finite natural number m, there is an oracle Turing machine program e ̃ with the following properties.

1. PA proves that the sequence enumerated by e ̃ , with any oracle, is finite. And it is empty when computed in the standard model N with oracle 0⁽ⁿ⁾.

2. For any model M of PA⁺ in which e ̃ , using oracle 0^(m) of M, enumerates a finite sequence s, possibly nonstandard, and any t ∈ M extending s, there is a Σ _m -elementary end extension of M to a model N ⊧ PA⁺ in which e ̃ , with oracle 0^(m) of N, enumerates exactly t.

Theorem 5.

(Main Theorem 1). Every countable model of PA has a pointwise-definable end extension satisfying PA. Indeed, for any particular m, there is a Σ _m -elementary such pointwise-definable end extension.

Proof.

Fix any countable model M₀ of PA. By the universal algorithm, we can find a countable end extension model M₁ where the last entry of the universal algorithm is any desired object a₀ of M₀. Next, by Theorem 4 we may find a countable Σ₁-elementary end extension M₂, such that the last element on the universal Σ₂-sequence is any desired element a₁ of M₁. Because the extension is Σ₁-elementary, the operation of the universal algorithm is frozen – no new numbers will appear on the sequence – and so a₀ is still last on that sequence. And so on. At each stage n, we find by Theorem 4 a countable Σ _n -elementary end extension M_n+1 of M _n , such that any desired a _n from M _n is last on the Σ_n+1-definable universal sequence, and this will be preserved to the later models. The result is a progressively elementary tower of models:

Because the tower is progressively elementary, each model M _n will be Σ _n elementary in the limit model M* = ⋃ _n M _n , which will therefore be a model of PA. Every a _n is definable in M* as the last element on the Σ_n+1-definable universal sequence, since this is true in M_n+1 and was preserved to all the later extensions and the limit. Therefore, by undertaking a simple bookkeeping method we can arrange that the elements a _n exhaust M*. We should simply arrange that every element appearing in any of the models M _k is eventually made definable as one of the objects a _n . In this way, M* becomes pointwise definable, as desired.

To achieve a Σ _m -elementary end extension, we need simply start right in with the Σ_m+1-definable sequence at stage 0, and then proceed to Σ_m+2, and so on.

Every stage in this tower will be at least Σ _m -elementary. □

Theorem 6.

(Hamkins and Woodin 2017). There is a Σ₂ definable finite sequence of sets

1. ZFC proves that the sequence is finite, and it is empty in any transitive model of ZFC.

2. If M is a countable model of ZFC in which the sequence is s and t is any finite extension of s in M, then there is a top extension of M to a model N⊧ZFC in which φ defines exactly t.

Theorem 7.

(Hamkins and Williams 2022). For any c.e. theory ZF ‾ extending ZF, there is a Σ₁-definable finite sequence

1. ZF proves that the sequence is finite, and it is empty in any transitive model of ZF ‾ .

2. If M is a countable model of ZF with an inner model W ⊧ ZF ‾ and the sequence defined in M is s, then for any finite extension t in M, there is a covering end extension of M to a model N ⊧ ZF ‾ in which the sequence is exactly t.

Theorem 8.

For any c.e. theory ZFC ‾ extending ZFC and any natural number m, there is a Σ_m+1 definable finite sequence of ordinals

1. ZF proves that the sequence is finite, and it is empty in any transitive model of ZFC ‾ .

2. If the sequence is s in a countable model M ⊧ ZFC ‾ , then for every finite sequence t extending s in M there is a Σ _m elementary end extension M ≺ Σ m N ⊧ ZFC ‾ such that the sequence is t in the extension N. (When m ≥ 1, this is therefore a top extension.)

Proof.

It will suffice to verify the adding-one extension property, where t = s^⌢α extends the sequence s by adding just one ordinal. The reason is that if there is a definable sequence with the adding-one extension property, then by interpreting those ordinals as encoding finite sequences to be concatenated, we can derive a new definable sequence with the universal extension property as stated.

As in both (Hamkins and Williams 2022; Hamkins and Woodin 2017), I shall describe two set-theoretic processes, A and B. These processes are intended to be run inside, respectively, ω-nonstandard models and ω-standard models and will then be merged in a way that provides a single definition fulfilling the desired properties. We may assume m ≥ 1, since the Σ₁ definable universal finite sequence of Hamkins and Williams (2022) already fulfils the case m = 0, as every end extension is Σ₀-elementary. Fix a specific enumeration of the ZFC ‾ theory, and let ZFC ‾ k refer to the theory arising from the first k enumerated axioms.

Process A . Let me start by describing process A, intended to be run inside ω-nonstandard models as an internal process, using whatever nonstandard natural numbers are to be found there. The process proceeds in a sequence of stages, placing one object onto the sequence at each successful stage. Stage n succeeds and α _n is defined, if there is a Σ _m -correct cardinal δ _n , larger than α _n and all earlier δ _i , and a natural number k _n , smaller than all earlier k _i , such that the structure V δ n , ∈ has no covering Σ _m -elementary end extension to a model N , ∈ N satisfying ZFC ‾ k n , inside which process A places this very ordinal α _n onto the sequence at stage n as the next and last element. Note that the earlier stages of the sequence are absolute to V δ n since it is correct about Σ _m -correctness and about whether a given V δ i has a Σ _m -elementary covering end extension to a model of a certain theory. In slogan form: we place an ordinal onto the sequence, if we find a Σ _m correct rank initial segment of the universe having no covering Σ _m -elementary end extension in which we would have done so as the next and last element. The process officially accepts and uses the triple (δ _n , α _n , k _n ) occurring lexically least. In other words, we first minimize δ _n , such that for some α _n < δ _n there is such a k _n for which the requirement is fulfilled.

The map n ↦ α _n , I claim, is Σ_m+1 definable. To see this, note first that the property “V _δ is Σ _m correct” has complexity Π _m , since it is equivalent to the assertion (using the universal Π _m -truth predicate) that every Π _m assertion true in V _δ is actually true. The relation y = V _δ is Π₁ in y and δ, and to say that V _δ has no end extension of the required form is a further Π₁ assertion. Thus, to verify α _n , we ask for the existence of a cardinal δ, an ordinal α, a natural number k, and a set y such that y = V _δ and y is Σ _m correct, such that y has no end extension to a structure satisfying a certain theory, and all lexically smaller (δ′, α′, k′) do give rise to such extensions (and indeed, they do so inside V _δ ). This is altogether an existential followed by a Π _m assertion, and hence has complexity Σ_m+1.

Although the definition may at first appear circular – we define α _n , after all, by reference to the definition of the process A sequence inside N – one may use the Gödel–Carnap fixed point lemma to find a definition ψ(n, α) for the map n ↦ α _n that solves this recursion, allowing ψ as described to refer to its own Gödel code ⌜ψ⌝. The same method is used in Hamkins and Woodin (2017), Hamkins and Williams (2022), and one should view this as analogous to the common use of the Kleene recursion theorem in computability-theoretic arguments, including its use in the universal algorithm (Hamkins 2018; Woodin 2011).

Since the natural numbers k _n are descending, there will be only finitely many successful stages, and so the sequence will be finite.

I claim that in the relevant models, every k _n arising from a successful stage is nonstandard; in particular, ω-standard models (those having no nonstandard natural numbers) will have no successful process A stages. To see this, consider any countable model of set theory M ⊧ ZFC ‾ , and let n be the last successful stage in M. For any standard k, the theory ZFC ‾ k reflects to some Σ_m+1 correct cardinal δ in M above δ _n . Consequently, ⟨ V δ M , ∈ W ⟩ ⊧ ZFC ‾ k and thinks object α _n was added at stage n as the last object on the sequence, since this is true in M. Furthermore, V δ M is a Σ _m -elementary end extension of V δ n M , as both are Σ _m -correct in M. If k _n ≤ k, this would violate the success of stage n in M, since V δ n would have had such a topped end extension after all. Therefore k < k _n for every standard k, and so k _n is nonstandard. In particular, in any transitive model of ZFC ‾ , the sequence will have no successful stages.

It remains to verify the extension property. Let M be an ω-nonstandard countable model of ZFC ‾ in which the sequence is s. Let n be the first unsuccessful stage. Let k be nonstandard, but smaller than all previous k _i . Since stage n did not succeed, it follows that for any ordinal α and any Σ _m correct cardinal δ above α and all earlier δ _i , the structure ⟨ V δ , ∈ ⟩ M does have a Σ _m -elementary topped end extension in M to a model satisfying ZFC ‾ k + “ordinal α was placed onto the sequence at stage n, the last successful stage,” since otherwise stage n would have succeeded. By the Keisler-Morley theorem, there is an elementary end extension M ≺ M⁺, and so n is also the first unsuccessful stage in M⁺ and the two models agree on the outcome of the earlier stages. Let δ be a Σ _m correct cardinal of M⁺ above M. Thus, by what I’ve just explained, for every ordinal α < δ, in particular every ordinal α in M, the structure ⟨ V δ , ∈ ⟩ M + has a Σ _m elementary covering end extension ⟨ N , ∈ N ⟩ ⊧ ZFC ‾ k in which the A sequence adds the ordinal α at stage n as the last successful stage. Since δ is Σ _m correct in M⁺, it is a Σ _m -elementary end extension of M, and so N is a Σ _m -elementary end extension of M realizing α as the next and last element on the sequence, as desired.

Process B . Let me turn now to process B, intended to be run as an internal process inside ω-standard models, using whatever (possibly nonstandard) ordinals are to be found there. The process proceeds in a sequence of stages, just as before, with each successful stage adding one ordinal α _n to the sequence. Stage n is successful and α _n is defined, if there is a Σ _m correct cardinal δ _n above α _n and all earlier δ _i and a countable ordinal λ _n , smaller than all earlier λ _i , such that the structure ⟨ V δ n , ∈ ⟩ M has no covering Σ _m -elementary end extension to a model ⟨N, ∈ ^N ⟩ satisfying ZFC ‾ + “process B places ordinal α _n on the sequence at stage n, the last successful stage,” and furthermore, it has no such end extension even in the forcing extension of M collapsing V δ n to become countable, and finally, the assertion about the real coding V δ n and α _n that there is no such end extension in that forcing extension, which is a Π 1 1 statement, has rank λ _n in the canonical representation of Π 1 1 -assertions by well-founded trees.

Once again, the circularity in the definition can be removed by the Gödel–Carnap fixed-point lemma. Since the ordinals λ _n are descending, there will be only finitely many successful stages, and so the sequence is finite.

And once again, the map n ↦ α _n is Σ_m+1 definable, since one asks for a cardinal δ _n and ordinals α _n and λ _n , and a set y, such that y = V _δ is Σ _m correct, and furthermore such that it is forced that this set has no end extension in the collapse forcing to a model of a certain theory, and further the rank of that Π 1 1 assertion is λ _n . Except for the Σ _m correctness, which has complexity Π _m , the rest of this can be verified inside any sufficiently large rank-initial segment of the universe and hence has complexity Σ₂, making the complexity Σ_m+1 altogether.

I claim that the ordinals λ _n in the relevant models are nonstandard. Suppose that M is a countable model of ZFC ‾ and let n be the last successful stage in M. This stage was successful because there was a Σ _m correct cardinal δ _n and ordinal α _n , such that in the forcing extension collapsing V δ n the structure ⟨ V δ n , ∈ ⟩ M had no Σ _m -elementary covering end extension to a model ⟨N, ∈ ^N ⟩ in which the B sequence added α _n as the next and final stage, and such that the rank of this assertion is λ _n . But since M itself is such a covering Σ _m elementary end extension of V δ n M in which α _n was added as the last element of the sequence, the statement that there is no such end extension is not actually true. Thus, the Π 1 1 assertion in the forcing extension of M making V δ n countable is not actually true, even though this model thought it was true. Since this is an ω-standard model, the tree it builds to represent this Π 1 1 statement is the same as we would build external to M, and so the tree is not actually well-founded, although it was found to have rank λ _n inside the model. So λ _n must be nonstandard. Consequently, the sequence is empty in any transitive model of ZFC ‾ .

Let me finally prove the extension property for process B in countable ω-standard models M ⊧ ZFC ‾ . Suppose the sequence is s in M. Let n be the first unsuccessful stage, and consider any ordinal α in M. By the Keisler–Morley theorem, there is a countable elementary end extension of M to a model M⁺. Let δ be a Σ _m correct cardinal in M⁺ above M, and let M⁺ [G] be a forcing extension in which V δ M + is made countable. If ⟨ V δ , ∈ ⟩ M + has a Σ _m -elementary covering end extension to a model of ZFC ‾ in which ordinal α is placed onto the sequence at stage n as the last successful stage, then we’d be done, since this would also serve as the desired end extension of the original model M. So let me assume toward contradiction that there is no such end extension of ⟨ V δ , ∈ ⟩ M + . This is a Π 1 1 -assertion about the generic real coding V δ M + and α in M⁺ [G], and this Π 1 1 statement therefore has some ordinal rank λ there in the representation of Π 1 1 -assertions by well-founded trees. Since the statement is really true (in the set-theoretic background of our universe), it follows that λ must be in the well-founded part of M⁺ [G]. In particular, λ < λ _i all i < n, since those ordinals are nonstandard. But this implies that stage n would have been successful in M⁺, contrary to assumption. So V δ M + must have had the desired end extension after all, and so I’ve verified the extension property of process B for the relevant countable ω-standard models of set theory.

Process C. I shall now merge processes A and B into a single process C, providing a single uniform Σ_m+1-definition that will work in all the relevant countable models of set theory. Process C proceeds in stages. At each successful stage, it will place one new object onto the sequence, either for an A-reason or for a B-reason. The A-reasons will involve data (δ _n , α _n , k _n ) as in process A, and the B-reasons will involve data (δ _n , α _n , λ _n ) as in process B, and in each case we shall insist that δ _n is a Σ _m correct cardinal above all the other data and the earlier δ _i and furthermore that k _n is below all earlier A-reason stage k _i , if this is an A-reason stage, and λ _n is below all earlier B-reason stage λ _i , if this is a B-reason stage. The A reason stages are successful, if ⟨ V δ n , ∈ ⟩ has no covering Σ _m -elementary end extension to a model ⟨ N , ∈ N ⟩ ⊧ ZFC ‾ k n in which process C adds α _n as the next and last element; the B reason stages are successful, if ⟨ V δ n , ∈ ⟩ has no covering Σ _m -elementary end extension to a model ⟨ N , ∈ N ⟩ ⊧ ZFC ‾ in which process C places α _n as the next and last element on the sequence, and furthermore there is no such extension even in the forcing extension making V δ n countable and the rank of this Π 1 1 assertion is λ _n . Once there has been a successful A-reason stage, then we proceed further only with A-reason stages.

To complete the argument, we observe that the analysis of processes A and B goes through for process C. The map n ↦ α _n is Σ_m+1 definable for similar reasons as in processes A and B. The sequence is finite since the k _n and λ _n can go down only finitely many times. At any A-reason stage, the number k _n must be nonstandard, and at any B-reason stage, the ordinal λ _n must be nonstandard, and so the sequence is empty in any transitive model of ZFC ‾ . In any ω-nonstandard model, we achieve the extension property for process C for the same reasons as process A; and in any ω-standard model, we achieve the extension property for process C just as with process B. □

Theorem 9.

For any c.e. theory ZFC ‾ extending ZFC + V = HOD and any natural number m, there is a Σ_m+1 definable finite sequence of sets

1. ZF proves that the sequence is finite, and it is empty in any transitive model of ZFC ‾ .

2. If the sequence is s in a countable model M ⊧ ZFC ‾ , then for every finite sequence t extending s in M there is a Σ _m elementary end extension M ≺ Σ m N ⊧ ZFC ‾ such that the sequence is t in the extension N.

Proof.

If V = HOD is part of the theory ZFC ‾ , then this theorem is a consequence of Theorem 8, since from the universal finite sequence of ordinals, we can use the definable well-ordering of the universe to pick out for each ordinal α the αth set in that definable order. The αth element of HOD is Δ₂ definable from α, and so from the definable ordinal sequence we extract the sequence of definable sets. And since we have the universal extension property for Σ _m -elementary end extensions for the ordinal sequence, we can thereby put any desired set object as the next element on the derived sequence, thereby establishing this theorem. □

Theorem 10.

(Main Theorem 2). Every countable model of ZF has a pointwise-definable end extension satisfying ZFC + V = L.

Theorem 11.

Every countable model of ZF with an inner model of a c.e. theory ZFC ‾ that includes V = HOD has a pointwise-definable end extension satisfying ZFC ‾ .

Theorem 12.

Every countable model of ZF with an inner model of a c.e. theory ZFC ‾ has an end extension to a model of ZFC ‾ that is a Paris model.

Theorem 13.

Let m be any particular finite natural number.

1. Every countable model M of a c.e. theory ZFC ‾ extending ZFC has a Σ _m elementary end extension

   M ≺ Σ m N

   to a Paris model N ⊧ ZFC ‾ , that is, one in which every ordinal is definable without parameters.

2. Consequently, every countable model M of a c.e. theory ZFC ‾ extending ZFC + V = HOD has a Σ _m -elementary end extension M ≺ Σ m N to a pointwise-definable model N ⊧ ZFC ‾ .

Proof.

Fix any countable model M₀ of ZF. By Theorem 8, we can find a Σ _m -elementary end extension model M₁ where the last entry of the universal Σ_m+1-sequence is any desired ordinal α₀ of M₀. Continuing iteratively just as in Theorem 5, we apply Theorem 8 again to find a Σ_m+1-elementary countable end extension M₂, such the last element on the universal Σ_m+2-sequence is any desired ordinal α₁ of M₁. Note that the Σ_m+1-elementarity means that we have preserved the previous Σ_m+1 definable universal sequence, and so α₀ is still last on that sequence. And so on. At each stage n, we find by Theorem 8 a Σ_m+n elementary end extension M_n+1 of M _n , such that any desired ordinal α _n from M _n is last on the Σ_m+n+1-definable universal sequence, and this will be preserved to the later models. The result is an progressively elementary tower of countable models:

Because the tower is progressively elementary, each model M _n will be Σ_m+n elementary in the union model M* = ⋃ _n M _n , which therefore will be a model of the desired theory ZFC ‾ , as well as Σ _m -elementary extension of the original model M₀. Every ordinal α _n is definable in M* as the last element on the Σ_m+n+1-definable universal sequence, since this is true in M_n+1 and preserved by all the later extensions. Therefore, since these are all countable models, a simple bookkeeping method will suffice for us to arrange that the elements α _n exhaust the ordinals of M*, which will therefore be the desired target Paris model, fulfilling statement (1).

For statement (2), we simply observe that every Paris model of V = HOD is pointwise definable, since every object is definable with ordinal parameters, but the ordinals themselves are definable without parameters. □

Corollary 14.

If a theory ZFC ‾ supports universal definable finite sequence of arbitrary sets with the universal extension property with respect to Σ _m -elementary end extensions of countable models of ZFC ‾ , for every m, then it proves V = HOD.

Proof.

Suppose that for every m we had a definable universal finite sequence with the universal extension property for adding any particular set to the sequence in some Σ _m -elementary end extension of a given countable model of ZFC ‾ . Let M₀ be any countable model of ZFC ‾ . Using the method of Main Theorem 2 (Theorem 10), we can build a progressively elementary tower of models

where m₀ ≥ 2 and at each stage we impose a degree of elementarity Σ m n exceeding the complexity of the definitions of the previous sequences, and where furthermore by bookkeeping we arrange that the union model M is pointwise definable. Since the tower is progressively elementary, the limit model M is a model of ZFC ‾ . Since it is pointwise definable, it satisfies V = HOD. Since M 0 ≺ Σ 2 M , it follows that M₀ is also a model of V = HOD, since this statement has complexity Π₃. So every model of ZFC ‾ must be a model of V = HOD, and the corollary is proved. □

Theorem 15.

The modal validities true in the countable models of ZFC under Σ _m -elementary end extensions form exactly the modal theory S4.

Proof.

The modal theory S4 is valid in any potentialist system. For the converse, it suffices as in [2018, Theorem 30], [2017, Theorem 8], or [2021, Theorem 11], to provide a railyard labelings of the finite tree pre-orders. This can be undertaken with the Σ_m+1-definable universal finite sequence, just as in those arguments with the analogous universal sequences. Namely, one interprets the objects on the universal sequence as instructions for climbing the finite tree preorder, and this will align Σ _m -elementary end-extensional possibility with accessibility in the tree preorder. It follows that the potentialist validities will be contained within the validities of the finite tree preorder frames, and this is S4, as desired. □

Theorem 16.

No model of ZF has a maximal Σ_m+1-theory (allowing natural number parameters), even with respect to fixing the Σ _m diagram of the model. That is, new Σ_m+1 facts can become true about natural numbers in a Σ _m -elementary end extension.

Proof.

This is immediate from the Σ_m+1-definable universal extension property, since the universal sequence can be extended to have one additional successful stage, which is a new Σ_m+1 fact that becomes true in a Σ _m -elementary end extension. □