On representations of intended structures in foundational theories Neil Barton∗, Moritz Müller†, Mihai Prunescu‡ 8 June 2020§ Abstract Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power and meta-theoretic properties when comparing first-order and second-order logic. Introduction This paper addresses the philosophical question of how well foundational mathematical theories are able to represent mathematical structures. Much of mathematical practice concerns the study of particular structures. Famous examples are the arithmetical structure of ∗Fachbereich Philosophie, University of Konstanz. E-mail: neil.barton@unikonstanz.de. †Universitat Politècnica de Catalunya. E-mail: moritz@cs.upc.edu. ‡University of Bucharest, Faculty of Mathematics and Informatics; and Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 5, P. O. Box 1-764, RO-014700 Bucharest, Romania. E-mail: mihai.prunescu@imar.ro, mihai.prunescu@gmail.com §Order of authors is alphabetical. The first author is grateful to the VolkswagenStiftung for their support through the project Forcing: Conceptual Change in the Foundations of Mathematics. The second author is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant agreement ERC-2014-CoG 648276 (AUTAR). 1 the natural numbers (N, +,×, 0, 1,<), the ordered field of the reals (R, +,×, 0, 1,<), or the field of complex numbers (C, +,×, 0, 1). The study of these structures is conducted mainly informally, such as the manner of reasoning we see in mathematical journals. This fact concerning mathematical practice is coupled with the existence of foundational theories. There are various such theories, first and foremost set theory ZFC, but also category theory, and more recently homotopy type theory. There are many features we might want a foundational theory to have, but one is to provide a generous arena in which the wide variety of mathematical objects can be studied.1 Roughly, this means that the foundational theory can encode or formalise all our informal mathematical discourse about the 'usual' objects of mathematics. In this way, if one had sufficient patience and time, once could formalise all theorems of informal mathematics as theorems within one's favourite foundational theory. The starting question of this short paper is: What is the desired relationship between informal and formalised mathematics? Being a very general question we restrict attention to the informal study of concrete structures, like the natural, real or complex numbers mentioned above; in a philosophical context these are often referred to as intended structures. Now, it is one thing to be able to formalise some piece of informal mathematics any-old-how, and quite another to do so faithfully. We would like the intuitive meaning of the formal statements to be similar to the intuitive meaning of the informal statements. For motivational purposes let us roughly distinguish two approaches to the foundations of mathematics: the axiomatic and the genetic method (see [Rav, 2008]). The first, chiefly embodied by Hilbert, replaces the intended structure by a set of axioms we argue are (or take to be) true. When done in first-order logic this approach is often incomplete (by Gödel's results). When done in higher-order logics, we lose various pleasant meta-theoretic properties, and so whilst of philosophical interest it has less practical value. The genetic approach is via construction. Instead of asserting axioms for the intended structure, one first constructs the structure in question by finding an object coding it in one's foundational theory, and then one asks about its properties. Theories with a high degree of interpretive power are able to translate some mathematical constructions into first-order definitions inside the theory. For example, ZFC can mimic the classical constructions of the natural, real or complex 1We refer to Maddy [Maddy, 2019] for an assessment of what we want a foundational theory to do for us. The term 'generous arena' is her term. 2 numbers by defining formulas. In a little more detail (we provide a full outline in §2) formalisation of the mathematical study of an intended structure S typically proceeds in two steps. First, S is represented by a formal object R of the foundational theory (in the case of ZFC, a set or a class). Second, the informal talk about S is translated to formal talk about R. Thus, in every model of ZFC we will find an avatar of the natural, real or complex numbers. Here, we are concerned with the first step; the choice ofR.2 We will focus mainly on this style of doing mathematics: by first constructing the structures and then examining their properties, and especially their first-order theory. We also restrict attention to first-order theories F to be our foundational theories (such as ZFC). Our proposal is to analyse one dimension of the faithfulness or similarity of meaning of a formalisation as dependent upon the similarity of S and what we define by R. We thus arrive at a more specific formulation of our question: what kind of similarity of S and R should we aim for, or at least hope for? A notion of similarity of obvious interest in this context is elementary equivalence. Our main claim then reads as follows (see Theorem 14 for a precise statement): Main Claim. Let F be suitable first-order foundational theory. Given a particular analysis of faithfulness in terms of elementary equivalence, an intended structure can be faithfully represented in F if and only if its (first-order) theory is decidable and F knows some decision algorithm for it. For our example structures, this implies that (R, +,×, 0, 1,<) and (C, +,×, 0, 1) are faithfully representable, but (N, +,×, 0, 1,<) is not. On the positive side this shows that foundational theories have an especially good grip on decidable parts of informal mathematics. Our main interest in the claim is, however, on the negative side. Many intended structures have undecidable theories, and so their study cannot be faithfully formalised in our sense. Moreover, the underlying assessment of faithfulness via elementary equivalence seems to be a fairly modest requirement on the representation of a structure, philosophically speaking. Outline: In (§1) we recall some basics about translations between first-order theories; in particular formalising the study of some intended structure in a foundational theory. We then motivate one way of understanding the idea of a faithful such formalisation that we shall 2In particular, we are not concerned with the relationship between informal proofs and formal derivations. 3 call absolute representability; an intended structure is absolutely representable when its representatives in models of F are elementarily equivalent to it. In (§2) we establish our main claim by proving Theorem 14. We then (§3) outline an application of our results, specifically by comparing properties of first-order and higher-order resources in characterising structures. Finally (§4) we conclude and present some open questions. 1 Absolute representability In this section we set up some key notions and motivate the formal definition we shall use, namely absolute representability. Recall, a first-order language consists of a set of relation symbols and function symbols, each having an associated natural number called its arity; we view constant symbols as nullary function symbols. First, we fix a countable such language L and a first-order L -structure SL : this is our informal intended structure.3 We also fix a consistent computably enumerable first-order theory F: this is our foundational theory. We shall add another assumption on F later when needed. Examples for SL to keep in mind are (N, +,×, 0, 1,<), (R, +,×, 0, 1,<) or (C, +,×, 0, 1), the example to keep in mind for F is ZFC; we assume ZFC is consistent. We employ a standard definition (see e.g. [Ebbinghaus et al., 1994, Chapter VIII] 4) of how our intended structure SL is represented in F: Definition 1. A representation R of an L -structure in F is a sequence of formulas in the language of F, namely a formula ψU(x) such that F proves ∃xψU(x), and for every r-ary relation symbol S ∈ L a formula ψS(x1, . . . ,xr) and for every r-ary function symbol f ∈ L a formula ψf (x1, . . . ,xr, y) such that F proves: ∀x1 * * * ∀xr∀y∀y′(ψf (x1, . . . ,xr, y) ∧ ψf (x1, . . . ,xr, y′)→ y = y′) ∧ ∀x1 * * * ∀xr(ψU(x1) ∧ . . . ∧ ψU(xr)→ ∃y(ψU(y) ∧ ψf (x1, . . . ,xr, y))) Such a representation definably singles out an L -structure in every model of the foundational theory F as follows. 3Here, we sidestep the discussion as to whether informal mathematics is firstorder or higher-order and content ourselves with the claim that its first-order part is a substantial one. Note any restriction of attention can only strengthen the negative interpretation of our main claim. 4There are more general versions that allow ψU (x) to have a tuple of free variables and/or add a formula ψ= interpreting the equality relation; the choice is a matter of no consequence for us, our choice is for the sake of simplicity. 4 Definition 2. LetM be a model of F (with universe denoted M ). The L -structure R(M) has universe U := {a ∈M | M |= ψU(a)} and interprets an r-ary relation symbol S ∈ L by {(a1, . . . , ar) ∈ U r | M |= ψS(a1, . . . , ar)}, and an r-ary function symbol f ∈ L by the function with the graph {((a1, . . . , ar), b) ∈ U r × U | M |= ψf (a1, . . . , ar, b)}. ThatR(M) is a well-defined L -structure follows from the assumptions on what F proves about R in Definition 1, namely, the universe U is non-empty and ψf really defines the graph of some function on U . Example 3. The usual representation RZFCN of (N, +,×, 0, 1,<) in ZFC is given taking for ψU(x) the formula x ∈ ω (understood as a formula in the language {∈} of ZFC) that defines the finite von Neumann ordinals; the formula ψ<(x1,x2) is x1∈x2, the formulas ψ+(x1,x2, y),ψ×(x1,x2, y) state the recursive definitions of addition and multiplication, and the formulas ψ0(y) and ψ1(y) are y=∅ and y={∅}, respectively. The models RZFCN (M), for M |= ZFC, are called ZFC-standard models of arithmetic in [Hamkins and Yang, 2013]. We refer to this paper and the references therein for some information about these structures. Given a representation R of our intended structure SL in our foundational theory F, it is straightforward to translate first-order talk about SL into F. The following is folklore (cf. [Ebbinghaus et al., 1994]): Lemma 4. Let R be a representation of an L -structure in F. For every L sentence φ there is a sentence R(φ) in the language of F such that for all modelsM of F: R(M) |= φ⇐⇒M |= R(φ). (1) Moreover, the map φ 7→ R(φ) is computable. Proof. (Sketch) We recall the proof for relational L . For atoms define R(x=y) := x=y and R(S(x)) := φS(x) for S ∈ L a relation symbol. Then proceed recursively, R(¬φ) := ¬R(φ), R(φ ∧ ψ) := R(φ) ∧ R(ψ) and R(∀xφ) := ∀x(φU(x)→ R(φ)). Remark 5. The proof sketch defines R(¬φ) = ¬R(φ), a property of the map φ 7→ R(φ) that we are going to use. Slightly more generally 5 we could use only that F proves (R(¬φ) ↔ ¬R(φ)) for every φ. This follows from (1) alone: let M be a model of F; then M |= R(¬φ) if and only if R(M) 6|= φ by (1), if and only ifM |= ¬R(φ) by (1) again. Similarly, we have the equality R(φ ∧ ψ) = R(φ) ∧ R(ψ) by the proof sketch and F-provable equivalence by (1) alone. The properties of the structureR(M) can vary significantly according to the modelM of F. This provides a situation in whichR identifies very different structures according to the first-order model we live in. The following question is then salient: Question. How similar can we make our foundational representative to the intended structure? More precisely, for a given notion of similarity ∼, does there exist a representation R of L -structures in F such that SL ∼ R(M) for all modelsM of F? Here, by a notion of similarity we mean an equivalence relation on L -structures. The finer this equivalence relation, the stronger the corresponding notion of representability. Obviously, taking the identity for ∼ results in an empty concept: no structure is identically representable in F. Taking isomorphism for ∼ means asking whether our intended structure SL is isomorphically representable in F, i.e., whether there exists a representation R such that R(M) ∼= SL for allM |= F. This suggestion for ∼ is naive because it is a quick consequence of the Compactness Theorem that:5 Proposition 6. Only finite L -structures are isomorphically representable in F. Hence isomorphic representability is a far too strong notion (at least as far as first-order logic is concerned). Philosophers and logicians often analyse a spectrum of similarity notions far coarser than isomorphism.6 We examine the prospects of choosing elementary equivalence: recall, two L -structures A,B are elementarily equivalent if they satisfy the same first-order L -sentences, i.e., Th(A) = Th(B), or equivalently, Th(A) ⊆ Th(B). Here, Th(A) denotes the first-order theory of A, i.e., the set of first-order sentences true in A. The corresponding notion of representability reads as follows: 5This has been noted across the structuralist literature, but is pressed particularly strongly throughout Stewart Shapiro's seminal defence of second-order logic in [Shapiro, 1991]. 6For example, partial isomorphisms, partial isomorphism calibrated by ordinals, elementary equivalence for various logics and fragments thereof, bisimilarity, homomorphic equivalence. . . the list is long. What is the right similarity notion depends on the topic under consideration. 6 Definition 7. SL is absolutely representable in F if there exists a representation R of an L -structure in F such that Th(SL ) = Th(R(M)) for all modelsM of F; in this case, we say R absolutely represents SL in F. Example 8. RZFCN from Example 3 does not absolutely represent (N, +,×, 0, 1,<) in ZFC. Indeed, let Con(ZFC) be an arithmetical sentence expressing the consistency of ZFC. Then Con(ZFC) is true in (N, +,×, 0, 1,<) but fails in some ZFC-standard models of arithmetic by Gödel's Second Incompleteness Theorem. Given the naturality of elementary equivalence, the question which structures are absolutely representable in F might deserve our mathematical curiosity. The above example hints at serious limitations, and we shall exactly delineate them in the next section where we establish our main claim from the Introduction. For now, we mention two reasons to find the notion philosophically interesting. First: Clearly one goal of the informal mathematical investigation of the intended structure SL is to find out what is true in SL , and firstorder truth is undoubtedly an important part of it. It thus seems that an absolute representation is a clear desideratum for the foundational theory. It states that first-order truth in the intended structure does not vary with different assumptions on the model of the foundational theory we are living in. Second: Absolute representation ensures a certain level of stability in the informal mathematical investigation of SL with respect to changes in the foundational theory. Thereby it provides comfort to the working mathematician who is not willing to restrict his or her investigations to F alone. For example, if we (consistently) expand F by adding more axioms, absolute representability of a structure in SL ensures that we do not change F's beliefs about what holds in SL by doing so.7 2 Absolute representability and decidability In this section we establish our main claim from the Introduction. We need the following lemma: 7It is important that we compare first-order truth of the intended structure SL and its formal counterparts R(M) in the informal meta-language. An analogous notion inside F = ZFC would state that Th(R(M)) as defined inM (which, assuming the universe of R(M) is a set, can be done) does not vary with M. This now includes non-standard sentences and ceases to be a property of R: this theory can vary withM even when keeping R(M) fixed; we refer to [Hamkins and Yang, 2013] for precise statements. 7 Lemma 9. Let R be a representation of an L -structure in F. Then R absolutely represents SL in F if and only if Th(SL ) = { φ | F ` R(φ) } . (2) Proof. Assume R absolutely represents SL in F. To show ⊆ in (2), let φ ∈ Th(SL ). We have to show that F proves R(φ): letM be a model of F, so φ ∈ Th(R(M)) = Th(SL ) by absolute representation, that is, R(M) |= φ, so M |= R(φ) by Lemma 4. To show ⊇ in (2) let φ /∈ Th(SL ). Then ¬φ ∈ Th(SL ), so F proves R(¬φ) = ¬R(φ) by the inclusion just proved. Hence F 6` R(φ) because F is consistent. Conversely, assume (2) and let M |= F. We have to show that R(M) |= Th(SL ). But, by Lemma 4, R(M) models the right-hand-side of (2). Proposition 10. If SL is absolutely representable in F, then Th(SL ) is decidable. Proof. Given as input an L -sentence φ compute R(φ) and ¬R(φ) = R(¬φ) and enumerate all consequences of F (which we assumed to be computably enumerable). By Lemma 9, exactly one of R(φ) and ¬R(φ) is eventually enumerated, and we accept or reject our input accordingly. In Example 8 we saw that a particular representation RZFCN is not an absolute representation. We can now say more: Example 11. The structures (N, +,×, 0, 1,<) and (N, +,×), being undecidable (see [Tarski, 1968]), are not absolutely representable in F. We now prove a partial converse to the above under an additional assumption on F: Assume there is a representation RFN of (N, +,×, 0, 1,<) in F such that F proves RFN(Q) where Q is the conjunction of the finitely many axioms of Robinson arithmetic. For any F worth calling a foundational theory, this surely is less than a minimal requirement (and clearly met by ZFC). Given this assumption, we note the following direct consequence of Lemma 4. Lemma 12. Let φ be an arithmetical sentence such that Q proves φ. Then F proves RFN(φ). Proof. LetM |= F. ThenM |= RFN(Q), so RFN(M) |= Q by Lemma 4, so RFN(M) |= φ as Q ` φ, soM |= RFN(φ) by Lemma 4. 8 Let A be an algorithm (i.e. a Turing machine) that halts on every input. For natural numbers n,m ∈ N, we write A(n) = m to express that A on input n halts with output m. It is well-known that there is an arithmetical formula "A(x)=y" with free variables x, y such that for all n,m ∈ N A(n) = m =⇒ Q ` "A(ṅ)=ṁ", A(n) 6= m =⇒ Q ` ¬"A(ṅ)=ṁ", (3) where ṅ denotes a canonical term for n (say, 0 := 0, 1 := 1, 2 := 1 + 1, 3 := 2 + 1, . . .). We now present our notion of what it means for F to "know" an algorithm deciding Th(Sφ). Let pφq denote the Gödel number of an L -sentence φ. Definition 13. Let R be a representation of an L -structure in F and let A be an algorithm. F pointwise verifies A with respect to R if for every L -sentence φ: F ` ( R(φ)↔ RFN ( "A(  pφq)=1" )) . (4) We should remark here that this definition is rather weak among various reasonable notions for what it means that F "knows" some algorithm. Note that the algorithm A and the inputs φ are standard (given in the meta-language), and F is asked to provide a proof of correctness "pointwise", i.e., separately for every standard φ. Alternative notions could quantify the algorithm and/or the sentences inside F. For example, we might require that F proves some sentence expressing "there exists an algorithm such that for all L -sentences. . . " where the witnessing algorithm might be non-standard but has to work also for nonstandard sentences. We should also note that our definition does not require F to prove "A halts on all inputs". We view Theorem 14 below as evidence that our notion is the right one in our context. As its proof shows, F knows in our sense any algorithm deciding the theory of an absolutely representable structure (Corollary 15). So, indeed, the notion we have defined does not require much. Philosophically speaking, we might view this knowability condition as a mere technicality, and regard our result as showing that for all practical purposes absolute representability and decidability are equivalent. The following establishes our main claim from the Introduction. Theorem 14. Let R be a representation of an L -structure in F. Then R absolutely represents SL in F if and only if there exists an algorithm A deciding Th(SL ) and F pointwise verifies A wrt R. 9 Proof. For the forward direction, assume R absolutely represents SL in F. By Proposition 10, there is an algorithm A deciding Th(SL ). We claim that F pointwise verifies A wrt R. Let φ be an L -sentence. We show (4) by distinguishing cases. – Case φ ∈ Th(SL ). Then A(pφq) = 1 as A decides Th(SL ), so Q ` "A(  pφq)=1" by (3), so F ` RFN("A(  pφq)=1") by Lemma 12. But F ` R(φ) by (2) of Lemma 9. Hence F proves both sides of the equivalence in (4). – Case φ /∈ Th(SL ). Then A(pφq) 6= 1 as A decides Th(SL ), so Q ` ¬"A(  pφq)=1" by (3), so F proves RFN(¬"A(  pφq)=1") by Lemma 12, and this sentence equals ¬RFN("A(  pφq)=1"). But ¬φ ∈ Th(SL ), so F ` ¬R(φ) using (2) and R(¬φ) = ¬R(φ). Hence F refutes both sides of the equivalence in (4). For the converse direction, assume A decides Th(SL ) and F pointwise verifies A wrt R. We verify (2) of Lemma 9. For ⊆, let φ ∈ Th(SL ). As A decides Th(SL ), we have A(pφq) = 1, so Q ` "A(  pφq)=1" by (3), so F ` RFN ( "A(  pφq)=1" ) by Lemma 12, so F ` R(φ) by (4). For ⊇, let φ /∈ Th(SL ). Then ¬φ ∈ Th(SL ), so F proves R(¬φ) = ¬R(φ) by the above. Then F 6` R(φ) as F is consistent. The proof shows: Corollary 15. Let R be a representation of an L -structure in F and let A be an algorithm. If R absolutely represents SL in F and A decides Th(SL ), then F pointwise verifies A wrt R. Concerning our example structures we get: Example 16. The structures (N, +), (N,×), (R, +,×, 0, 1,<) and (C, +,×, 0, 1) have decidable theories [Presburger and Jabcquette, 1991, Mostowski, 1952, Tarski, 1998], and ZFC pointwise verifies their decision algorithms with respect to their standard representations. Hence, these structures are absolutely representable in ZFC. 3 An application to foundational debates We have seen thus far that an intended structure is absolutely representable in a first-order foundational theory F if and only if its (firstorder) theory is decidable and F knows some decision procedure for it. In this section we'll discuss an application of this observation to the 10 debate between proponents of first-order versus higher-order foundations. An important debate in the philosophy of logic and mathematics is whether foundations should be conducted in first-order or higherorder logic (or, if one is more tolerant in outlook, which logic is suited for what purposes). Throughout this paper, we have explicitly restricted our attention to first-order theories-both with respect to the foundational theory F under consideration and the theory of informal mathematics that we are trying to formalise in F. Many authors argue that our foundational theory should contain expressive resources greater than first-order, since many notions cannot be characterised up to isomorphism in first-order logic.8 All of finiteness, natural number, real number, and various infinite wellorderings evade characterisation. Logics with greater than first-order resources at their disposal, by contrast, are able to characterise some of these notions at the expense of pleasing meta-theoretic properties, namely compactness and Löwenheim-Skolem by Lindström's theorem [Lindström, 1969], and specifically completeness with respect to a finitary proof system. There is thus a trade-off between expressive power and the smoothness of transition between validity and proof. Our results inform this trade-off by providing bounds on when a first-order foundational theory can capture truth in an intended structure. Whilst it is clearly true that for an infinite structure SL , asking for isomorphic representation of SL is too much, nonetheless our results show that there are precise conditions on which a first-order theory can be omniscient concerning truth in SL . This shows that for a certain class of infinite structures, even first-order logic can have a good deal of traction (structurally speaking). On the other side of the coin, we have shown that a foundation which is both first-order and computably enumerable has limits in absolutely representing theories. The meta-theoretic advantages given by these two features have their price. Not only will any first-order foundational theory F fail to determine the cardinality of an intended infinite structure SL , but if the theory of SL is undecidable F loses traction on truth in SL too. 8This is argued in various places, but [Shapiro, 1991] is one of the strongest advocates of the use of higher-order resources for certain purposes in foundations. 11 4 Conclusion Recall, we asked for a representation R such that SL and R(M) are similar for all modelsM of F. Taking similarity as elementary equivalence, we saw that this requires Th(SL ) to be decidable. If it is not, it is natural to ask for weaker notions of representability. There are many possibilities and we briefly discuss one of them, namely the one obtained by weakening the equality in (2) of Lemma 9 to an inclusion: call R a sound representation of SL in F if{ φ | F ` R(φ) } ⊆ Th(SL ). (5) Roughly said, F proves only true first-order sentences about SL . Clearly, the working mathematician studying SL would reject any foundational theory not providing such a representation. Proposition 17. Let R be a representation of L -structures in F. Then R is a sound representation of SL in F if and only if there exists a modelM of F such that R(M) is elementarily equivalent to SL . Proof. Assume (5) holds. It suffices to show that the theory F ∪ { R(φ) | φ ∈ Th(SL ) } is consistent. Indeed, a modelM of this theory has the property that R(M) |= φ for all φ ∈ Th(SL ) by Lemma 4, so R(M) and SL are elementarily equivalent. The claimed consistency follows from compactness: if the theory above is inconsistent, then there are finitely many φ1, . . . ,φk ∈ Th(SL ) such that F refutes R(φ1)∧ . . .∧R(φk) = R(ψ) for ψ := φ1 ∧ . . . ∧ φk (see Remark 5); as F proves ¬R(ψ) = R(¬ψ) and ¬ψ /∈ Th(SL ), this contradicts (5). Conversely, if (5) fails, then there exists φ /∈ Th(SL ) such that F proves R(φ). By Lemma 4, R(M) |= φ for every modelM of F while SL 6|= φ, so R(M) and SL are not elementarily equivalent. Thus, asking for a sound representation is asking for a special model M of F, namely one such that R(M) and SL are elementarily equivalent. This makes sense also for other notions of similarity, and in particular for isomorphism. For example, in the case of (N, +,×, 0, 1,<) the latter asks F to have an ω-model. It is thus philosophically justified to ask F to be more than just consistent, but it is unclear how much more one should or can ask for. Another way to weaken the notion of representability, in order to make it apply to structures with undecidable theories, is to consider only "intended" models M of F, or an expansion thereof formulated using greater than first-order resources. For example, restricting M to transitive standard models of F = ZFC makes RZFCN 12 an isomorphic representation of (N, +,×, 0, 1,<); on the other hand, (R, +,×, 0, 1,<) is absolutely but not isomorphically representable. By contrast, if we instead formulate ZFC in quasi-weak second-order logic (where second-order variables are stipulated to range over countable relations), (R, +,×, 0, 1,<) becomes isomorphically representable. We therefore ask: Given different "semantic" extensions of F, what natural structures are absolutely representable but not isomorphically representable?9 References [Barton, S] Barton, N. (S). Structural relativity and informal rigour. Manuscript under review. [Ebbinghaus et al., 1994] Ebbinghaus, H.-D., Flum, J., and Thomas, W. (1994). Mathematical Logic. Springer New York. Second Edition. [Hamkins and Yang, 2013] Hamkins, J. D. and Yang, R. (2013). Satisfaction is not absolute. arXiv:1312.0670v1 [math.LO]. 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