Internal Categoricity Arguments

• Internal categoricity arguments are methods to show that any two realizations of a theory are isomorphic using resources internal to the object's language.
• They contrast with external categoricity by relying solely on the deductive system, thereby avoiding stronger metatheoretic assumptions.
• These arguments reinforce structuralist views by establishing intersubjective determinacy and formal coordination without invoking external semantic predicates.

Searching arXiv for recent and foundational papers on internal categoricity. arXiv search query: "internal categoricity Väänänen first-order second-order philosophy mathematics" Internal categoricity arguments are arguments in which the claim that any two realizations of a theory are isomorphic is formulated and derived within the object-language and proof system of the relevant logic, rather than established by quantifying over models in an external set-theoretic metatheory. In Jouko Väänänen’s formulation, this yields a proof-theoretic notion that applies to both first- and second-order axiomatizations, recovers the classical categorical results associated with Dedekind, Hilbert, Huntington, Peano, Veblen, and Zermelo, and is proposed as the appropriate replacement for the traditional picture in which categoricity depends essentially on a hierarchy of stronger and stronger metatheories (Väänänen, 2020).

1. Historical setting and motivating problems

The modern interest in internal categoricity arose against a background in which classical categoricity arguments had become philosophically unstable. Early second-order axiomatizations of arithmetic, geometry, analysis, and set theory were presented as categorical, but twentieth-century model theory made non-categoricity ubiquitous in first-order settings through Löwenheim–Skolem and compactness, while second-order categoricity came to depend on full semantics and thus on substantial set-theoretic assumptions. The resulting contrast between earlier categorical practice and later metalogical results is a central point of departure for Väänänen’s reconstruction (Väänänen, 2020).

Button and Walsh situate the internal turn within a broader debate involving Benacerraf, Putnam, Parsons, McGee, Shapiro, and Lavine. Benacerraf’s push-through construction was taken to show that there are many set-theoretic realizations of the natural numbers, supporting the structuralist thought that only the progression-structure matters. Putnam’s permutation argument then threatened determinacy of reference more generally by suggesting that any model-theoretic account of reference permits non-trivial permutations preserving truth. On this picture, an appeal to a preferred semantics is “just more theory,” and that further theory is itself vulnerable to model-theoretic reinterpretation. Internal categoricity was developed partly as a response to precisely this predicament: it seeks a categoricity result that does not presuppose a disputed semantic clause for “full” second-order quantification (Button et al., 2015).

This historical development also explains why internal categoricity has been taken to have a specifically structuralist significance. The target is not uniqueness of a distinguished implementation of arithmetic or set theory, but uniqueness up to isomorphism under a derivation that can be carried out within the relevant deductive apparatus itself. A plausible implication is that the internal approach shifts the focus from metaphysical individuation of mathematical objects to formally provable coordination between realizations.

2. External and internal categoricity

The standard external notion is semantic. If $L$ is a first- or second-order language and $T$ a set of $L$-sentences, then $T$ is categorical exactly if for every pair of $L$-models $\mathfrak{M},\mathfrak{N}\models T$ there is an isomorphism $\mathfrak{M}\cong\mathfrak{N}$. This is a metalogical condition, evaluated “from the outside,” by quantifying over models in a background set theory (Väänänen, 2020).

Internal categoricity replaces that external quantification by a single formula of second-order logic. Fix a finite vocabulary $L$ and a closed second-order $L$-formula $\varphi$. One passes to two disjoint copies $T$0 and $T$1, adds fresh unary predicates $T$2 and a fresh binary function symbol $T$3, and defines $T$4 to assert closure of $T$5 under the function symbols of $T$6. If $T$7 is an $T$8-sentence, then $T$9 is the relativization of $L$0 to $L$1, with first-order quantifiers ranging over $L$2 and second-order variables over relations on $L$3. Let $L$4 assert that $L$5 is a bijection respecting all relation and function symbols of the two copies. The internalized categoricity sentence is then:

$L$6

A second-order sentence $L$7 is internally categorical exactly if $L$8 is provable in the deductive system of full second-order logic with comprehension axioms. Equivalently, by a Henkin-style completeness theorem, $L$9 is internally categorical iff $T$0 is valid in every Henkin-model of second-order logic (Väänänen, 2020).

The contrast with the external notion is exact. External categoricity asks whether all models of a theory are isomorphic in a surrounding metatheory. Internal categoricity asks whether second-order logic itself proves that any two relativized realizations of the theory are isomorphic. Väänänen’s comparison is asymmetric: in general, $T$1 implies external categoricity, but the converse can fail if the external proof relies on a stronger metatheory than the proof system of second-order logic itself. This is why internal categoricity is described as weaker than categoricity in the first-order case and stronger in the second-order case (Väänänen, 2020).

3. Second-order internal categoricity

The principal second-order examples are classical. Väänänen identifies the following second-order axiomatizations as internally categorical: Dedekind–Peano arithmetic $T$2, a second-order characterization $T$3 of an infinite set, a theory $T$4 axiomatizing $T$5, and a second-order axiomatization $T$6 of the ordered field of reals using field axioms, order, and a least-upper-bound completeness axiom. In each case, the standard model-theoretic proof of categoricity can be formalized entirely within second-order logic using comprehension and elementary properties of the relevant operations, yielding a finite derivation of the internal categoricity sentence (Väänänen, 2020).

For arithmetic, the relevant second-order Peano formula can be presented as

$T$7

Parsons-style presentations emphasize that one may write a single second-order sentence asserting that any two triples $T$8 and $T$9 satisfying the relativized Peano axioms are connected by an isomorphism $L$0. The proof proceeds internally by defining $L$1 by second-order recursion, using comprehension to collect the ordered pairs satisfying the recursive clause, and then proving uniqueness and bijectivity by induction. No semantic predicate $L$2 and no separate axiom of “fullness” enter the formal derivation (Button et al., 2015).

Set theory is more delicate. Väänänen states that second-order ZFC$L$3 is quasi-categorical in the sense that, once the height is fixed to an inaccessible $L$4, one can derive that any two ZFC$L$5-models of height $L$6 are isomorphic (Väänänen, 2020). Button and Walsh present the related second-order internal quasi-categoricity result for Zermelo’s set theory in the form

$L$7

where $L$8 says that $L$9 is an isomorphism from one model onto an initial segment of the other. They also record McGee’s stronger internal categoricity result for pure sets in second-order $\mathfrak{M},\mathfrak{N}\models T$0, where the isomorphism is asserted on the pure-set parts of the universes (Button et al., 2015).

These cases show why Väänänen treats internal categoricity as recovering the early categorical axiomatizations without recreating an infinite regress of stronger metatheories. The proof of categoricity is no longer exported to a higher semantic level; it appears as a finite derivation within second-order logic itself (Väänänen, 2020).

4. First-order internal categoricity

In first-order logic the setting changes fundamentally, because no infinite first-order theory is externally categorical. Internal categoricity is therefore reformulated so that the theory “sees” two copies of its own nonlogical vocabulary in a single ambient structure. This makes the first-order notion strictly weaker than external categoricity: the requirement is not that every pair of models in the metatheory be isomorphic, but that two copies present within the combined language be provably coordinated by a definable isomorphism (Väänänen, 2020).

For arithmetic, let $\mathfrak{M},\mathfrak{N}\models T$1 be the usual first-order Peano axioms together with the induction schema for all formulas in the extended language $\mathfrak{M},\mathfrak{N}\models T$2, and similarly for $\mathfrak{M},\mathfrak{N}\models T$3. Väänänen defines an explicit first-order formula

$\mathfrak{M},\mathfrak{N}\models T$4

saying that the unique function $\mathfrak{M},\mathfrak{N}\models T$5 defined by coding initial segments is a ring-isomorphism from $\mathfrak{M},\mathfrak{N}\models T$6 onto $\mathfrak{M},\mathfrak{N}\models T$7. The theorem is that first-order logic proves

$\mathfrak{M},\mathfrak{N}\models T$8

Thus, although non-standard models exist externally, the two visible arithmetic structures must agree up to a definable isomorphism (Väänänen, 2020).

The corresponding result for set theory uses a combined language with two membership relations. If $\mathfrak{M},\mathfrak{N}\models T$9 denotes ZFC in a language that permits formulas containing both $\mathfrak{M}\cong\mathfrak{N}$0 and $\mathfrak{M}\cong\mathfrak{N}$1 in the Separation and Replacement schemas, and $\mathfrak{M}\cong\mathfrak{N}$2 is defined symmetrically, then from

$\mathfrak{M}\cong\mathfrak{N}$3

one can derive a first-order formula $\mathfrak{M}\cong\mathfrak{N}$4 asserting that the two membership relations define isomorphic models on the same domain. Väänänen summarizes the upshot by saying that first-order arithmetic and set theory are “categorical in their own vicinity” (Väänänen, 2020).

A later first-order generalization is given by the notion of $\mathfrak{M}\cong\mathfrak{N}$5-definiteness. Fix a base language $\mathfrak{M}\cong\mathfrak{N}$6, expand it to $\mathfrak{M}\cong\mathfrak{N}$7, and let $\mathfrak{M}\cong\mathfrak{N}$8 be a scheme in the new predicates. Then $\mathfrak{M}\cong\mathfrak{N}$9 is $L$0-definite exactly when, in $L$1, any two expansions $L$2 and $L$3 of the same $L$4-structure satisfying the same instantiation of the scheme are indiscernible by every $L$5-formula; in particular the predicates coincide extensionally. Internal categoricity over $L$6 is then defined by requiring that any two $L$7-structures satisfying $L$8 be isomorphic via an isomorphism fixing $L$9 pointwise, and strong internal categoricity is shown to coincide exactly with $L$0-definiteness. The same framework introduces “intolerance” as the case of definiteness over the empty theory (Gruza et al., 26 Nov 2025).

5. Philosophical role, achievements, and limitations

Internal categoricity arguments have been taken to secure a form of intersubjective determinacy. Button and Walsh formulate this in explicitly practical terms: if two practitioners, “Kurt” and “Michael,” each work with their own predicates and function symbols satisfying the internal arithmetic axioms, they can combine their languages and derive that the two systems are isomorphic. In this sense, internal categoricity explains why mathematical discourse is stable under harmless renaming and why arithmetic or set-theoretic practice can be coordinated structurally (Button et al., 2015).

What such arguments do not provide is equally important. They do not define a semantics for the claim that a predicate really picks out the intended domain, and they do not themselves supply a satisfaction relation $L$1. Accordingly, they do not directly answer the skeptic who questions whether second-order quantifiers range over all subsets one intends. Button and Walsh therefore distinguish the result achieved by internal arguments from a full semantic account of truth and reference: the achievement is relative or intersubjective determinacy of truth-value, not a direct semantic solution to the problem of reference (Button et al., 2015).

Several objections and limitations follow from this diagnosis. One is the domain-restriction worry, developed by Incurvati against McGee’s proposal, concerning whether one speaker can simply import another speaker’s quantifiers as unrestricted. Another is Parsons’s appeal to “acquiescence in the mother tongue,” which critics have described as trivial or question-begging. A third concerns the difference between quasi-categoricity and full categoricity for set theory: Zermelo’s initial-segment result leaves open how far the hierarchy extends, and Isaacson reads this as expressing a genuine extensibility of the set-theoretic universe. These disputes do not negate the formal results, but they do constrain what those results can legitimately be said to establish (Button et al., 2015).

A plausible implication is that the force of internal categoricity is strongest when the target claim is structural agreement rather than semantic absolutism. It shows that two admissible realizations cannot diverge except up to isomorphism inside the formal framework, but it does not, by itself, settle contested questions about the interpretation of that framework.

6. Later generalizations and axiom-sensitivity

Recent work has developed internal categoricity in two distinct directions. One direction systematizes the first-order phenomenon. The $L$2-definiteness framework is presented as a unified setting for internal categoricity, strong internal categoricity, and intolerance, together with robustness results showing invariance under definitional extensions, syntactic replacements, conjunctions and weakening, addition of parameters, and conservative extensions. Within this framework, the induction scheme in arithmetic is shown to be $L$3-definite, hence strongly internally categorical, while the replacement scheme in $L$4 is shown not to be definite: there can be two distinct class-functions each satisfying the replacement scheme but disagreeing on some ordered pair. The contrast is intended to map a spectrum of definiteness properties rather than a single binary division (Gruza et al., 26 Nov 2025).

The other direction investigates internal categoricity in alternative foundational settings. Meadows studies John Steel’s generic-multiverse theory $L$5, formulated in a two-sorted first-order language with a set-sort, a world-sort, and a single binary relation symbol $L$6 ranging over set-member-of-set and set-member-of-world. Under the assumption that there is a countable transitive model of ZFC, the standard axiomatization $L$7 fails internal categoricity: there are two distinct models of $L$8 with the same set-universe but different world-universes. By contrast, a variant $L$9, obtained by adding Global $\varphi$0 and a World-domination schema, is internally categorical; moreover, $\varphi$1, so the contrast is one of axiomatization rather than deductive strength. Meadows’ stated philosophical conclusion is that internal categoricity per se does not enforce fixity of subject matter: even when $\varphi$2 is internally categorical, the continuum hypothesis remains neither decided nor expressible in a robust way in the intended multiverse setting (Meadows, 28 Aug 2025).

This later literature suggests that internal categoricity is best viewed not as a single decisive solution to foundational indeterminacy, but as a family of formal phenomena concerning the extent to which a theory pins down its nonlogical symbols by proof alone. In second-order settings it serves as a proof-theoretic strengthening of classical categoricity; in first-order settings it yields local or scheme-relative uniqueness; and in newer applications it reveals a pronounced sensitivity to formulation, background theory, and the kinds of structures a theory permits (Väänänen, 2020).