Frege Arithmetic (Hume's Principle)

Origin. Frege's Grundlagen (1884) §63 states the principle and Grundgesetze (1893, 1903) derives arithmetic from it — but by way of Basic Law V, which Russell's paradox destroyed. Crispin Wright (Frege's Conception of Numbers as Objects, 1983) observed that the derivation only ever used Hume's Principle, and that HP is consistent where Law V is not. Boolos formalized and named the result: Frege's Theorem. Heck's work on Grundgesetze established that Frege himself had the derivation isolated.

Models. Second-order logic with one added operator and one added axiom. The operator #F sends a concept to its number; the axiom says two concepts have the same number exactly when they are equinumerous. From this alone, with no set theory and no arithmetical primitives, second-order Peano arithmetic follows. The abstraction principle does the work Basic Law V was supposed to do and does not extend to a universal comprehension, which is why it survives.

Formalism.

Base: full second-order logic with impredicative comprehension.

Language: the second-order language plus a term-forming operator # taking a monadic concept variable to an object: #F.

Hume's Principle (HP): #F = #G ↔ F ≈ G where F ≈ G abbreviates the second-order statement that some relation is a bijection between the F's and the G's: ∃R [∀x(Fx → ∃!y(Gy ∧ Rxy)) ∧ ∀y(Gy → ∃!x(Fx ∧ Rxy))]

FA = second-order logic + HP. One axiom.

Frege's Theorem: FA interprets PA₂ (second-order Peano arithmetic). The derivation: 0 := #[x : x ≠ x] Precedes(m, n) := ∃F ∃y (Fy ∧ n = #F ∧ m = #[x : Fx ∧ x ≠ y]) ℕ := the weak ancestral of Precedes starting from 0 (definable in SOL) Then: 0 is a number, successor is functional and injective on ℕ, 0 has no predecessor, and induction holds — all provable in FA. The existence of successors turns on Frege's device of counting the numbers up to n, which requires no additional axiom.

Consistency: FA is equiconsistent with PA₂, hence with Z₂. Boolos gave the model: the natural numbers plus one extra object, with # interpreted as cardinality for finite concepts and the extra object for infinite ones.

Contrast with Basic Law V: Law V: ε̂F = ε̂G ↔ ∀x(Fx ↔ Gx) — inconsistent in second-order logic (Russell). HP: #F = #G ↔ F ≈ G — consistent, and sufficient. Both are abstraction principles of the form §F = §G ↔ E(F, G) for an equivalence E. The difference is that Law V's right side is too fine — it forces an injection from concepts to objects — while HP's is coarse enough to be satisfiable.

The Bad Company problem: not every abstraction principle is admissible. The Nuisance Principle and Boolos's parity principle are individually consistent and jointly inconsistent with HP; George Boolos's "New V" and Kit Fine's work on the general theory attempt a criterion. No agreed criterion exists.

Symbols.

Metatheory. FA interprets PA₂ and is interpretable in it, so the two are mutually interpretable and equiconsistent — the neo-logicist claim is therefore not a claim about strength but about what counts as logic. That is where the argument lives: HP is consistent, sufficient, and one axiom, but whether an abstraction principle is analytic, and whether impredicative second-order logic is logic, are the disputed premises, and Frege's Theorem settles neither. The Bad Company problem is the technical residue: consistency of an abstraction principle is not a sufficient condition for admissibility, since consistent principles can be pairwise inconsistent, and no criterion has been established that admits HP and excludes the rest without circularity.

Applies to. Neo-logicism and the philosophy of arithmetic — the entire abstractionist programme. The general theory of abstraction principles, including applications to real analysis (Hale) and set theory (Boolos's New V). The reconstruction of Grundgesetze, which is the historical case that Frege had the theorem. Second-order logic's expressive strength, as an instance where one axiom over SOL yields a full arithmetic.

Limitations. The base is full impredicative second-order logic, which is not neutral: without impredicative comprehension the derivation of successor's existence fails, and predicative FA is much weaker than PA₂. HP is consistent but not obviously analytic, and the Bad Company problem makes "consistent abstraction principle" an inadequate criterion of legitimacy. The interpretation is of PA₂, not PA, so the arithmetic delivered comes with second-order logic's incompleteness attached. Frege's own project fails regardless: Law V is inconsistent, and the theorem is a rescue of the derivation, not of the Grundgesetze.

© 2026 Lingenic LLC