Remarks

Cross-cutting remarks on subtleties that span several of the logic pages — the kind of foundational fine print that doesn't belong to any single system but matters for understanding how they fit together.

Contents

1. Set Theory and First-Order Logic — the apparent circularity between first-order logic and set theory (ZFC), resolved via the object-theory/metatheory distinction; the foundational ordering of syntax, set theory, and semantics; and first-order logic's canonical status (Lindström's theorem).
2. Models, Truth, and Metalanguage — why model and truth talk is irreducibly metalinguistic (Tarski's undefinability of truth), and the object-level/meta-level "two gazes" picture of a structure.
3. The Metalanguage Hierarchy and Semantics in Mathematics — Tarski's tower of object/meta levels and whether the regress is vicious, and what "semantics" means in mathematical practice (interpretation in a structure, meaning conferred from the metatheory).
4. What Justifies Con(ZFC)? — why the consistency of ZFC cannot be proved non-circularly, and the convergence of intuitive (iterative conception), quasi-empirical (track record), structural, and metaphysical considerations that warrant it.
5. What the Metalanguage for First-Order Logic Assumes — the precise metatheoretic commitments of FOL metatheory: the meta-logic, the reverse-mathematics ledger (PRA / WKL₀ / ZF+BPI), the axiom-of-choice subtlety (BPI ⊕ full AC), and why $\mathrm{Con}(\mathrm{ZFC})$ is not assumed.
6. The Axiom of Choice and Excluded Middle (Diaconescu's Theorem) — why, over intuitionistic logic, the full set-theoretic axiom of choice implies the law of excluded middle, and why type-theoretic and weaker (countable/dependent) choice escape this.