Rigorous and load-bearing. Tarski's semantic definition of truth and the model theory it launched are foundational — they underwrite logical consequence, database query semantics, and the formal verification of hardware and software.
You can't say "true" from inside. Tarski's undefinability theorem (a sibling of [[kurt-godel]]): no sufficiently rich language can define its own truth predicate. "True" must be spoken in a higher metalanguage — truth is stratified.
Banach–Tarski, read honestly. A solid ball can be cut into pieces and reassembled into two identical balls — a real theorem, but it needs the Axiom of Choice and non-measurable "pieces" with no volume. A truth about idealized sets, not matter.
A sentence isn't true or false in a vacuum; it's true in a model — a particular structure that says which basic facts hold. Take the formula (p ∧ q) → r. Toggle which atoms p, q, r hold in the current world, and watch the sentence's truth value. Then click the preset models: the same sentence is true in one and false in another. Truth is a relation between language and structure — Tarski's central insight.
φ is true in M" means M makes φ come out true — and different models disagree. This is the T-schema made operational: "(p∧q)→r" is true in M iff (p∧q)→r holds in M. ⚑ Every database query ("is this true of my data?"), every model-checker verifying a chip or protocol, every type-soundness proof is computing exactly this relation — truth-in-a-structure, Tarski-style."‘Snow is white’ is true if, and only if, snow is white." — Tarski's Convention T, 1933
"This sentence is false." If it's true, it's false; if false, true — the Liar paradox. Tarski's diagnosis: the trouble is a language trying to talk about its own truth. His theorem proves no rich language can contain its own truth predicate, so "true" must be spoken in a metalanguage one level above the object language it judges. Stratify the levels, and the Liar simply can't be formed.
Truth = satisfaction in a structure (1933). Model theory — the study of which sentences hold in which structures — became half of modern logic and the semantics of computing.
No rich language defines its own truth predicate (Tarski's theorem) — a semantic twin of Gödel's incompleteness. Truth outruns any single system that tries to capture it from inside.
Gate kept on. Tarski's work is rigorous and foundational — but two things deserve honest framing. First, the scope of "truth": Tarski defined truth for formal languages, not the messy truth of ordinary speech, and his definition makes truth explicitly relative to a model and a metalanguage — it is a precise mathematical relation, not a theory of Truth-with-a-capital-T about reality. Second, the famous Banach–Tarski paradox (with Stefan Banach, 1924): yes, a solid ball can be decomposed into finitely many pieces and reassembled into two balls of the same size — but this is a theorem about idealized point-sets, requiring the Axiom of Choice and "pieces" so pathological they have no definable volume; it says something deep about infinity and measure, and nothing about cutting up real matter. Pop accounts that make it sound like a duplication machine are wrong. The man: a Polish Jew who happened to be lecturing in the United States in September 1939 when Germany invaded Poland — the trip saved his life, though much of his family was murdered in the Holocaust; he became the great founder of logic at Berkeley, training a school that shaped the field for generations. The definition of truth, and the proof that no language can hold its own, are his. ⚑ With Tarski the LOGIKĒ foundations rank rests: from [[thales-of-miletus]]' first proof through the gate, the limits, and now the meaning of truth itself.