undefinability of truth
Tarski's Theorem states that no first-order logical theory has a predicate defining the Godel numbers of statements which are true of the natural numbers. In other words, a first-order theory which includes arithmetic can't define its own truth predicate. This is a fundamental limitation on mathematics, because it implies no formal system is powerful enough to define its own semantics.
This concept has the prerequisites:
- Know the statement of Tarski's Theorem and why this implies a formal language can't define its own truth predicate
- Prove Tarski's Theorem
Core resources (read/watch one of the following)
→ A Mathematical Introduction to Logic
An undergraduate textbook in mathematical logic, with proofs.
Location: Start of Section 3.5, "Incompleteness and undecidability," pages 234-236
→ A Course in Mathematical Logic
A graduate textbook in mathematical logic.
Location: Section 7.4, "Tarski's Theorem," pages 327-331
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