Tarski’s Indefinability Theorem

Tarski’s Indefinability Theorem

Tarski’s indefinability theorem is a fundamental result in mathematical logic and the philosophy of language. It demonstrates a profound limitation regarding the definition of truth within formal systems.

Overview

The theorem, formulated by Alfred Tarski, asserts that for any sufficiently rich formal language, the concept of ‘truth’ for sentences within that language cannot be defined within the language itself without leading to contradictions. To define truth consistently, one must step outside the language into a meta-language.

Key Concepts

The theorem relies on several key concepts:

• Formal Language: A precisely defined language with a fixed syntax and semantics.
• Truth Predicate: A property that applies to sentences of the language, indicating whether they are true.
• Consistency: The absence of contradictions.
• Meta-language: A language used to talk about another language (the object language).

Deep Dive

Tarski showed that if a language is strong enough to express basic arithmetic, then any consistent definition of truth for that language, formulated within the language itself, must be incomplete. A common technique to demonstrate this involves constructing a sentence that essentially says ‘This sentence is false’ (a variant of the Liar paradox). If truth were definable within the language, this sentence could be shown to be both true and false, leading to a contradiction.

Applications

The theorem has significant implications for:

• Formal semantics
• Foundations of mathematics
• Philosophy of language
• Computer science (e.g., in formal verification)

Challenges & Misconceptions

A common misconception is that the theorem implies truth is ultimately undefinable. Instead, it highlights the necessity of hierarchies of languages (object language and meta-language) for discussing truth rigorously. It does not preclude defining truth in a more powerful meta-language.

FAQs

Q: What is the main consequence of Tarski’s theorem?
A: It shows that a consistent theory of truth for a language requires a higher-level language to define it.

Q: Does this mean we can’t talk about truth?
A: No, we can talk about truth, but we must use a language that is distinct from the language whose truth we are discussing.