What is truth? - Mathematician explains | Joel David Hamkins and Lex Fridman

Quick Overview

The discussion between Joel David Hamkins and Lex Fridman centers on the fundamental distinction in logic between "truth" (semantic satisfaction within a model) and "provability" (syntactic derivation from axioms), a concept highlighted by Gödel's Incompleteness Theorems and Alfred Tarski's disquotational theory of truth, which separates the language being talked about from the language used to talk about it.

Key Points: Provability is defined as the syntactic derivation of a formula from axioms using inference rules, while Truth is the semantic satisfaction of a formula within a specific model. The discussion references Kurt Gödel's Incompleteness Theorems, which imply that any consistent formal system rich enough for basic arithmetic contains true statements that cannot be proven within the system, and cannot prove its own consistency. Alfred Tarski's disquotational theory of truth is introduced, stating that a sentence is true if and only if the content of the sentence is the case (e.g., "snow is white" is true if and only if snow is white), achieved by removing quotation marks from the assertion. The inability to algorithmically decide the provability of every true statement (the decision problem for arithmetic) is directly equivalent to the Halting Problem, a fundamental limitation in computability theory. Classical proof systems are characterized as being both 'sound' (proofs preserve truth) and 'complete' (all truths are provable), properties that Gödel's Incompleteness Theorems show cannot be simultaneously held by sufficiently complex formal systems.

Context: This video features a deep dive into foundational concepts of mathematical logic between Lex Fridman and mathematician Joel David Hamkins. The core conversation revolves around clarifying the difference between 'truth' and 'provability' in formal systems, drawing heavily on the work of 20th-century logicians like Kurt Gödel and Alfred Tarski, and contrasting these concepts with the historical goals of mathematicians like David Hilbert.