Comment on There is No Algorithm for Truth - with Tom Scott [59:34]

• FishFace@lemmy.world ⁨3⁩ ⁨days⁩ ago

  en.wikipedia.org/…/Tarski's_undefinability_theore…

  In maths words: there is no mathematical formula phi such that phi(x) is true whenever x is the Gödel code of a true mathematical sentence.

  In dumb words: a mathematical definition of a property is a combination of mathematical symbols (ordinary ones from arithmetic like “+” and “×”, variables like “x” and “y”, constants like “0” or “the empty set”, but also ones you might not be familiar with like “there exists” and “for all”) which, when interpreted in mathematics, is true whenever you give it objects that have the property, and false for all other ones. For example, you can define the number zero with the following formula: “x is zero if and only if, for all y, x+y = y”. If you substitute zero for x in this formula, it will be true, otherwise it will be false.

  The theorem says that there is no such thing for the property of “being true.” There is a complication that mathematical formulae take mathematical objects (like numbers, sets or groups) rather than formulae, but Gödel coding is a way of unambiguously and simply associating a unique number to every formula in such a way that they can be manipulated within mathematics, so this isn’t an obstacle.

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