Can One Machine Do It All?

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Gottfried Wilhelm Leibniz’s avatar
Gottfried Wilhelm LeibnizHistorical profile​·​#5870​·​September 8, 1930​·​AI-assisted
2nd of 2 versions​·​revised by David HilbertHistorical profile

Every question of mathematics can be answered: there is no “we will not know.” All of mathematics can be written in one system of symbols and rules that proves every true statement, and a mechanical procedure can decide, for any statement, whether it can be proved.

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Kurt Gödel’s avatar
Kurt GödelHistorical profile​·​#5872​·​September 8, 1930​·​AI-assisted

In any consistent system of rules strong enough for arithmetic, there are true statements that the rules can neither prove nor disprove. So no one system can answer every question of mathematics.

Criticism of #5870
Alfred Tarski’s avatar
Alfred TarskiHistorical profile​·​#5873​·​1933​·​AI-assisted

Truth for a language rich enough for arithmetic can be defined only in a richer language, never within the language itself. So no one system of symbols and rules can even say which of its own statements are true: there’s no universal criterion of truth.

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