Shaking the Foundations of Mathematics

For most of our history, we believed that all mathematical truths required valid proof. However, in the early 20th century, an Austrian logician and mathematician named Kurt Gödel was about to shake those foundations by proving that some mathematical truths cannot be proven.

In language, we already knew this was the case. For example, the sentence “This statement is false” is an example of something that cannot be proven true because if it were true, it would contradict itself and be false. We thought mathematics was different and that we could find a finite set of axioms to deduce all mathematical theorems, a goal pursued by mathematicians like David Hilbert.

Gödel's Incompleteness Theorems, first published in 1931, demonstrated that Hilbert's program was inherently unachievable. Any system complex enough to include arithmetic would inevitably contain true propositions that cannot be proven within the confines of its own rules. Gödel did this by cleverly encoding statements about arithmetic into numbers, showing that certain propositions are undecidable. He proved that while everything in mathematics is true or false, some factual statements cannot be proven.

Gödel's work challenged the very foundations of mathematical logic and set limitations on what can be achieved through arithmetic and logical systems, inspiring future work on computation by Alan Turing. To achieve this, Gödel had to challenge the status quo that all truths can be proven and creatively use the area of language to create a code demonstrating that not every truth can be proven.

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