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Hi I am looking not to understand the Incompleteness Theorem, but to find out more about how and what this has effected the mathematics world.

I am in high school, in Honors Algebra II, and I am doing a project about Kurt Gödel and one of the areas we need to research is their major contributions to mathematics.

If anyone can help me find out more about this in a fairly simple way that would be great.

thank you.

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Feferman's article (referenced in the comments) is great for the technical impact of Gödel's incompleteness theorem, but he explicitly doesn't go into its historical/philosophical impact, probably because this tale has been told so often.

The Wikipedia articles on Hilbert's program and Intuitionism, and the Foundational Crises section of the Foundations of Mathematics articles should give you a good quick intro to the historical context. Gödel's incompleteness theorem showed that Hilbert's program could not be carried out. Also check out the linked articles in the Stanford Encyclopedia of Philosophy.

Constance Reid's delightful biography Hilbert, Chapters 18-23, will give you a very good feel for what was going on at the time.

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