**Kurt Gödel (1906 – 1978)**

If you read the last two sections on the axiom of choice and the continuum hypothesis, you might, after you get over the initial shock and even concern that mathematics is on shaky ground, be willing to accept both the AC and CH. So now you think, OK let’s get on with it and get back to proving theorems. Our logical system is consistent with ZF + AC + CH. Whew!

*“Not so fast!”* says Gödel, who, in 1931 showed that in any logical system there are statements (like the AOC and CH) which can neither be proved or disproved. So, even if you accept the AC and CH to the ZF system, there will be* other* statements which you can never prove or disprove, using the logical rules of that system. Don’t misinterpret this result. If you are working on a mathematics proof and can’t do it. It is probably because you lack the necessary tools or just the right clever idea. There is a big difference between you can’t prove it (you’re not clever enough) and it can’t be proved – or disproved (Goödel).

Mathematics will always be a work in progress! In addition, it is not as perfect as you think it is.

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