Welcome at the entrance of a world without logic

The beauty of the notion of description :

1. Show that the set of positive integers that can't be described by at most 999 999 bytes is non-empty, and that it has a smallest element a.
2. Show that a cannot be described in less than one million bytes.
3. Show that, eventually, a can be described in less than one million bytes.
4. Convince yourself that you are not very consistent today.

OK, same player, shoot again.

1. Consider the following sentence S : "This sentence is improvable".
2. Show that if S is provable, there exists a provable sentence which is false (inconsistency).
3. Show that if S if unprovable, there exists a true sentence which is unprovable (incompleteness).
4. Convince yourself that the law of excluded middle, well, is not that healthy.

If you are interested in this kind of stuff, well, some people are getting paid for it ; please visit Gregory Chaitin's website. Here you have a text in french : "toward the unknowable".

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