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For those interested: Assume a formal language that can model the natural numbers. Consider the set of all provable statements. This is countable because it’s
by moomin 5y ago
For those interested:
Assume a formal language that can model the natural numbers.
Consider the set of all provable statements. This is countable because it’s a subset of the list of strings of a finite number of characters, which is countable.
Consider the statement “Statement N is false”. This is both a valid statement and
cannot appear in the list.
Godel proved significantly more than that, but that’s the basic result. It’s basically the same as the diagonal argument of real numbers being uncountable and it’s also basically the halting problem.
- harmoat 5y agoSince these 3 results are so similar are they a subset of a more general concept?
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- moomin 5y agoArguably the halting problem is just a subset of Godel where every theorem is "This does or does not halt". Someone's written a paper going the other way: https://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf https://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf The crucial trick is that you can plug in an entry as data to another "program/theorem". This is extremely interesting, because there are limited logics and limited computation models where Godel and the Halting Problem do not apply. Not so sure about the diagonal argument.
- alisonkisk 5y agoIt's "This cannot be proved.", where "this" is defined via the mathematical coding system he created. The intuitive idea is very simple, but Godel's achievement in this theorem was in finding a way to rigorously define the word "this".
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