What does Godël’s Incompleteness Theorem actually mean and imply? I just saw Ted-Ed’s video on this topic and didn’t fully understand what it means or what the implications of this are.
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I like to use the following example.
Let’s say we have a library and are creating useful indexes for it.
We make a red book that lists all the library books that refer to the themselves. Encyclopedias, dictionaries, etc.
We make a green book that lists all the library books that don’t refer to themselves. That will be most of them.
Now, logicians and mathematicians started the 20th century working on the rules of logic that would let us decide for every statement whether it was true or false. Somewhere God should have an answer book listing all the true statements in our system of logic. For any function we would like to be able to determine whether F(x) or not F(x) is in God’s answer book.
Russel’s paradox asks: “Does the green book lists itself?” We can’t answer “yes,” because if it does it shouldn’t. We can’t answer “no” because if it doesn’t, it should. We can’t find either F(x) or not F(x) in God’s answer book. Oops.
And there is another question we could ask. “Does the red book list itself?” If it does…great! “F(red book)” goes in God’s answer book. If it doesn’t…great! It shouldn’t! “Not F(red book)” goes in God’s answer book.
Oops. We don’t want both F(x) and notF(x) to be true.
You probably think “this book refers to itself” sounds like a really weird function. Maybe we can just leave it out of our system of logic? Sweep the problem under the rug.
But Gödel proved that any formal logic with enough tools to include statements like (1+2=3) will have this problem.
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