# Why is incompleteness important?

Or take Russel's paradox. Either the barber does or doesn't shave himself -- that's all there is. How you describe it is an artificial construct.

Godel's theorem is like dividing by zero and declaring arithmetic incomplete.

It seems to me that all it means is, humans can construct artificial incomplete formal systems. You might as well assert that TRUE = FALSE and be amazed at the "paradox."

Is there some utility in believing that arithmetic is incomplete under pathological infinite recursion?

• Well, I think most mathematicians would agree that there is some utility in understanding that mathematics cannot be axiomatized. To understand the utility, it might help to read some relevant history, e.g. plato.stanford.edu/entries/hilbert-program . One take-away was that to even come up with a consistent theory of sets, one had to somehow figure out how to understand and formalize what you point at with the term "pathological recursion". Mar 31 at 22:28
• This question is not about computer science but about logic and foundations of mathematics. Please delete it here and ask it on math.stackexchange.com where it belongs. While you're at it, it would help if you explain why you think there is anything "pathological" in recursion. using an unexplained non-mathematical phrase like that just disqualifies your question as being about mathematics. Mar 31 at 22:32
• I'll just delete it; the hell with points. I speak lambda calculus as good as the next hacker, but you don't need it to convey the concept of a singularity due to nonterminating recursion involving a non-well-defined predicate. I'm not making it about terminology. That trivializes it. But I'll delete this, as I seem to be handling a social context fault interrupt again, and I hate it. I asked here because I associate incompleteness with redundant explainations of its brilliance in almost every CS class. Apr 1 at 12:06
• Oops, it won't let me delete it. oh well. i officially abandon it. Apr 1 at 12:14
• Hey, I was permanently banned from the logic stack xch for asking this exact same question. Is it any wonder that I avoid saying what are seemingly-normal things to non-autstics because they get inexplicably outraged? Apr 1 at 12:23

Russell's paradox showed that a proposed set theory, proposed for the foundation of mathematics itself, contained a contradiction:

Let $$R = \{x \mid x \notin x \}$$, then $$R \in R$$ if and only if $$R \notin R$$.

Any consistent formal system $$F$$ within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of $$F$$ which can neither be proved nor disproved in $$F$$.