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Godel's theorem implies that in a consistent system there are unprovable truths. I've always wondered if those truths could be things no one would care about a
by dmoney 16y ago
Godel's theorem implies that in a consistent system there are unprovable truths. I've always wondered if those truths could be things no one would care about anyway. Are statements like "this statement is unprovable" useful for anything other than proving the Incompleteness theorem?
- dgordon 16y agoThe halting problem seems kind of important. Also, at least two of Hilbert's problems -- the continuum hypothesis and solving general Diophantine equations -- turned out to be undecidable (the latter because it reduces to the halting problem.) Edit: Er, actually, the Diophantine equation question would be uncomputable (like the halting problem), not undecidable. The ideas are closely related, though.
- dmoney 16y agoI meant statements which state that the statement itself is unprovable. Or are you saying that the halting problem exists because of incompleteness?