Turing’s halting problem Danny Brown

Barber paradox

Barber program BARBER(p) If SHAVE(p,p) = N Then SHAVE(barber,p) = Y Else SHAVE(barber,p) = N END

Barber program What happens when we compute BARBER(barber)?

Barber program BARBER(barber) If SHAVE(barber,barber) = N Then SHAVE(barber,barber) = Y Else SHAVE(barber,barber) = N END

Barber program …so what’s the conclusion?

Halting program TEST(p) If HALT(p,p) = N Then HALT(test,p) = Y Else HALT(test,p) = N END

Halting program TEST(p) If HALT(p,p) = N Then HALT(test,p) = Y STOP Else LOOP END

Halting program TEST(test)…?

Halting program TEST(test) If HALT(test,test) = N Then HALT(test,test) = Y STOP Else LOOP END

Halting program …so what’s the conclusion?

Cantor’s diagonal argument # … # … # … # … ………………… …

Cantor’s diagonal argument # … # … # … # … ………………… …

Cantor’s diagonal argument # … # … # … # … ………………… #N …

Continuum Hypothesis

Can’t be disproved (Godel 1940) …and can’t be proved (Cohen 1963)