1. Turing’s halting problem Danny Brown

2. Barber paradox

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

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

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

6. Barber program …so what’s the conclusion?

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

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

9. Halting program TEST(test)…?

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

11. Halting program …so what’s the conclusion?

12. Cantor’s diagonal argument #10.3521… #20.1150… #30.6241… #40.5388… ………………… …

13. Cantor’s diagonal argument #10.3521… #20.1150… #30.6241… #40.5388… ………………… …

14. Cantor’s diagonal argument #10.3521… #20.1150… #30.6241… #40.5388… ………………… #N+10.1211…

15. Continuum Hypothesis

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