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Turing’s halting problem Danny Brown

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Barber paradox

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

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

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

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Barber program …so what’s the conclusion?

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

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

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Halting program TEST(test)…?

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

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Halting program …so what’s the conclusion?

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

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

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

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Continuum Hypothesis

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

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