Set Theory: doubtful: This statement is false. Russell Paradox is - PowerPoint PPT Presentation

Mathematics for Computer Science Self application MIT 6.042J/18.062J Self application is notoriously Set Theory: doubtful: “This statement is false.” Russell Paradox is it true or false? Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.1 russell.2 Self membership Self membership The list (setcar (second L) L) L = (0 1 2) L · · L · · · · · · · · 1 2 1 2 0 0 Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.5 russell.6 1

Self membership Self membership Lists are member of themselves: (setcar (second L) L) L = (0 L 2) L · · L · · · · · · · · 0 2 0 2 Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.7 russell.8 Self membership Self application compose procedures Lists are member of themselves: L = (0 (0 (0…2) 2) 2) (define (compose f g) (define (h x) · · L · · · (f (g x))) h) 2 0 Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.9 russell.11 2

Self application Self application compose procedures compose procedures (define (comp2 f) ((compose square add1) 3) (compose f f)) � 16 ( = (3 + 1) 2 ) ((compose square square) 3) ((comp2 square) 3) � 81 ( = (3 2 ) 2 ) � 81 Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.12 russell.13 Self application Russell’s Paradox { } = ∈ ets| ∉ ∉ Let W :: s S s s apply procedure to itself:   ∈ ∉ so s W s s IFF (((comp2 comp2) add1) 3)   � 7 Now let s be W, and (((comp2 comp2) square) 3) reach a contradiction: � 43046721 (= 3 16 )   ∈ ∉ ∉ W W IFF W W   Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.14 russell.15 3

...but paradox is buggy Disaster: Math is broken! I am the Pope, Assumes that W is a set! Assumes that W is a set! ⎡ s ∈ W IFF s s ⎤ ∉ Pigs fly, ⎣ ⎦ and verified programs for all sets s …can only substitute crash... W for s if W is a set Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.16 russell.17 ...but paradox is buggy Zermelo-Frankel Set Theory Assumes that W is a set! Assumes that W is a set! No simple answer, but the We can avoid the paradox, axioms of Zermelo-Frankel if we deny that W is a set! along with the Choice axiom …which raises the key question: (ZFC) do a pretty good job. just which well-defined collections are sets? Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 russell.18 russell.19 4

MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 20 15 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.