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

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russell.1 Albert R Meyer, March 4, 2015

Mathematics for Computer Science

MIT 6.042J/18.062J

Set Theory: Russell Paradox

russell.2 Albert R Meyer, March 4, 2015

Self application

Self application is notoriously doubtful:

“This statement is false.” is it true or false?

russell.5 Albert R Meyer, March 4, 2015

Self membership

The list

L = (0 1 2) L · ·

· · ·

1 2

russell.6 Albert R Meyer, March 4, 2015

Self membership

(setcar (second L) L)

L · ·

· · ·

1 2

2

russell.7 Albert R Meyer, March 4, 2015

Self membership

(setcar (second L) L)

L · ·

· · ·

2

russell.8 Albert R Meyer, March 4, 2015

Self membership Lists are member of themselves:

L = (0 L 2)

L · ·

· ·

·

2

russell.9 Albert R Meyer, March 4, 2015

Self membership Lists are member of themselves:

L = (0 (0 (0…2) 2) 2)

·

L

·

· ·

·

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russell.11 Albert R Meyer, March 4, 2015

Self application

compose procedures (define (compose f g) (define (h x) (f (g x))) h)

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russell.12 Albert R Meyer, March 4, 2015

Self application

compose procedures ((compose square add1) 3) 16 ( = (3 + 1)2 ) ((compose square square) 3) 81 ( = (32)2 )

russell.13 Albert R Meyer, March 4, 2015

Self application

compose procedures (define (comp2 f) (compose f f)) ((comp2 square) 3) 81

russell.14 Albert R Meyer, March 4, 2015

Self application

apply procedure to itself: (((comp2 comp2) add1) 3) 7 (((comp2 comp2) square) 3) 43046721 (= 316)

russell.15 Albert R Meyer, March 4, 2015

Russell’s Paradox

Let W :: s S = ∈ ets| ∉ s s

{ }

Now let s be W, and reach a contradiction: ∉

IFF

s s W so s   ∈   ∉ ∈ W W IFF ∉ W W     ∉

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russell.16 Albert R Meyer, March 4, 2015

Disaster: Math is broken!

I am the Pope, Pigs fly, and verified programs crash...

russell.17 Albert R Meyer, March 4, 2015

s ⎤ ⎦

...but paradox is buggy

Assumes that W is a set!

⎡s ∈W IFF ∉ ⎣ s

for all sets s …can only substitute W for s if W is a set

russell.18 Albert R Meyer, March 4, 2015

...but paradox is buggy

Assumes that W is a set! We can avoid the paradox, if we deny that W is a set!

…which raises the key question: just which well-defined collections are sets?

russell.19 Albert R Meyer, March 4, 2015

Zermelo-Frankel Set Theory

No simple answer, but the axioms of Zermelo-Frankel along with the Choice axiom (ZFC) do a pretty good job.

Assumes that W is a set! Assumes that W is a set!

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