Axioms
Mathematics for Computer Science
MIT 6.042J/18.062J
Equality
∀x[x ∈ y ↔ x ∈ z] → y = z
Set Theory
Albert R Meyer,
February 19, 2010
Power set
∀x∃p∀s. s ⊆ x ↔ s ∈ p
lec 3F.1
Russell’s Paradox
February 19, 2010
February 19, 2010
I am the Pope,
Pigs fly,
and verified programs
crash...
lec 3F.3
Albert R Meyer,
February 19, 2010
...but paradox is buggy
...but paradox is buggy
Assumes that W is a set!
Assumes that W is a set!
We can avoid the paradox,
if we deny that W is a set!
for all sets s
…can only substitute
W for s if W is a set
Albert R Meyer,
February 19, 2010
lec 3F.2
Disaster: Math is broken!
Now let s be W, and
reach a contradiction:
Albert R Meyer,
Albert R Meyer,
lec 3F.4
…which raises the key question:
just which well-defined
collections are sets?
lec 3F.5
Albert R Meyer,
February 19, 2010
lec 3F.6
1
Zermelo-Frankel Set Theory
Zermelo-Frankel Set Theory
According to ZF, the elements
of a set have to be “simpler”
than the set itself. In
particular,
No simple answer, but the
axioms of Zermelo-Frankel
along with the Choice axiom
(ZFC) do a pretty good job.
Albert R Meyer,
February 19, 2010
no set is a member of itself.
lec 3F.8
Albert R Meyer,
This implies that
(1) the collection of all sets is not
a set, and
(2) W equals the collection of all
sets …which is why it’s not a set
February 19, 2010
lec 3F.10
no surjection from A to pow(A)
W::= {a ∈A | a ∉ f(a)}, so
a ∈W iff a ∉ f(a).
f a surj, so W=f(a0), some a0 ∈A.
So, a0 ∈f(a0) iff a0 ∉ f(a0).
February 19, 2010
Are infinite sets the “same size”?
NO, by Russell paradox variant:
Theorem: No surjective function
from A to pow(A),
even for infinite A
Albert R Meyer,
February 19, 2010
lec 3F.11
{0,1}ω is uncountable
A is countable iff can be
listed a0,a1,a2,….
same as surj fcn: N→A
Pf by contradiction: suppose
surj fcn f:A→pow(A). Let
Albert R Meyer,
lec 3F.9
infinite sizes
Zermelo-Frankel Set Theory
Albert R Meyer,
February 19, 2010
So {0,1}ω is uncountable, because
N → {0,1}ω → pow(N)





surj
lec 3F.12
surj
Albert R Meyer,
bij
February 19, 2010
lec 3F.13
2
Team Problems
Problems
1―3
Albert R Meyer,
February 19, 2010
lec 3F.16
3
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6.042J / 18.062J Mathematics for Computer Science
Spring 2010
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