Gödel's Completeness Theorem
Power & Limits of Logic
Three Profound Theorems
about
Mathematical Logic
Thm 1, good news:
only need to know* a few axioms
& rules, to prove all validities.
*Theoretically only: having more
rules is convenient.
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.1
Axioms & Inference Rules
September 16, 2005
lec 2F.3
Copyright ©Albert R. Meyer, 2005.
September 16, 2005
lec 2F.4
Three Profound Theorems
Thm 3, Worse News:
if we stick to domain, N, with
predicates x + y = z,
x · y = z,
We won't prove these Theorems.
Their proofs usually require half
a term in an intro logic course
after 6.042.
then no proof system can
prove all the true assertions!
September 16, 2005
lec 2F.2
Thm 2, Bad News: there is no
procedure that determines when
quantified assertions are valid
(in contrast to propositional
formulas).
Gödel's Incompleteness Theorem
for Arithmetic
Copyright © Albert R. Meyer, 2005.
September 16, 2005
Cannot Determine Validity
Namely, starting from a few
propositional & simple
predicate validities, every valid
assertion can be proved using
just UG and modus ponens
repeatedly!
Copyright © Albert R. Meyer, 2005.
Copyright ©Albert R. Meyer, 2005.
lec 2F.5
Copyright ©Albert R. Meyer, 2005.
September 16, 2005
lec 2F.6
1
Sets
Mathematics for Computer Science
MIT 6.042J/18.062J
Informally,
a set is a collection of mathematical
objects with the collection treated as
a single mathematical object.
This is circular of course: what’s a
collection?
Sets & Functions
Copyright © Albert R. Meyer, 2005.
lec 2F.7
September 16, 2005
Copyright ©Albert R. Meyer, 2005.
Some sets
lec 2F.8
September 16, 2005
Some sets
Real numbers,
R
complex numbers, C
integers, Z
∅
the emptyset,
{7, “Albert R.”, π/2, Τ}
A set with 4 elements: two numbers,
a string, and a Boolean value.
Same as
the set
of
all sets
of integers , pow(Z)
{“Albert R.”,7,Τ, π/2}
the power set
-- order doesn’t matter
Copyright © Albert R. Meyer, 2005.
lec 2F.9
September 16, 2005
Copyright ©Albert R. Meyer, 2005.
Membership
x∈ A
Membership
x∈ A
x is an element of A
π/2 ∈ {7, “Albert R.”,π/2, Τ}
π/3 ∉ {7, “Albert R.”,π/2, Τ}
14/2 ∈ {7, “Albert R.”,π/2, Τ}
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.10
September 16, 2005
lec 2F.11
x is a member of A
shorter: “x is in A”
7∈Z
Copyright © Albert R. Meyer, 2005.
2/3 ∉ Z
September 16, 2005
Z ∈pow(R)
lec 2F.12
2
In or Not In
Containment
A⊆ B
{7, π/2, 7} is same as {7, π/2}
An element, like 7, is in a set,
or not in the set.
(No notion of being in the set
more than once)
A is a subset of B
A is contained in B
Every element of A is also an
element of B.
Z⊆R, R⊆C, {3}⊆{5,7,3}
∅⊆ every set
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.13
Copyright © Albert R. Meyer, 2005.
Team Problem
lec 2F.14
September 16, 2005
Defining Sets
The set of elements, x, in A
Problem 1
Copyright © Albert R. Meyer, 2005.
September 16, 2005
such that P(x) is true.
{x ∈ A | P( x)}
lec 2F.15
Copyright © Albert R. Meyer, 2005.
lec 2F.16
September 16, 2005
New sets from old
Defining Sets
The set of even integers:
{n ∈ Z | n is even}
A
{n ∈ Z | ∃m ∈ Z. n = 2m}
Copyright © Albert R. Meyer, 2005.
September 16, 2005
B
Venn Diagram for A and B
lec 2F.17
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.19
3
union
A
intersection
A
B
A ∪ B ::= {x | ( x ∈ A) ∨ ( x ∈ B)}
Copyright © Albert R. Meyer, 2005.
lec 2F.20
September 16, 2005
B
A ∩ B ::= {x | x ∈ A ∧ x ∈ B}
Copyright © Albert R. Meyer, 2005.
set difference
complement
D −A known
A=
A
domain D
B
A
A − B ::= {x | ( x ∈ A) ∧ ( x ∉ B)}
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.22
A ::= {x ∈ D | x ∉ A} = D − A
Copyright © Albert R. Meyer, 2005.
power set
pow({a, b}) = {{a, b} , {a} , {b} , ∅}
September 16, 2005
September 16, 2005
lec 2F.23
Quickie
pow ( A) ::= {S | S ⊆ A}
Copyright © Albert R. Meyer, 2005.
lec 2F.21
September 16, 2005
lec 2F.24
What is Pow(∅)?
Ans: {∅}
What is Pow(Pow(∅))?
Ans: {{∅}, ∅}
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.25
4
Russell’s Paradox
Russell’s Paradox
Let W ::= {S ∈ Sets | S ∉ S }
So the collection, Sets, of all
sets, cannot be considered a
set!
so S ∈ W ↔ S ∉ S
Let S be W and reach a
contradiction:
W ∈W ↔ W ∉W
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.26
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.27
f :A→ B
Mathematics for Computer Science
MIT 6.042J/18.062J
function, f, from set A to set B
associates an element, f ( a ) ∈ B
with an element a ∈ A.
Functions
Example: f
is the string-length
function: f(“aabd”)=4
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.31
Copyright © Albert R. Meyer, 2005.
f :A→ B
September 16, 2005
lec 2F.32
g :R2 → R
1
x− y
domain(g) = all pairs of reals
g ( x, y ) =
f is the string-length function.
A, the domain of f,
is the set of strings.
B, the codomain of f, is N
Copyright © Albert R. Meyer, 2005.
September 16, 2005
codomain(g)= all reals
But g is partial:
not defined on all pairs of reals
lec 2F.33
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.34
5
g′ : D → R
g ′( x, y ) =
Total functions
1
x− y
f : A → B is total
iff every element of A is
assigned a B-value by f
where D = R − {( x, y ) | y = x}
Then g´ is total:
defined on all pairs in domain D.
2
Copyright © Albert R. Meyer, 2005.
September 16, 2005
∀a ∈ A ∃b ∈ B. f (a ) = b
lec 2F.35
Copyright © Albert R. Meyer, 2005.
September 16, 2005
Onto functions
1-1 functions
f : A → B is onto
f : A → B is 1-1
iff every element of B is
f of something
iff every element of B is
f of at most 1 thing
∀b ∈ B ∃a ∈ A. f (a ) = b
a surjection
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.37
Bijections
an injection
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.38
f : A → B is a bijection iff it is
an exact correspondence:
it is all those good things:
total, onto, and 1-1
September 16, 2005
∀a, a′ ∈ A.( f (a ) = f (a ')) → (a = a ')
Bijections
f : A → B is a bijection iff
Copyright © Albert R. Meyer, 2005.
lec 2F.36
f( )=
A
lec 2F.39
Copyright © Albert R. Meyer, 2005.
B
September 16, 2005
lec 2F.40
6
Team Problem
Bijections
exactly one arrow out
exactly one arrow in
f( )=
A
Copyright © Albert R. Meyer, 2005.
B
September 16, 2005
lec 2F.41
Problem 2
Copyright © Albert R. Meyer, 2005.
September 16, 2005
lec 2F.42
7