Axiomatic Systems Proving 1+1=2 Russell's Paradox

1. cs1120 Fall 2011
Class 34:
David Evans Proving
11 - 11 - 11
Unprovability

2. Plan
Explanation of PS8 Options
Unprovability
Any questions about the interpreter?
Comments about Ivan’s lecture Wednesday
2

3. 3

4. Problem Set 8
Option J: Option C: Option W:
Aazda (Charme) Conveying Web
Interpreter in Computing Application
Java + static type
checking
Only Option J automatically satisfies the
prerequisite for taking cs2110 this Spring.
If you indicated interest in Computer
Science major on your PS0 survey you are
expected to do Option J. If you prefer to
cs2110: do a different option, must provide a
Software Development Methods convincing reason why.
4

5. Meta-Circularity?
Much of the course so far:
Getting comfortable with recursive definitions
Learning to write programs to do (almost) anything
Starting today and next week:
Getting un-comfortable with recursive definitions
Things no program can do!

6. Computer Science/Mathematics
Computer Science (Imperative Knowledge)
Monday
Are there (well-defined) problems that
cannot be solved by any procedure?
Mathematics (Declarative Knowledge)
Today
Are there true conjectures that cannot be the
shown using any proof?

7. Mechanical Reasoning
Aristotle (~350BC): Organon
Codify logical deduction with rules of inference
(syllogisms)
Every A is a P
Premises
X is an A
X is a P Conclusion
Every human is mortal.
Gödel is human.
Gödel is mortal.

8. Euclid (~300BC): Elements Newton (1687):
Reduce geometry to a few Philosophiæ Naturalis
axioms and derive the rest by Principia Mathematica
following rules Reduce the motion of objects
(including planets) to following
axioms (laws) mechanically

9. Mechanical Reasoning
1800s – mathematicians work on codifying
“laws of reasoning”
Augustus De Morgan (1806-1871)
George Boole (1815-1864) De Morgan’s laws
Laws of Thought proof by induction

10. Bertrand Russell (1872-1970)
1910-1913: Principia Mathematica
(with Alfred Whitehead)
1918: Imprisoned for pacifism
1950: Nobel Prize in Literature
1955: Russell-Einstein Manifesto
1967: War Crimes in Vietnam
Note: this is the same Russell who wrote In Praise of Idleness!

11. When Einstein
said, “Great spirits
have always
encountered violent
opposition from
mediocre minds.”
he was talking about
Bertrand Russell.

12. All true statements
about numbers

13. Perfect Axiomatic System
Derives all true
statements, and no false
statements starting from a
finite number of axioms
and following mechanical
inference rules.

14. Incomplete Axiomatic System
incomplete
Derives
some, but not all true
statements, and no false
statements starting from a
finite number of axioms
and following mechanical
inference rules.

15. Inconsistent Axiomatic System
Derives all true
statements, and some false
statements starting from a
finite number of axioms
and following mechanical
inference rules.
some false
statements

16. Principia Mathematica [1910]
2000 pages
Attempted to
axiomatize
Alfred Whitehead Bertrand Russell
mathematical
(1861-1947) (1872-1970) reasoning
Claimed to be complete and consistent:
All true theorems could be derived
No falsehoods could be derived

17. Proving 1+1 = 2
17

18. More Understandable Proof
Define the natural numbers
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19. More Understandable Proof
Define the natural numbers
Peano’ s Postulates:
N is the smallest set satisfying these postulates:
P1. 1 is in N .
P2. If x is in N , then its "successor" (succ x) is in N .
P3. There is no x such that (succ x) = 1.
P4. If x is not 1, then there is a y in N such that (succ y) = x.
P5. If S is a subset of N , 1 is in S, and the implication
(x in S=> (succ x) in S) holds, then S=N.
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20. Proving 1+1 = 2
N is the smallest set satisfying
Define +: N × N  N these postulates:
1. 1 is in N .
2. If x is in N , then its "successor"
(succ x) is in N .
3. There is no x such that (succ x) =
1.
4. If x is not 1, then there is a y in
N such that (succ y) = x.
5. If S is a subset of N , 1 is in
S, and the implication (x in S=>
(succ x) in S) holds, then S=N.
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21. Proving 1+1 = 2
N is the smallest set satisfying
Define +: N × N  N these postulates:
1. 1 is in N .
2. If x is in N , then its "successor"
Call the inputs a and b. (succ x) is in N .
3. There is no x such that (succ x) =
1.
If b is equal to 1: 4. If x is not 1, then there is a y in
(+ a b) = (succ a) N such that (succ y) = x.
5. If S is a subset of N , 1 is in
Otherwise: S, and the implication (x in S=>
by P4, there exists (succ x) in S) holds, then S=N.
c such that b = (succ c)
(+ a b) = (succ (+ a c))
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22. Now the Proof!
Definition of (+ a b):
If b is equal to 1:
(+ a b) = (succ a)
Otherwise:
by P4, there exists
c such that b = (succ c)
(+ a b) = (succ (+ a c))
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23. Now the Proof!
“2” = (succ 1) Definition of (+ a b):
If b is equal to 1:
1 + 1 = (succ 1) (+ a b) = (succ a)
By definition of +, Otherwise:
by P4, there exists
1 + 1 = (succ 1) = “2” c such that b = (succ c)
(+ a b) = (succ (+ a c))
QED!
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24. Bertrand Russell, My Philosophical Development, 1959
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25. Russell’s Paradox
Some sets are not members of themselves
e.g., set of all Jeffersonians
Some sets are members of themselves
e.g., set of all things that are non-Jeffersonian
S = the set of all sets that are not members of
themselves
Is S a member of itself?

26. Russell’s Paradox
S = set of all sets that are not members of
themselves
Is S a member of itself?

27. Russell’s Paradox
S = set of all sets that are not members of
themselves
Is S a member of itself?
If S is an element of S, then S is a member of itself
and should not be in S.
If S is not an element of S, then S is not a member
of itself, and should be in S.

28. Ban Self-Reference?
Principia Mathematica attempted to resolve this
paragraph by banning self-reference
Every set has a type
The lowest type of set can contain only
“objects”, not “sets”
The next type of set can contain objects and sets of
objects, but not sets of sets

29. Russell’s Resolution (?)
Set ::= Setn
Set0 ::= { x | x is an Object }
Setn ::= { x | x is an Object or a Setn - 1 }
S: Setn
Is S a member of itself?
No, it is a Setn so, it can’t be a member of a Setn

30. Epimenides Paradox
Epidenides (a Cretan):
“All Cretans are liars.”
Equivalently:
“This statement is false.”
Russell’s types can help with the
set paradox, but not with these.

31. Gödel’s “Solution”
All consistent axiomatic formulations of
number theory include undecidable
propositions.
undecidable: cannot be proven either
true or false inside the system.

32. The Information, Chapter 6
Kurt Gödel
Born 1906 in Brno (now
Czech Republic, then
Austria-Hungary)
1931: publishes Über formal
unentscheidbare Sätze der
Principia Mathematica und
verwandter Systeme (On
Formally Undecidable
Propositions of Principia
Mathematica and Related
Systems)

33. 1939: flees Vienna
Institute for Advanced
Study, Princeton
Died in 1978 –
convinced everything
was poisoned and
refused to eat

34. Charge
Today:
Incompleteness: there are theorems that
cannot be proven
Monday
Uncomputability: there are problems that
cannot be solved by any algorithm
Wednesday: PS7 Due