Transcript The P=NP problem

Logic and Set Theory
Actual infinity
Aristotle distinguished actual vs potential infinities
actual infinity: elements exist together simultaneously
potential : elements exist only consecutively over time.
In mathematics, actual infinity is the notion that all
numbers (natural, real etc.) can be enumerated in
some sense sufficiently definite for them to form a set.
• the abstraction of actual infinity involves the
acceptance of infinite entities, such as the set of all
natural numbers as given objects
• The term "actual" in is synonymous with definite,
completed, extended or existential, but not to be
mistaken for physically existing.
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• Gauss’ (1831) summary of common view: * I protest against the
use of infinite magnitude as something completed, which is never
permissible in mathematics. Infinity is merely a way of speaking,
the true meaning being a limit which certain ratios approach
indefinitely close, while others are permitted to increase without
restriction..
• The drastic change was initialized by Bolzano and Cantor in the
19th century
• Bolzano (c. 1830s) introduced the notion of “set” * A multitude …
with the property that every finite set … is only a part of it, I will
call an infinite multitude
• Georg Cantor (c. 1870s) developed “set theory”; distinguished 3
realms of infinity: infinity of God, of nature, and of mathematics.
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* The numbers are a free creation of human mind. (R. Dedekind)
Infinity
• In mathematical analysis, it is absolutely
essential to deal with infinite quantities in a
rigorous way.
• E.g., there are infinitely many numbers
between 0 and 1, but we can say that the
length of the interval [0,1] equals 1?
• Can we say that 0.999…=1?
• Can we say that ½+1/4+1/8+…=1?
• The natural numbers 1,2,3… are infinite
• The rational numbers are equal in number to
the whole numbers.
• The real numbers exceed the number of
Countable Uncountable sets
Real numbers and decimal
and binary representations
Equivalence of sets
• Two sets A and B are equivalent if there is
a rule that assigns to each element of A a
unique element of B
• In a sense, equivalent sets have the same
number of elements
• Continuum hypothesis: There are no sets
that have more elements than the whole
numbers and fewer than the real numbers.
Logic and Set Theory
• How do we deduce mathematical
theorems that depend on properties of
sets?
• Boole: Boolean logic
• DeMorgan: Laws of inclusion and
exclusion
Propositional calculus
• Assign truth values to “atomic”
propositions
• Calculate truth of a compound statement
based on truth values of atoms
Boole (1815-1864) and
DeMorgan (1806–1871)
• Boolean algebra
Boole (1815-1864) and
DeMorgan (1806–1871)
• De Morgan’s laws:
• not (P and Q) = (not P) or (not Q)
• not (P or Q) = (not P) and (not Q)
Modus ponens ("mode that affirms”)
Truth table version
Other logical deduction rules
Continuum hypothesis
• There is no set whose size is strictly
between that of the integers and that of
the real numbers.
• CH and AC are consistent with ZF
Logic and infinite sets
• Problems arise when we try to apply
predicate logic to statements about “all of
the members of a set”
• “Second order logic”
Russell’s paradox
• Suppose that, for any formal criterion, a
set S exists whose members are exactly
those objects satisfying the criterion
• Can’t do this IF a there is set S
containing exactly the sets that are not
members of themselves.
• If S qualifies as a member of itself, it
would contradict its own definition as a
set containing sets that are not members
of themselves.
• If S is not a member of itself, it would
qualify as a member of itself by the
same definition.
• This contradiction is Russell's paradox.
The axiom of choice (AC)
• AC says that given any collection of bins,
each containing at least one object, it is
possible to select exactly one object from
each bin, even if there are infinitely many
bins and there is no "rule" for which object
to pick from each.
The Banach–Tarski paradox states that a solid ball
in 3-dimensional space can be split into several
non-overlapping pieces, which can then be put
back together in a different way to yield two
identical copies of the original ball.
Unlike most theorems in geometry, it depends in a
critical way on the axiom of choice in set theory.
Prisoners hat problem
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Prisoners are to be lined up facing forward. Each wears a black hat, or a
white (random)
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Warden starts from the back of the line, asking each prisoner his hat color,
then moving forward one by one. Prisoners are only allowed to say "black"
or "white."
They are executed if wrong and spared if right.
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Each prisoner can hear the guesses of other prisoners (*and their outcome).
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The prisoners learn of the warden's plan and the rules described above the
night before this is to take place and can therefore strategize beforehand.
What is their best plan?
instead of a finite number of prisoners, take a countably infinite number of
them (say, one for each natural number). Any attempt to extend the
solution for the previous problem to this one results in an infinite number
of executions.
Completeness
• Kurt Gödel in 1931,
• First incompleteness theorem:
• Any consistent, effectively generated
formal theory that proves certain basic
arithmetic truths, there is an arithmetical
statement that is true, but not provable in
the theory.
• Analogy: liar paradox: "This sentence is
false."