Survey of Math Quiz 2
If S has N elements than
P(S) has 2^n elements
N <
PN < PPN < PPPN....
Is this true for the infinite case? n < 2^n
YES! For any set (S) the power set P(S) is always bigger
Are there more levels of infinity than there are in a particular set?
Yes! Always
An infinite list is
a 1-1 correspondence
finite set
a part is always smaller than a whole
A set
any well defined collection of objects
The standard axioms of mathematics
aren't strong enough to settle the question of the Continuum Hypothesis
Rational numbers are
countable and have a 1-1 correspondence
definition of a 1-1 correspondence
everything in set a corresponds to one and only one element in set b
Cantor's diagonal method
1 changes to 2 everything that's not 1 changes to 1 The diagonal number is 0.09036 Anti-diagonal is x= 0.11111...
What following sets have the same infinite size
1) Natural numbers 2) Any infinite subset of N 3) All integers 4) Rational numbers
George Canton
19th/20th century German Mathematician Agreed with the second horn- there are equal amounts of perfect squares and numbers Rejected the first horn saying that just because not ALL numbers are perfect squares does NOT prove there are less perfect squares than numbers
Is L1 L2 the same size?
Not in length, but yes in cardinality (same number of points)
Continuum Hypothesis
Cantor conjectured that R=2nd level of infinity
Cantor's Method to find a set NOT on the list
Find C = all words NOT in a corresponding set
S and N can be put into a 1-1 correspondence
IF S can be put into an infinite line or is an infinite subset of N
L1=
L2 in cardinality
Rational numbers in between 0 and 1 are the same size as
N
many infinite sets have the same cardinality (size) as
N
R's are bigger than
N's. The Diagonal argument
Can an anti-diagonal be on the list?
NO
Can S and P(S) be put in a 1-1 correspondence?
NO!
Can all infinite sequences of colored circles and N be put into a 1-1 correspondence?
NO!
Is N and R the only 2 types of infinity?
NO! Any infinite set always has a larger set
Can R (real numbers) be put into a 1-1 correspondence with N?
NO! R is unlistable
Infinitely many rational numbers between two integers - is this a higher type of infinity?
NO! Rational numbers an be put into a 1-1 correspondence with N
Is X rational?
NO! X is IRRATIONAL
galileo's paradox
first horn- most numbers aren't perfect squares (seems to be there are less perfect squares than numbers) Second horn- every number has a square (seems to be equal amount of squares and numbers)
An infinite set can be put into a 1-1 correspondence with N
if and only if you can list the elements
Anti-diagonal
if it differs from the diagonal in each digit
we use a 1-1 correspondence to compare
infinite sets
between any two integers there are
infinitely many rational numbers (0-1)
There are infinitely many levels of
infinity
George Canton
proved that not all infinite sets have the same carndality (1-1 correspondence) R and N can't be put into a 1-1 correspondence.
Kurt Godel
proved the Continuum Hypothesis could not be DISPROVED using 'the standard axioms of mathematics'
Paul Cohen
proved the Continuum Hypothesis could not be PROVED using 'the standard axiom of mathematics'
C
set of all elements NOT in their corresponding subset
basic principle
sets a and b can be put into a 1-1 correspondence if they have the same number of elements (same size)
cardinality
the number of elements in a set
gaileo's conclusion
the qualities "equal, greater, or less" CAN ONLY be applied to the finite and NOT the infinite
any infinite subset is always
the same size as N
The subset of N will always be
the same size as any smaller type of infinity
N represents
the smallest type of infinity
Hilbert's Hotel
we can always fit in one more person by moving guests around NOT by kicking them out
Intuitive - proof
when the details need to be verified