copyright © 2014 curt hill cardinality of infinite sets there be monsters here! at least serious...

Copyright © 2014 Curt Hill

Cardinality of Infinite Sets

There be monsters here!At least serious weirdness!

Cardinality• Recall that the cardinality of a set

is merely the number of members in a set

• This makes perfect sense for finite sets, but what about infinite sets?

• We may compare the sizes of such infinite sets by attempting a one to one correspondence between the two

• It gets a little weird here and our intuition does not always help


Some Definitions• Sets A and B have the same

cardinality iff there is a one to one correspondence between their members

• Finite sets are obviously countable• The notation for cardinality is the

same as absolute value• If A = {1, 3, 4, 5, 9}

then |A|=5


Infinite Countable Sets• An infinite set, A, is countable iff

there is a one to one correspondence between A and the positive integers– We refer to this cardinality, – Last symbol is the Hebrew Aleph,

read Aleph null or aleph naught• If this is not the case then the

infinite set is uncountable– There is an hierarchy of alephs


Counter Intuitive• We would normally think that:

– If A B then |A|< |B|• This is true for finite sets but not

necessarily for infinite sets• Consider the positive even integers• It is a subset of the positive

integers• Yet it is one to one with the

positive integers– Thus is a countably infinite set and

has the similar cardinalityCopyright © 2014 Curt Hill

Countable• The function f(x) = 2x where x is a

positive integer• This a mapping from positive

integers to positive even integers• This mapping is one to one• The definition of countable is now

met for the even positives, so the cardinality of positive evens is


Hilbert’s Grand Hotel• This paradox is attributed to David

Hilbert• There is a hotel with infinite rooms• Even when the hotel is “full” we

can always add one more guest– They take room 1 and everyone else

moves down one room• This boils down to the notion that

adding one to infinity does not change infinity: +1 = – Recall that infinity is not a real

number Copyright © 2014 Curt Hill

Another Countable• The rationals are countable as well• Use a matrix of integers

– One axis is the numerator– The other the denominator– Duplicate values ignored

• In a diagonal way enumerate each rational– That is set them in one to one with

positive integers


Count ‘em


1 2 3 4 51 1/1 2/1 3/1 4/1 5/12 1/2 2/2 3/2 4/2 5/23 1/3 2/3 3/3 4/3 5/34 1/4 2/4 3/4 4/4 5/45 1/5 2/5 3/5 4/5 5/5Start at 1/1 and diagonally count each non-duplicate. 1/1 is 1, 2/1 is 2, 1/2 is 3, 1/3 is 4, 3/1 is 5 …

Very Interesting!• What we now see is three infinite

sets with the same cardinality• A is the positive evens

A Z+ Q yet |A| | Z+ | |Q|– All are

• Thus, it is hard to think about cardinality of infinite sets as exactly the same as set size

• Much more similar to big O notation• Where many details are largely

ignored


Uncountable Sets• If there are countable sets then

there must be uncountable sets• The real numbers is such an

infinite set – Cardinality

• The book supplies a proof by contradiction– This will be similar– See next slides


Reals Uncountable 0• There exists a theorem that states

0.9999… is the same as 1.0– The idea of the proof is that as the 9s

go to infinity the limit of the difference is zero

– In other words however small you want the difference between two distinct reals to be we can make the difference between these two less

• An uncountable set cannot be a subset of a countable set


Reals Uncountable 1• Assume that the reals between 0

and 1 are countable• Then there is a sequence r1, r2, r3,

…• This sequence must have the

property that ri < ri+1

• Each rn has a decimal expansion that looks like this:0.d1d2d3d4d5…where each d is a digit


Reals Uncountable 2• Next look at any adjacent pair of

reals, rn and rn+1

• These two must be different at some di

– If they are not we have numbered identicals

– We also disallow that the lower one is followed by infinite 9s and the higher one by infinite zeros which would be two representions of the same number


Reals Uncountable 3• Now rn must have a non-nine

following di call it dj

– Otherwise we violated the no identicals rule

• Create a new real rk that is rn with dj incremented by 1

• We now have rn < rk < rn+1 which contradicts our original assertion

• In fact we can insert an infinity of such numbers by incrementing the digits after dj Copyright © 2014 Curt Hill

Reals Uncountable Addendum

• In the last screen there was the argument that there must be a non-nine in the sequence

• The symmetrical argument is that there must be a non zero following the rn+1

• Since we disallowed a …9999… followed by …0000… we can shift the argument to the second rather than first


Results• If A and B are countable then AB

is also countable• If |A||B| |A||B| then |A|=|B|

– Schröder-Bernstein Theorem• The existence of uncomputable

functions– Functions that cannot be generated

by program• Continuum hypothesis

– No cardinality numbers between


Exercises• 2.5

– 1, 3, 5, 17, 23


copyright © 2014 curt hill cardinality of infinite sets there be monsters here! at least serious...

Documents