The Continuum Hypothesis:
A Look at the 20th Century History of the Real
Numbers from Cantor to Cohen/Scott/Solovay.
Initial Questions (pretest)
What is a counting number? a natural number? an integer?
Ans:

counting numbers: 1,2,3,...

natural numbers: 0,1,2,3,...

integer: ...3,2,1,0,1,2,3,
...
What is the likelihood (probability)
of choosing a counting number at random from all the integers between 10
and 10?
What is a rational number? an algebraic number? a real
number?
Ans:

rational number: k/n where k
is an integer and n is a counting number.
examples: 3/5, 2/7, 36/9

algebraic number: a solution to an
equation of the form a_{0} + a_{1}x
+ a_{2}x^{2} + ... + a_{n}x^{n}
= 0 where a_{0} , a_{1 }, a_{2}
, ... , a_{n}.are integers, not all 0.
examples: 3/5 [(3)
+5x = 0] ; sqrt(2) [ (2) + x^{2}
= 0] ; sqrt(1) [ 1 + x^{2} = 0]

real number (?) : can be expressed
as a decimal.
examples: 3/5 = .6;
3/7 = 0.42857142857142857142857142857143... ;
sqrt(2) = 1.4142135623730950488016887242097...;
ln(2) = 0.69314718055994530941723212145818...
;
pi = 3.1415926535897932384626433832795...
What is the likelihood (probability)
of choosing a counting number at random from all the rational numbers between
10 and 10?
Ans.: 0
What is the likelihood (probability)
of choosing a rational number from all the real numbers between 10 and
10?
Ans: 0
Arrange the following sets of numbers from smallest to
largest:
natural numbers 
algebraic real numbers 
integers 
real numbers 
transcendental real numbers 
rational numbers 

natural numbers

integers

rational numbers

algebraic real numbers

transcendental real numbers

real numbers