The Continuum Hypothesis:
A Look at the 20th Century History of the Real Numbers from Cantor to Cohen/Scott/Solovay.

Initial Questions (pre-test)

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 a0  +  a1x + a2x2 + ... + anxn = 0 where a0 , a1 , a2 , ... , an.are integers, not all 0.

• examples:  3/5  [(-3) +5x = 0] ;   sqrt(2)   [ (-2) + x2  = 0]  ; sqrt(-1)   [ 1 + x2  = 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
1. natural numbers
2. integers
3. rational numbers
4. algebraic real numbers
5. transcendental real  numbers
6. real numbers