Wednesday, November 29, 2006
Lesson Plan #1. Relationships: Analogy and Resemblance
Relationships: analogy, resemblance
In education, we teach you what kind of assumptions you can make.
Everyone has seen the kid who sees a bird and then thinks that he can fly.
This is false.
Unless he can generate sufficient lift to overcome gravity. Orbits, bernouli, etc.
Now: correspondence. When are two things equal? Equivalent: right and left hand.
Cardinal number (e.g. 5 or infinity) natural numbers vs. transfinite cardinal numbers
Name an infinite set: the set of all natural numbers
Denumberably infinite set is given a name: aleph-null.
Other examples: all even numbers (show correspondence)
All pairs of natural numbers; [therefore rational fractions]
Arranged diagonally: ((1,1)>(2,1)>(1,2)>(1,3)>(2,2)>(3,1)>(4,1))
Aren’t all infinite sets denumerably infinite then?
No: real numbers
Start simple: between 0 and 1
Make a correspondence: 0.20746….=1
Until you have a countably infinite list. But now construct a new decimal whose first digit is different from 1, whose second is different from 2, whose third is different from 3, etc. Continue until you have a decimal that is different from the infinite number already given. Obviously you could do this at least 8 other times, and then a new order would suffice to let you do it again, and again, and again…
Thus the set of natural numbers and the set of real numbers are not equivalent
The set of real numbers is called transcendental numbers. It is equivalent both to a line (or any size) and the plane(!), or a space of any dimension.
A new tool for making bigger infinities: The set of all subsets.