Chris Discenzauncountable@gmail.com

Size Matters, Even for InfinityHow Big is Your Infinity?Infinity: Not Just a Midsize SedanInfinity: You Can’t Get There from Here! Infinity and BeyondInfinity: It can be Really Small, too.

Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language, and forthwith it means something entirely different.

Goethe 

Caution

Caution

The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell.

St. Augustine

The Joy of Sets• A Set is a collection of objects with a rule that

specifies how the object was chosen to be in that set.

• Example:Set of all drinks at Angels and Kings is

Where A is the set of all objects at Angels and Kings and b can stand for beer bottles or wine glasses or flask in a paper bag.

}:{ beveragealcoholicanisbAbB ∈=

Counting and Cardinality• Suppose we wanted to

know how many people are in this bar.

• We wouldn’t want to count them because they keep moving around.

• Stop moving! I’m trying to count you.

• Instead, start handing each person a card.

You may need more than one deck.

What does size mean?• One to one implies the set are the same size• Many to one means one is bigger

Example: drinks to people (you)

• Fingers: even and odd

• Conclusion: Half is less than the whole.

• However, the size of the even numbers is equal to size of all numbers.

• Corollary: You can’t clone yourself by plucking every other cell from your body.

Many to one One to one

Paintings by some famous unknown artist

Hilbert’s Hotel

• Infinitely many rooms

• All rooms are occupied.

• Can they accommodate you?

• How about infinitely many friends to come stay at the hotel.

• How are they accommodated?

What about not whole numbers?• Mathematically speaking these are called fractions.• But we’re at a bar so let’s call them rational numbers.• How many are there? • Let’s start counting them from the beginning. The first

one is 0 then…According to

Archimedes, there is not a next rational

number.

• There are infinitely many rationals between 0 and 1

• To name a few: 1/2 1/3 1/4 1/5…1/100 …1/1,000,000,000,000

• Or if you prefer decimals .5 .33333333… .25 .2 .01 .000001

• Between any two rational numbers, you can find infinitely more.

• This suggests that the size of the rationals should be much bigger than the set of natural numbers.

Let’s try to visualize this.

Sorry.

Georg Cantor 1845 - present

Why don’t we go back to my place

and talk about transfinite induction?

Sets of numbers

Suppose that I am a complete list of the real numbers.

The a,b,c’s are the digits of the decimal numbers. For instance, one of them could be 2.71828183... or 4.000000... Now consider the real number:

This is the real number that consists of all the diagonal digits in our above list.

But now let us change each of these digits to another number.

For example if the number

is 2.718..., then change the 2 to any other number like 6.

Change the 7 to 3, the 1 to 5 and so forth for the rest of the decimal places.

Then this example would give us the real number 6.354...

K718.2

K354.6bbbb

bbbbThis number is different from all the others in the above list.

But that means we have found a leftover real number!

This suggests that there must be more real numbers than natural numbers.

Even though there are infinitely many of both, somehow the real numbers are bigger in size than the natural numbers.

What does this mean?

We have two sets with infinitely many things in them, yet one set is bigger than the other.

So in other words, there are different sizes of infinity!

So what has two thumbs and is greater than the set of real numbers?

<This Guy!

and

The set of all subsets of the real numbers: the power set of R.

The set of all functions: )(xfy =

ℜ

Infinity and Beyond?

The Continuum Hypothesis

ℜ<<ℵ ?0

...210 <ℵ<ℵ<ℵ

Keith Delvin, The Joy of Sets.John Yarnelle, An Introduction to Transfinite Mathematics.Various Disreputable Internet Sites.Erin Thompson

References

• The Muller-Lyer illusion is one of the most famous optical illusions. It was created by German psychiatrist Franz Muller-Lyer in 1889. The Muller-Lyer illusion consists of two arrow-like figures, one with both ends pointing in, and the other with both ends pointing out. When asked to judge the lengths of the two lines, which are equal, viewers will typically claim that the line with outward pointing arrows is longer.

Unfortunately in 1891 a young psychiatrist called Funk Fritz-Gruber invented an illusion of much greater significance, Fritz-Gruber invented the worlds first physical illusion, he took Muller-Lyer's illusion and made it so that even if you measured the lines they would appear to be of different sizes, but really they are both the same size.Muller-Lyer was devastated and dedicated the rest of his life to working out how Fritz-Gruber's illusion worked.

Sire, I had no need of that hypothesis.-Pierre-Simon Laplace