Infinity (An excellent place to end)

Significant events:

Aristotle: Potential infinity. Infinity that could in principle be, but in practice never was.
Galileo:  Sowed the seeds of all that was to come (1600)
Cantor:    Put infinity on a firm logical foundation (1900)


Three kinds of infinity: Religious, Mathematical, Physical

Truth table: Philosopher minterms
Aristotle minterm   000
Russel    minterm   010
Cantor    minterm   111
Aquinus   minterm   100
Bob         minterm   X1X       (don't cares, not a philosopher anyway)


Symbol   : John Wallis around 1650 (Inventor of Integral Calculus)

Early romans used a similar symbol for 1000, a big number.

Galilao: Came closest to our current notion of infinity.
Considered the question whether there were more integers than squares.

The squares are certainly less dense than integers.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ...

Amazingly in 15something: Galilao's argument "every number has a square, we can pair the first square with the first number, the second with the second number, and so on. We shall never run out of squares until we run out of numbers, since we will never run out of numbers we will never run out of squares. Thus showing we have "as many" squares as numbers."

"I see no other decision, but to say that all Numbers are infinite; Squares are infinite; and that neither is the multitude of Squares less than  all Numbers, nor this greater than that; and in conclusion, that the
Attributes of Equality, Majority, and Minority have no place in Infinities, but only in terminate quantities"

300 years later 1880: Georg Cantor developed a precise
definition of an infinite set.

"An infinite set, is one that can be placed into one to one correspondence with a proper part of itself".

or

Some of the parts is equal to the whole :)

This required a significant amount of mental and physical persistence as it tends to go against an axiom of Euclid, where, the "whole is greater than the part".  Cantor's work around was that the axiom did not apply to infinite quantities.

This leads immediately to seeing that the number of rational points on a line 1 inch long are the same as the number of points on a line of any length.


Some of Cantor's arguments are the most elegant for infinite sets.

Here is a proof that rational numbers are "equal" to the integers. That is, can be placed in one to one correspondence with the integers.

Write the fractions with the sum of the numerator and denominator equal to 2, i.e 1/1

Write the fractions with the sum of the numerator and denominator equal to 3, i.e 1/2 and 2/1

Write the fractions with the sum of the numerator and denominator equal to 4, i.e 1/3, 2/2, 3/1
etc. (eventually you get all rational numbers)

Write them out as 1/1 1/2  2/1 1/3 2/2 3/1  ... which has a one to one correspondence with the integers. Done!



Cantors proof of an infinity larger than the counting numbers.

Consider the infinite fraction (real numbers that are not rational)
Suppose they are lined up with the counting numbers. Proof by contradiction.
1   0.435783425708765809...
2   0.573456627669679867...
3   0.674354615645652615...
4   0.567545645789431432...
5   0.454433443466675668...
6   0.676575645648986564...
7   0.122135467619578989...
etc.

You can find another by writing a number that is different from the above along the diagonal.

Candidate: 0.3467298.... will not show up in any list paired with the integers as it is different than all of them.

The infinity of real numbers is "stronger" than the infinity of the integers.

The set of of decimal fractions is non-denumerable.

Cantor demonstrated many other uncommon notions.
For example: The number of points in the plane is the same as the number of points on a line.  Pick a point on a unit square.

e.g. (0.34534543522... , 0.65453454325..)
The corresponding point on the unit line.
0.364554354354453453222... one number from each of the above shuffled.
Obviously, the reverse is a similar mapping. Establishing a one to one correspondence.

etc. to higher dimensions!  Obviously counter intuitive.


Alephzero,  first transfinite cardinal number, a cardinal number about infinite sets (positive integers).

Cantor stated i) that this was the smallest transfinite number and ii) for every transfinite number there exists a next larger transfinite cardinal.

Cantor called the transfinite cardinal number associated with the reals c, "the number of the continuum".  He knew it was Alephsomething.
 Symbolically, the continuum hypothesis is that Alephone = c.

Gödel's and Cohen's established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable

The next largest transfinite cardinal is Alephone.

A generalized version of the continuum hypothesis is 2alepha = alepha+1, which is also undecidable.

Ignoring the undecidability, here is a simple analog.

One final existence proof of Cantor's
(penultimate actually)

What do we know? We know there are Integers, Rationals, Reals. We know there are more Reals than Integers. We know there are algebraic numbers, id est, numbers  that are solutions to algebraic equations, such as  x2 = 2.  We know  there are numbers that are not algebraic, like e and pi. How do these  transcendental numbers  compare with  the  algebraic  numbers?  That is the question?

Are there more algebraic numbers like root2 or more numbers like pi?

Cantor's existence proof of transcendental numbers.

For an algebraic equation define the height h of the equation as the sum of the magnitude of the coefficients and the equation's degree n.

This is clearly a finite number.  n is clearly less that h. The number of algebraic equations of height h = (h+1)h+1. These can be placed in one to one correspondence with the the Integers. Each equation of degree n can have at most n solutions. These solutions (algebraic numbers) can be listed again in a one to one correspondence with the Integers, and omitting duplicates is a one to one mapping. Thus the algebraic numbers are Aleph-zero.

However the continuum c is greater that Aleph-zero. c is made up of algebraic and transcendental, hence the transcendentals are the same as c, or Aleph-one.

Amazingly although we only know of a handful of transcendentals, such as e and pi. If you were to pick a real number at random from c it would most likely with extremely high probability be transcendental.


Final Thoughts: What about numbers that are not computable?
Quiz 4: Are there more computable numbers like 5, root2, pi, e than non computable numbers?

As it turns out the computable numbers are denumerable, id est, countable, id est, Aleph-zero. The non computable are of the same size as the continuum c, likely Aleph-one.

Amazingly although we only know lots of computable numbers, such as 5, root2, e and pi. If you were to pick a real number at random from c it would most likely with extremely high probability be non computable.

That brings us to the end, thank you for your time.
References:
  • "Number", Tobiaz Dantzig, 4th Edition , 1954. (Masterpiece Science Edition, 2005)
  •  The web of course.
  • "From Zero to Infinity", Constance Reid, 1955. (50th Anniversary Edition, 2005)
  • "The Infinite Book", John D. Barrow, 2005, not as good, but British.
  • "One, Two, Three, Infinity", George Gamow, 1961 Edition, 1988 Dover.
  • "Concepts of Modern Mathematics",  Ian Stewart Dover 1995 edition 


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