dinsdag 27 januari 2015

Two ways of "counting infinity"

Do these sets have the same size? :

{1, 2, 3} and {1, 4, 9}

Is your answer the same if we extend the sets indefinitely, like this? :

{1, 2, 3, 4, ...} or {1, 4, 9, 16, ...}

Probably you determined the size by counting the number of elements in answering the first question. The sets {1, 2, 3} and {1, 4, 9} have each 3 elements so the answer to the first question is: yes, the two sets have the same size.

How to count the two infinite sets of the second question? This is impossible using conventional 1, 2, 3, --- counting. It would not end. Mathematician Cantor therefore proposed a method to find out whether two infinite sets have the same size, a way of “counting infinity”:

If we can pair all the elements of two sets than they have the same size. 

Because we can pair like this: {1, 1}, {2, 4}, {3, 9}, {4, 16}, ... Cantor's answer to the second question is the same as to the first: yes, the two sets have the same size. His method also works for finite sets, by the way. As such it is a generalization of the finite case.

But wait... something strange is going on. The set {1, 4, 9, 16, ...} is a proper sub set, a part of {1, 2, 3, 4, …} (not the same, hence “proper”). Nevertheless they have the same size. How can that be?

A mathematician's response might be that we should not speak of the size of a set but of its cardinality instead. That's the word Cantor introduced. This makes clear that we are dealing with the mathematician's world which differs from the ordinary world. In the ordinary world proper sub sets of a set are always smaller than the set - and infinite sets cannot be counted in this world. In mathematics proper sub sets of an infinite set can have the same cardinality as that set. Not the same size, which is just an everyday, non mathematical word.

So in the mathematical world it seems that we gain something - cardinality as a measure for the size of sets - and we loose something: that proper sub sets of a set are always 'smaller' than that set.

This is a kind of friction I was uneasy with for decades. I realized myself all too well the difference between 'size' and 'cardinality' of above and especially the more or less arbitrary choice of Cantor's pairing principle. Would another, maybe better choice be possible?

It felt as an impasse to me but probably not to William Byers, the author of the book 'How mathematicians think - using ambiguity, contradiction, and paradox to create mathematics'. His main point is that ambiguity etc. - things that somehow do not feel right - stimulate the formation of new mathematical ideas. Difficulties advance mathematics.

This is not an idea pertaining exclusively to mathematics. For example, koans are the Zen equivalent given by zen masters to their pupils. These are paradoxical puzzles like: “"What is the sound of one hand clapping?" To solve these one has to widen one's view of the world. A poetic, musical analog is this line from Leonard Cohen's song "Anthem": “There is a crack in everything. That's how the light gets in.” Imperfections are necessary for advancement, seems to be the suggestion.

Back to the original problem regarding infinity. How was it solved? About 100 years after Cantor the theory of “numerosities” was proposed by Vieri Benci. Like cardinality the numerosity is a measure for the size of a set. An important difference is that proper sub sets of a set have always a smaller numerosity than that set (while the cardinality can be the same).

Some more information can be found here: http://www.newappsblog.com/2014/03/counting-infinities.html. From the discussion there it appears to me that numerosity is a more complicated concept than cardinality – but I can be wrong.

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