Wednesday, January 11, 2006

A Mathematical Postulate

This is probably covered elsewhere by better mathematicians than I, but it's something that I have come up with. I think that every infinite series (integers, reals, 2d co-ordinates, 3d-coordinates) can be mapped on a 1-to-1 basis with eachother. That is to say, I think there is a 1-1 mapping between the set of positive integers, and the set of positive reals.

Here is my proposed method. Let us suppose that real numbers are represented decimally, and that things like irrational numbers can be approximated with arbitrary precision decimally. That is to say, pi can be represented by 3.14159... to any accuracy.

Any decimal number can be represted uniquely by an integer.

Let every even digit count as the "whole number part" (in reverse order) and every odd digit count as the "fractional" part.

3.14159 would then be represented by 3104010509. 245.3 would be represented by 53402. 34.98765 would be represented by 4938070605.

In this way, you can see that there is a way to translate losslessly between these two number series.

However, in order to represent a decimal number to precision X, you need 2X integer digits. This number I shall call the "order" of the infinite set. The integers are order one, and the reals are order two.

Consider, then, a 2D co-ordinate system of infinite length along each axis. What you have then is an order 4 system. In order to "pack" the set into an integer mapping, you need every integer to map uniquely onto a pair of real numbers.

In this mapping, you would a sequence of four numbers. Let X and Y refer to the co-ordinate axes, wPart refer to whole number parts, fPart refer to fractional number parts, and n refer to the position along the number line.

The integer would then be a series of four-digit identifiers, like : )nXwPart, nXfPart, nYwPart, nYfPart)....

A philosophy lecturer of mine used a similar concept to try and prove that the reals were a *larger* set than the integers -- that is to say that there are *more* reals than integers. I would accept that the reals are a larger set in the sense that their "order" is greater, but I can't see that given one can find a 1-to-1 mapping between the reals and the integers, that there are truly any more of them.



Anonymous Anonymous said...

"That is to say, pi can be represented by 3.14159... to any accuracy."

Yes, but any representation is not actually pi - it is an approximate representation of pi.

1/13/2006 10:48:00 AM  
Blogger MFG said...

We are dealing here with the question of whether some infinite quantities are bigger than others!

There are parts of modern physics where physicists assume some "infinities" are bigger than others - so they can "regauge" or "re-normalise" to solve really hard Maths problems - but it is a fudge. (A simple case of an answer that is arbitrarily near enough being good enough!)

It seems logical to me that there is only *one* infinite quantity - like there is only one *zero.* The only thing that changes is how quickly we are travelling towards the infinite quantity.

(Your Maths games here go far towards proving the existence of *one* infinite quantity!)

Cheers! MFG.

1/13/2006 12:33:00 PM  
Blogger Mitchell said...

Your mapping only covers real numbers whose decimal representation terminates: 3, 3.1, 3.14... but never pi itself. All such real numbers are rational, and it's already known that there is a 1-1 mapping between integers and rationals. The problem comes from the nonterminating, nonrepeating irrationals, which make up the bulk of the reals (the rationals are a subset of "measure zero" in the reals). Every integer has a finite number of digits, every irrational has an infinite number of digits. This is why mappings like yours will not work - to apply them to the irrationals, you would need "integers" with infinitely many digits.

You should read about the diagonalization argument, which is Cantor's original proof-by-contradiction that a 1-1 mapping between reals and integers is impossible.

1/20/2006 09:34:00 AM  

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