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From: <GSLevy.domain.name.hidden>

Date: Fri, 10 Nov 2000 14:11:30 EST

This is in reply to Hal Ruhl idea that Nothing is unstable and expands to

encompass Everything.

I kind of agree with his idea but, being a lowly engineer instead of a

mathematician, I need to put it in familiar terms (high school level).

Taking the system of natural numbers under the operations of addition and

multiplication (and the associated axioms) we find that this system is

consistent i.e., 5+3 = (4+1) + 3 = 8 ... and we can solve equations such as

X+3 = 8. i.e., X = 5

However, the system is incomplete since X+8 = 3 has no solution within the

set. So we are forced to invent an extension to the natural numbers, that we

call the (positive and negative) integers and zero. The solution to X is then

-5.

This new system works fine i.e., 5*3 = 15 until we try to solve the equation

X*3 = 5. At this point we must again expand our set to include the rational

numbers to include the solution X=5/3.

So far so good. But what if we are faced with a series such as X = 1- 1/2 +

1/3 - 1/4.... No rational number can be equal to X. So we must again expand

the set to irrationals to include the solution X=Ln(2).

And so on.... To vectors, tensors, matrices, spinors, infinitesimals,

Cantor's infinities, and so on. And operators follow the same pattern:

subtraction, division, power, logarithms, derivatives, integrals....

As Goedel proved no system can ever be complete, so we must keep expanding

the set of our objects forever in search of completeness which is actually

never achieved.

If the physical world follows the same pattern as the mathematical world we

do end up, in the limit, with a plenitude in which every potentiality has an

actuality.

George Levy

Received on Fri Nov 10 2000 - 11:15:48 PST

Date: Fri, 10 Nov 2000 14:11:30 EST

This is in reply to Hal Ruhl idea that Nothing is unstable and expands to

encompass Everything.

I kind of agree with his idea but, being a lowly engineer instead of a

mathematician, I need to put it in familiar terms (high school level).

Taking the system of natural numbers under the operations of addition and

multiplication (and the associated axioms) we find that this system is

consistent i.e., 5+3 = (4+1) + 3 = 8 ... and we can solve equations such as

X+3 = 8. i.e., X = 5

However, the system is incomplete since X+8 = 3 has no solution within the

set. So we are forced to invent an extension to the natural numbers, that we

call the (positive and negative) integers and zero. The solution to X is then

-5.

This new system works fine i.e., 5*3 = 15 until we try to solve the equation

X*3 = 5. At this point we must again expand our set to include the rational

numbers to include the solution X=5/3.

So far so good. But what if we are faced with a series such as X = 1- 1/2 +

1/3 - 1/4.... No rational number can be equal to X. So we must again expand

the set to irrationals to include the solution X=Ln(2).

And so on.... To vectors, tensors, matrices, spinors, infinitesimals,

Cantor's infinities, and so on. And operators follow the same pattern:

subtraction, division, power, logarithms, derivatives, integrals....

As Goedel proved no system can ever be complete, so we must keep expanding

the set of our objects forever in search of completeness which is actually

never achieved.

If the physical world follows the same pattern as the mathematical world we

do end up, in the limit, with a plenitude in which every potentiality has an

actuality.

George Levy

Received on Fri Nov 10 2000 - 11:15:48 PST

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