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What About Liebnitz' "infinitessimals"?

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Posted by Mark on December 8, 2001 06:03:38 UTC

Mathematicians had a hard time swallowing the idea of "as close to zero as you can possibly get without actualy being zero".

However, dx, is nonetheless a common term in differential calculus. It is an infinitessimal increment, and for all practical purposes, equal to zero (about as close as you can get without actualy "being").

0/1 = 0

What is the reciprocal of 0?

1/0 = reciprocal of 0 = infinite (or undefined)

But perhaps we can define "zero" within the context of George Cantor's transfinite numbers...(??)

Just as infinite has several orders of magnitude (or cardinalities), the first being aleph-nought... perhaps zero can also have different orders of "infinitessimal-magnitude". It's just a thought.

Perhaps just as zero is indivisible, or "infinitely small", there can be different cardinalities of "infinitessimals".

This way aleph-nought does indeed have a reciprocal. The reciprocal of the first order of infinite happens to be "almost zero but not just quite".

1 - .9999999 = "almost zero but not just quite".

It can be shown that 1 - .99999 can be made to be smaller than any arbitrary real number, just as infinite can be shown to be larger than any arbitrary real number (which is the definition of "infinite").

If there are transfinite numbers, can there also be "trans-infinitessimal" numbers? Is the "dx" term used in calculus just such a number?

If 1/0 is infinite... then which cardinality of infinite?

Is the reciprocal of infinite, zero?

Which "cardinality" of zero?.....

See what I mean?

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