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Topic: Almost infinite
Replies: 19   Last Post: Mar 21, 2013 2:40 PM

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Jim Burns

Posts: 1,200
Registered: 12/6/04
Re: Almost infinite
Posted: Dec 13, 2012 8:12 AM
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David R Tribble wrote:
> We see the phrase "almost infinite" (or "nearly infinite", or "infinite
> for all practical purposes") in much literature for the layman, usually
> to describe a vastly large number of combinations or possibilities from
> a relatively large number of items. For example, all of the possible
> brain states for a human brain (comprising about 3 billion neurons), or
> all possible combinations of a million Lego blocks, etc.
> Obviously, these are in actuality just large finite numbers; having an
> infinite number of permutations of a set of objects would require the
> set to be infinite itself, or the number of possible states of each
> element would have to be infinite. Most uses of the term "infinite
> possibilities" or "almost infinite" are, in fact, just large finite
> numbers. All of which are, of course, less than infinity.
> But is there some mathematically meaningful definition of "almost
> infinite"? If we say that m is a "nearly infinite" number, where
> m < omega, but with m having some property that in general makes it
> larger than "almost all" finite n?
> Personally, I don't think there is such a definition; but then I would
> enjoy being proved wrong.

The key here is what we mean by infinite, of course.

The clearest way to think about the smallest infinity that I have
found is to imagine a one-ended chain, loop-free, etc --
literally infinite, not-ended in one direction.

I think that would mean a finite-but-almost-infinite chain
would have slightly more than one end, and that would imply
an integer between one and two.

In physics, properties are often calculated from the most
important effects and then corrections are made from less
and less important effects. The orbit of the Uranus can be
calculated to a good degree of accuracy with a straightforward
application of Newton's laws, but it took the assumption of
the existence of another major planet (Neptune, later found)
to account for some of the difference between prediction
and observation.

If Neptune were infinitely far from Uranus, then its
gravitational effect could be ignored. Since its gravitational
effect can be *almost* ignored, I suppose that one could say
that Neptune is *almost* infinitely far from Uranus.

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