Uncountably Finite

UncountablyFinite is the term I use for numbers which are larger than any reasonable human being can count to or understand, but which are still finite. e.g. number of people on the planet, gross national product, space on a hard drive, etc. They have funny mathematics (see below).

We can refer to humanly accessible numbers like 5, 200, even 10,000 as "small." Then UncountablyFinite UF numbers are "large", and (1/UF) numbers are "miniscule". --AlistairCockburn

I must take issue with these examples. Number of stars in all known galaxies, okay, *that* one can't be related to anything. There are approximately six billion people on the planet, 270 million of them in the US, 7 million of those in New York City, and 40,000 in my "small" town just outside the city. Alone these numbers may be meaningless, but in combination, they give each other meaning. Anyone who makes the effort can indeed appreciate the significance of large numbers. -- DanielKnapp

For me, the "interesting" sort of question that arises is: "Can all the food produced in the world feed all the people in the world?" The answer involves UF numbers. The answer to the question is one of: Easily, No Chance, or Just Barely. That's an awfully big variation in answer, with extreme consequences.

A second question I ran into was when reading about Velikovsky - he claimed that Mars flew by Earth in historic times and caused the sun to stand still (the biblical account) (n.b. if you want to discuss this topic, do that on a separate wiki page, please). When I read a pro- and con- discussion of that claim, the question was, "Could Mars have slowed Earth's rotation by that much?" The answer on the one side was Obviously, and on the other side was Obviously Not. The two answers differed because the two scientists disagreed on the rotational inertia of the earth by a number like 10^9. !!. Both Mars' and Earth's masses and energies are large (UF) numbers, and so the math yields drastically different answers.

A third example was the recent US Federal budget surplus and the tax rebate recently issued by the U.S. government. There are about 500,000,000 people in the US (I don't know about you, but I sure can't do any mental math with that figure). I don't recall the amount of the surplus, but it was a number that I also don't relate to. When they did the division and sent out the checks, we each got about $120, a very real amount useful for buying a couple of shoes. To me, that's an amazing piece of math, that UF/UF = $120. I mostly expected to get $0.02 or similarly nothing. --AlistairCockburn

UncountablyFinite numbers have an interesting arithmetic.

UF + UF = UF (large + large = large)
UF * UF = UF (large * large = large)
UF - UF = large positive, large negative, or small positive or negative (we can't tell which without computation)
UF / UF = large, miniscule or small (we can't tell which without computation)

-- AlistairCockburn

Example ?

UF * UF / UF = ?

 10^20 * 10^20 / 10^39 = 10? (small) 
 10^20 * 10^20 / 10^60 = 10^-20? (miniscule)
 10^60 * 10^60 / 10^20 = 10^100? (large)

The rule reads

 (UF * UF) / UF 
 = UF / UF 
 = large, small, or miniscule

What about (UF / UF) * UF?

That depends: what is miniscule times large? Could be miniscule, small or large.

Basically this is a restatement of what infinity is, only with smaller numbers. Read any calculus book for a list of fun indeterminates (0/0, infinity/infinity, infinity-infinity, 1^infinity, 0*infinity, etc.)

[Well, some calculus texts!]

Except that we're not talking about infinity, we're talking about finite numbers that show up in the real world.

The same math applies

Uh, I'm not seeing it, so I need to ask you to explain. I was taught that 0/0 and infinity/infinity are undefined, which is not the case with UF/UF. Please, if you will, tell again about the infinities math. Thanks.

Alrighty. I'll just translate some of the rules listed above.

UF + UF = UF (large + large = large)

infinity + infinity = infinity

UF * UF = UF (large * large = large)

infinity * infinity = infinity

UF - UF = large positive, large negative, or small positive or negative (we can't tell which without computation)

infinity - infinity is an indeterminate (hence no simplification)

UF / UF = large, miniscule or small (we can't tell which without computation)

infinity / infinity is an indeterminate (hence no simplification)

The indeterminate expressions quoted above do not stand for actual numbers. 0/0 does not exist. Instead, they are shorthand for various limit problems that may be solved by l'Hopital's Rule. For example, the limit of sin x / x as x tends to 0, or the limit of e^x / x^2 as x tends to positive infinity, or the limit of x log x as x tends to 0 from the right, or other similar problems. That's all those expressions mean.

Now, the material above with UF can actually be formalized as a variant of NonstandardAnalysis?, as Nelson's Internal Set Theory. But, it requires a lot of work to define this correctly. One doesn't name any particular number as being big; one just says that there are big numbers. -- EricJablow