factorial factorial

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factorial factorial

Post by dalehileman » Fri Feb 24, 2006 6:03 pm

Aproposism is a neologism and a portmanteau or blend as well, like "babelicious" and "dreadline"


Since there are about 50,000 common words, prefixes, and suffixes in the Mother Tongue, one cannot help wondering how many such combinations are possible. Barring triads and larger as well as multi-word expressions, it must approximate 50,000-factorial, a very big number indeed


I remember from my engineering courses what a factorial is; it multiplies by the next diminishing number; thus 5 factorial is 5 x 4 x 3 x 2 =120

In attempting to express an unimaginably large number (Ken should like this), is it correct to say "factorial factorial"

Thus (1) Would the latter mean 119 x 118....etc and (2) does it have a symbol to avoid repeated keystrokes

Thanks guys
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factorial factorial

Post by russcable » Fri Feb 24, 2006 9:22 pm

The symbol for factorial is ! as in 5! = 120. (5!)! = 120! = ~6.6895029134491270575881180540904e+198 (~ is approximately, e+198 is times 10 to the 198th power) = ~6.7 googol * 1 googol. Quite an impressive number but not unimaginably large.

(aleph-null!)! is pretty hard to imagine. Any math-philosophy majors know if it's bigger than aleph-one?

Babelicious is babe and delicious, dreadline is dread and deadline, apropos-ism is apropos and ??? You're just adding an ending, I don't think that counts. I put the dash in because Google spell check has it that way. Google doesn't like any of these other fancy words including (oddly) googol.
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Post by dalehileman » Sat Feb 25, 2006 12:26 am

Russ: Of course it is! I knew that!

Tho I have to admit I might have forgotten to use the parentheses

To me it's unimaginable
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Post by kagriffy » Sat Feb 25, 2006 1:12 am

Russ, do you mean to tell us that you actually UNDERSTOOD whatever the heck dale was asking??????? If so, perhaps you should really have your sanity checked! None of the rest of us has any idea what he's rambling on about! lol
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K. Allen Griffy
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factorial factorial

Post by dalehileman » Sat Feb 25, 2006 6:51 pm

Kag thank you for your encouragement as you aren't the first to so remark. However, when in the hope of creating more intelligible threads and followups, I had asked for specific instances of my correspondent's beffudlement, alas I was met with deafening silence
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Post by Ken Greenwald » Sun Feb 26, 2006 7:49 am

Russ, It should be noted that (5!)! is not the same as 5!! The latter is called the double factorial which is defined as n(n-2) . . . 5x3x1 if n is odd, and n(n-2) . . . 6x4x2 if n is even, and is equal to 1 for n = -1, 0. So that 5!! is equal to a measly ’15.’

Some infinities are bigger than others (see postinginfinite number of monkeys). ALEPH-0 is the smallest of the infinities and can be put into a one-to-one correspondence with the natural counting (whole) numbers. One would think that the set of rational numbers (all fractions, which includes whole numbers) would be a larger infinity than ALEPH-0, but it ain’t so. This defies normal logic, since it would seem that the subset, the whole numbers, should be smaller than the set of rational numbers from which it comes. But it just ain’t so and the proof is simple and is shown in many elementary math texts. By this reasoning the infinite set of whole numbers is of the same size (has the same cardinality or power) as the set of even whole number, as the set of odd whole numbers, as the set of all positive integers, as the set of all negative integers, as the set of all integers. Aleph-0 is called a ‘countable infinity’ since it can be put into a one-to-one correspondence with the counting numbers.

Aleph-1, on the other hand, is a much larger infinity because there aren’t enough counting numbers to ever be put into a one-to-one correspondence with it and this may also be easily proven and also appears in many elementary math texts. Aleph-1 is therefore known as an ‘uncountable infinity,’ an example of which is the set of ‘real numbers’ - the set of all rational numbers plus the irrational numbers (those numbers as ‘pi’ and the square root of 2 which can never be expressed as a fraction). The real numbers are just denser than the counting numbers and may be thought of as the points on a continuous line and thus Aleph-1 is often referred to as the cardinality of the continuum. And we again have the very weird result that the number of points on any segment of a line (or on an infinitely long line for that matter) has the same cardinality as the number of points on any subsegment of that line.

German mathematician (actually born in Russia) Georg Cantor (1845-1918) who was the mathematician that first came up with this theory of infinites (transfinite numbers), along with the various proofs, wrote to fellow mathematician Dedekind in 1877, “I see it, but I don’t believe it” and asked his friend to check his work. So all this in its day was difficult for even mathematicians to swallow, but in the years that followed, Cantor’s ideas on infinity, although counterintuitive, were fully accepted and today permeate almost all of mathematics.

It also turns out that from each cardinality of infinity, we can generate a next higher cardinality and so there is an infinite number of cardinalities. The number of subsets of any set is 2^n, where n = the number of elements in the set. And it also can be proven that the number of all subsets of any infinite set produces the infinite set of next higher cardinality. Thus the cardinality of the set of all subsets of ALEPH-0 is ALEPH-1 = 2^ALEPH-0, of all the subsets of ALEPH-1 is ALEPH-2 = 2^ALEPH-1, of all the subsets of ALEPH-2 is ALEPH-3 = 2^ALEPH-2, . . . .

Cantor raised the question of whether there is an infinite set with a cardinal number between that of the smallest infinite set ALEPH-0 and the larger infinite set of real numbers C (the continuum). This question first posed by Cantor himself became known as the ‘continuum hypothesis,’ which can be stated as Aleph-1 = C. And in 1900 this question was the first of 23 questions proposed as a challenge by the great German mathematician David Hilbert (1862–1943) to be solved in the 20th century.

Well thanks to the famous ‘theorem of undecidability’ of Kurt Gödel (1906-1978, U.S. logician born in Czechoslovakia.) known as Gödel’s Incompleteness Theorem, which amazingly states that there are questions within an axiomatic system that can be proven neither true nor false (e.g. there are things that are true that cannot be proven true, which is really disappointing, especially when a mathematician spends years trying to prove something that may be true but unprovable) and the work of American mathematician Paul Cohen (1934- ), it was established that the validity of the continuum hypothesis depends on the set of axioms one begins with and the question is therefore undecidable.

I’m not sure if there is any physical interpretation for an infinity above ALEPH-1, but I’ll leave you with what Einstein once jokingly said about infinity:
<“Only two things are infinite, the universe and human stupidity, and I’m not sure about the former.”>
______________________

Ken – February 25, 2006
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Post by Bobinwales » Sun Feb 26, 2006 12:41 pm

Hello Allen, I feel less lonely now.

"A much larger infinity"?

I understoond Einstein's joke though.
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factorial factorial

Post by russcable » Sun Feb 26, 2006 7:11 pm

Ken Greenwald wrote: Russ, It should be noted that (5!)! is not the same as 5!! The latter is called the double factorial which is defined as n(n-2) . . . 5x3x1 if n is odd, and n(n-2) . . . 6x4x2 if n is even, and is equal to 1 for n = -1, 0. So that 5!! is equal to a measly ’15.’
Interesting. I was only 2 credits short of having a double major in Math Science and I hadn't come across the double factorial before. I had merely (accidentally correctly evidentally) included the parentheses in my post to make it more obvious that I meant n bang bang and not n bang exclamation.
Googling, I find it is part of a whole series of functions denoted with an increasing number of bangs, i.e. n!!!!!... . I find this form of notation a distasteful addition to the parsing of algebraeic expressions both for human and computer parsers for different reasons, in other words, YUCK!!!
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Post by Edwin Ashworth » Tue Feb 28, 2006 8:28 pm

I'd never encountered the !! terminology either, Ken.
I came across an extension of the arrow-notation for powers, the original being used on calculators and computers to avoid the difficulty of trying to use superscripts to denote powers (I believe). I'll use an asterisk here - thus

5*2 = 5 x 5 = 25 ; 7*4 = 7 x 7 x 7 x 7 = 2401 etc

However, 5**2 and 7*******4 etc were very complex entities.
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