First page . . . Back to page 6
After developing the ordinal and cardinal theories to this point, Cantor could not determine whether c was distinct from ℵ1 or equal to it. Cantor tried for a long time to discover a set of points that had more than ℵ0 points but less than c (if found, he could say that this set had ℵ1 points, and c would be ℵ2 or larger). He couldn't find such a set, and then proposed what is now called the continuum hypothesis:
c is equal to ℵ1 ? (continuum hypothesis)
Cantor then tried to prove or disprove this hypothesis but never succeeded. Today, with the benefit of Gödel's results, it is not surprising to see why he had so much trouble: Cantor was attempting to combine or assimilate results from two different formal systems: the ordinal and cardinal types of counting.
In an ordinal system, 1 + X is not always equal to X + 1, but X × 2 is always greater than X. In a cardinal system, 1 + X equals X + 1 but X × 2 is not always greater than X. Another more formal way of saying this is that ordinal systems retain the property of a unique multiplicative identity and cardinal systems retain the property of commutativity — but neither retains both.
Gödel showed in 1940 that Cantor could not have disproved the continuum hypothesis using his axioms (which are now called "Zermelo Fraenkel set theory with the Axiom of Choice", often abbreviated ZFC), Paul Cohen showed in 1963 that Cantor could not have proved it either. For this work, Gödel and Cohen both did major new work in the field of metamathematics, which involves "modeling" mathematical axiom-proof systems with "bigger" systems.
So, at least in standard ZFC set theory, the continuum hypothesis must be declared to be true or false using a new axiom, or left undecided (as Cantor did). You get a different system of infinities each way. By the 1990's, most mathematicians preferred to define the continuum hypothesis as being false (mostly because of the usefulness of the results that can be derived). The implication is that (if you follow the preference of the mathematicians) c is greater than ℵ1.
Returning permanently to cardinal set theory, we proceed to higher infinities beyond c. The set of integers, and all other countable sets, has ℵ0 elements. A continuum (like a line) has c points, and the set of integer sequences also has c elements. The set of integer sequences is an example of something called a power set: the set of all subsets of some other set. Cantor showed that power sets always have more elements than the set from which they were constructed, and so generate another higher infinity.
Let S1 be a set with ℵ0 elements (like the set of integers)
Let S2 be the set of all countable ordinals
Let T be a set with c elements (like the set of points on a line)
Let T' be the set of all subsets of T (the power set of T).
Let T'' be the set of all subsets of T' (the power set of S').
etc.
If (as is more commonly assumed), Continuum Hypothesis is false, then we say:
ℵ0 is the order of S1. (The number of elements in S1).
ℵ1 is the order of S2.
ℵ2 is the next ordinal infinity after ℵ1.
ℵ3 is the next ordinal infinity after ℵ2
etc.
c is the order of T.
2c is the order of T'.
22c is the order of T''.
etc.
AND there is no proven relation between the two series, other than that
c is bigger than ℵ1.
In cardinal set theories it can be shown that that there are no infinities "in between" these. Any definition of an infinite quantity can be shown to be equivalent to a member of the power set sequence. Since Continuum Hypothesis taken to be false, c cannot be equivalent to ℵ1, but it could be ℵ2 or one of the higher ones. All of the higher power sets would then coincide in the same way. For example, if c were ℵ2, then 2c would be ℵ3 and so on.
Consider the order of the set T':
c* = 2c = order of set T'
This infinity is usually thought to be equal to the number of distinct sets of points in a Euclidean space. This is a little difficult to comprehend; an easier definition to comprehend is the number of distinct "wiggly lines" in two-dimensional space. A "wiggly line" in this case can be extremely convoluted, such that any level of magnification will show more and more wiggles (like a fractal, but not necessarily a self-similar fractal).
The next infinity after c* or 2c is c or 22c. There appears to be no useful geometrical definition or application (outside set theory) for this or any of the higher infinites. Whereas the first three infinities can be thought of as counting the number of integers, points, and curves in 2-d space, 22c doesn't appear to count anything geometrical. Anything we've found that can be counted is covered by one of the lower infinities.
This idea of only three useful infinities is hauntingly reminiscent of the (perhaps mythical) "one, two, three, many" of the Hottentots, bringing us full-circle back to class-0 numbers.
Finally consider the limit of these processes:
ℵ0, ℵ1, ℵ2, ℵ3, ... (ordinals)
ℵ0, c, 2c, 22c, ... (cardinals)
In each of these processes, imagine the infinity you "get to" as you carry the process on "forever". This includes any algorithmic process in which the number of steps is finite, working up to such things as ℵBB(n) where BB(n) is the busy beaver function and N is some gratuitous huge integer.
Since the infinities all have an integer subscript, the "number of infinities" (or number of classes, if you are working within an ordinal system) is ℵ0, and the "limit" of the process of defining higher infinities is the "ℵℵ0" class (ordinal system), or "2④ℵ0" (cardinal system).
Then you make another definition (still in a formal well-defined way) so you can talk about ℵℵ0 directly and thence move on to ℵw+1 or ℵℵ1 (depending on whether your larger formal system uses ordinal rules or cardinal rules, respectively). This process can be continued, and eventually formalized through another level of abstraction to construct even higher infinities. One of these is so big that is is equal to its own ℵ-number: theta = ℵtheta.
If you stay "within the system" while doing this process, by sticking to well-defined symbols, rules, axioms, etc. you can create more and more infinities, but you will always be working within a formal system of number theory or set theory.
However, all number theories and set theories are incomplete. It has been shown that by going outside the system you can demonstrate the existence of "inaccessible cardinals" or "inaccessible infinities", which are bigger than all of those producible through formal systems. This result is analagous to the computation-theory result of the uncomputable functions.
1 : http://www.sizes.com/numbers/big_numName.htm
2 : http://www.miakinen.net/vrac/nombres#lettres_zillions
3 : http://www.io.com/~iareth/bignum.html (Latin number names, some of the large examples like centumsedecillion)
4 : http://www.miakinen.net/vrac/zillions : page by Olivier Miakinen; and personal communication.
5 : http://www.graner.net/nicolas/nombres/wechsler.txt : note from Allan Wechsler
6 : (no web page) Bulletino di Bibliographia e di Storia delle Scienze matematiche e fisische. — Bologna volumes XIII, 1880, ISSN 9012-9458.
7 : (no web page) The Oxford English Dictionary (Second Edition), 1989, entry for million (vol. IX, pp. 784-785), sense 1. a. (a)
8 : http://www.linguistlist.org/issues/7/7-451.html
9 : (no web page) Le Nouveau Petit Robert (1993 edition), entry for the word billion (page 223); entry for the word trillion (page 2312)
10 : Conway, John Horton and Guy, Richard, The Book of Numbers, New York: Springer-Verlag, New York, 1996. ISBN 038797993X.
pp. 59-61 (Knuth up-arrow notation)
p. 60 (Ackermann numbers)
p. 61 (Conway chained-arrow notation)
p. 61 (Skewes's number)
pp. 61-62 (Graham's number)
pp. 266-276 (Cantor ordinal infinities)
pp. 277-282 (cardinal infinities and the continuum)
11 : Hawking, Stephen, God Created the Integers (an anthology of translated works of great mathematicians throughout history), pp. 971-1039 (Georg Cantor)
12 : http://www.toothycat.net/wiki/wiki.pl?CategoryMaths/BigNumbers Douglas Reay, commenting on discussion of formal theory of computation, toothycat.net wiki (created by Sergei and Morag Lewis), CategoryMaths, BigNumbers.
13 : http://www.math.ohio-state.edu/~friedman/ Web site of Harvey M. Friedman. In the "preprints, drafts and abstracts" is a paper Enormous Integers in Real Life, 2000, which summarizes several methods of producing large integers, related to combinatorics and theory of computation.
14 : Harvey Friedman, Long Finite Sequences, 1998. Available at the above website13.
15 : http://math.eretrandre.org/tetrationforum/showthread.php?tid=184 Henryk Trappman and Andrew Robbins, Tetration FAQ (online document) Note: A previous version was here
16 : Martin Gardner, <:The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems, W. W. Norton (2001), ISBN 0393020231. Graham's number: pp. 448-450; also appeared in Scientific American in 1977. Most Gardner material has been published multiple times, so you might find it in one or another of his earlier books.
17 : Knuth, Donald E., Coping With Finiteness, Science vol. 194 n. 4271 (Dec 1976), pp. 1235-1242.
18 : http://yudkowsky.net/singularity.html Eliezer Yudkowsky, Staring into the Singularity, web page (1996-2001).
19 : http://www.bipm.org/en/CGPM/db/19/4/ BIPM (Bureau International des Poids at Mesures), Resolution 4 of the 19th Meeting of the CGPM (1991) (as translated from the official French)
20 : http://jimvb.home.mindspring.com/unitsystem.htm Jim Blowers, Extended System of Units, web page.
21 : http://groups.google.com/group/sci.answers/browse_thread/thread/6cfbc3688fcbe192/d5ad476584d024b5 Alex Lopez-Ortiz, sci.math FAQ: Name of Large Numbers, "version 7.5", Feb 27 1998.
22 : Louis Epstein (through personal communication) cites usage of dea- in a 1985 edition of the Guinness Book of World Records.
23 : http://home.att.net/~numericana/answer/units.htm Gerard P. Michon's Numericana, Final Answers — Measurements and Units. (Has lots of details about real and bogus SI prefixes).
24 : http://groups.google.com/group/sci.math/browse_thread/thread/b6b75e8a51ba00a2/e3d0868922d3fc30 Alex Lopez-Ortiz, sci.math FAQ: Name of Large Numbers, "version 7.0", Nov 20, 1995.
25 : http://en.wikipedia.org/wiki?title=Talk:Names_of_large_numbers Wikipedia, discussion page for Names of large numbers (accessed on 2010 Feb 26th.
26 : Wikipedia, Long and short scales.
27 : http://www.polytope.net/hedrondude/illion.htm Page by Jonathan Bowers AKA "hedrondude" concerning large number names most ending in "-illion".
[28] Edward Brooks, The Philosophy of Arithmetic, 1904. Cited by [29].
[29] Dmitri Borgmann, Naming the numbers. Word Ways: the Journal of Recreational Linguistics 1 (1), pp. 28-31, 1968. Cover and contents are here and article is here.
[30] Rudolf Ondrejka, Renaming the numbers. Word Ways: the Journal of Recreational Linguistics 1 (2), pp. 89-93, 1968. Cover and contents are here and article is here.
[31] John Candelaria, Extending the number names. Word Ways: the Journal of Recreational Linguistics 8 (3), pp. 141-142, 1975. Cover and contents are here and article is here.
[32] John Candelaria, Renaming the extended number. Word Ways: the Journal of Recreational Linguistics 9 (1), p. 39, 1976. Cover and contents are here and article is here.
[33] Donald E. Knuth, Supernatural numbers. In The Mathematical Gardner, ed. David A. Klarner (1981).
[34] Georges Ifrah, The Universal History of Numbers, ISBN 0-471-37568-3. (1999).
[35] Donald E. Knuth, personal communication, 2010 Feb 26.
Scott Aaronson, Who Can Name the Bigger Number?, essay about how to win the often-contemplated contest; covers many of the topics discussed here.
Crandall, The Challenge of Large Numbers, Scientific American February 1997, pages 74-79
Davis, Philip and Hersh, Reuben. The Mathematical Experience, Birkhaeuser, 1981. pages 223-225 (infinities)
Davis, Philip J., The Lore of Large Numbers, New York: Random House, 1961
Dewdney, A.K., The Busy Beaver, in Mathematical Recreations column, Scientific American, April 1985, p. 30.
Gamow, George, One, Two, Three... Infinity: Facts and Speculations of Science, Viking, 1947 (reprinted in paperback by Dover, 1988). This was an early source for me and unfortunately gave me the impression that the ℵn series of infinities was equivalent to a power-set series, and also to the continuum power-set series.
Hofstadter, Douglas, Gödel, Escher Bach: An Eternal Golden Braid
Hudelson, Matt, Extremely Large Numbers
Kasner, Edward and Newman, James, Mathematics and the Imagination, Penguin, 1940
Knuth, Mathematics and Computer Science: Coping with Finiteness. Advances in our ability to compute are bringing us substantially closer to ultimate limitations., Science, 1976, pages 1235-1242
Knuth, Supernatural Numbers, in The Mathmatical Gardener, D. A. Klarner, ed., 1981
Kosara, Robert, The Ackermann Function
MacTutor history of Mathematics page on Chuquet
Matuszek, David, Ackermann's Function
McGough, Nancy, The Continuum Hypothesis (web pages)
Miller, George: The Magical Number Seven Plus or Minus Two: Some Limits on Our Capacity for Processing Information (1956)
Munafo, Robert, hypercalc (the Perl calculator program that handles numbers up to 10④10000000000)
Pilhofer, Frank, Googolplex and How to get a Googolplex
Rado, Tibor, "On non-computable functions", Bell System Tech. Journal vol. 41 (1962), pages 877-884. (busy beaver function)
Rucker, Rudy, Infinity and the Mind, 1980. (ordinal infinities: the relevant chapter was reproduced here the last time I checked.)
Spencer, Large Numbers and Unprovable Theorems, American Mathematical Monthly, 1983, pages 669-675
Steinhaus, Hugo, Mathematical Snapshots (3rd revised edition) 1983, pp. 28-29.
Stepney, Susan, Ackermann's function
Stepney, Susan, Big Numbers
Stepney, Susan, Graham's Number
Weisstein, Eric (ed.), Ackermann Function
Weisstein, Eric (ed.), Large Number
To Morgan Owens (packrat at mznet gen nz) for news of the Knuth -yllion names and the Busy Beaver function
Unconfirmed SI prefixes: Sci.Math FAQ, Alex Lopez-Ortiz, ed.
List of Notes:
Personal History
Large numbers have interested me almost all my life. At age 5 100 was the biggest number I knew, by age 6 it was 1000000, at age 7 I asked my Mom what was after 1000 and a million and she told me about the (lesser) billion and trillion (1012); at age 8 I learned about vigintillion (1063) in a book from the school library. I loved vigintillion so much I wrote it in the sand in the schoolyard:
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
prompting much harassment from the other children! I kept going: by age 10 I had invented higher dyadic operators and by age 13 I knew the Steinhaus-Moser notation. That was about as high as anyone had gone at the time, so I turned my attention to computers and began to write programs to manipulate large Class-2 numbers. My latest accomplishment in this area is hypercalc. It literally cannot overflow, except by dividing by zero.
If you like this you might also enjoy my numbers page.
Robert Munafo's home pages on HostMDS
(c) 1996-2010 Robert P. Munafo. about
contact
This work is licensed under a Creative Commons Attribution 2.5
License. Details here
s.13