re: the above: let’s agree that when it comes to silly questions or trivia or puzzles like this, that there exist a large number of good answers. “The answer…” ? I don’t think so. Maybe “the answer that the guy making it up was thinking of,” …ya know? I mean, don’t sell yourself short - NO one really has THE answer.
Don’t forget about tetration! Tetration, for those not in the know, is iterated exponentiation, in the same sense that exponentiation is iterated multiplication and multiplication is iterated addition. Tetration is normally represented by a superscript to the left of a number. [sup]y[/sup]x can be defined to mean “an exponentiated stack of y many x’s.”
[sup]9[/sup]9 represents 9[sup]9[sup]9[sup]9[sup]9[sup]9[sup]9[sup]9[sup]9[/sup][/sup][/sup][/sup][/sup][/sup][/sup][/sup] (note the nine nines in the series; that’s the way tetration works). This number is so unimaginably huge that scientific notation fails completely in expressing it. Even a googelplex is only 10[sup]10[sup]10[sup]2[/sup][/sup][/sup], or less than [sup]4[/sup]10 — enormously smaller than [sup]9[/sup]9.
I doubt this is the answer that the ask er of the original question had in mind, but it’s probably larger than whatever he or she was thinking of.
how about 1[sub]1[/sub]? (If I screwed that up its supposed to be 1 in base 1). I’m not entirely sure what the mathematical consensus on the value of that is, but it could be viewed as an infinity.
Well, I found a real example that uses the double factorial, but it wasn’t the example I was thinking of. I noticed it in a table of integrals, so I stopped looking. To wit:
/ .5Pi 2n+1 (2n)!!
| (sin x) dx = --------
/ 0 (2n+1)!!
(The Handbook of Chemistry and Physics, CRC Press, integral #650)
It also occurs to me that we can do even better than 9![sup]9![/sup]… We could take 9[sup]9![/sup]! . Or, if we want to be non-mathematical smart aleks, we can go
[sup]/¯¯\ /¯¯\
\__/ \__/
/ /
/ /[/sup]
which is a very large number, indeed.
Chronos:
Are you sure about the Aleph[sub]Aleph[/sub] part? It seems like an OK cardinal to me. I don’t think Russel’s Paradox negated the existence of any particular cardinals, rather, it just said that it was contradictory to speak of such things as the “set of all sets” or, more specifically, “the set of all sets that do not contain themselves”.
How about lim[sub]x->0[/sub] 1/x
There are two digits in the above expression. Can’t get much bigger than that!
Silly Nitpick:
Another answer is, you can make ANY number with two digits, if they are one and zero and are used in a binary system. You want to say using two numerals.
Just turn an “8” on it’s side. (That seems to be what we’re reaching for.)
If we must use a second digit, I would add a “+0”.
(Of course, we could take two sideways “8”'s and sum them, multiply them, make them and exponent or make them a tetration, but I’m not sure that those numbers would be defined, meaningful or even bigger.)
It’s evident from the posts immediately after mine that I missed the point entirely. There’s no need to pick my example out.
Plus, others are using an awful lot of flexibility in defining “digits”, or adding other items (!, lim, etc.). I’ve always heard of hex numbers described as hex “digits”. But it’s irrelevant, since I obviously missed the answer.
Well, ‘infinity’, itself, isn’t a number, but there are numbers which are infinitely large-- Those are the alephs that me, Cabbage, and The Ryan mentioned. They’re called transfinite cardinals. Aleph[sub]0[/sub] is the cardinality of the set of integers (the amount of integers), aleph[sub]1[/sub] is the cardinality of the set of all subsets of the integers, and also the cardinality of the set of all real numbers, aleph[sub]2[/sub] is the cardinality of the set of all subsets of a set of cardinality aleph[sub]1[/sub], etc. Interestingly, most operations on them are defined, but even tetration will just get you back the same number.
And no, Cabbage, I’m not certain about the aleph[sub]aleph[/sub] business, especially not if a real mathematician says otherwise. That was just a wild guess from a non-mathematician.
Actually, that’s the (generalized) continuum hypothesis. Aleph[sub]0[/sub] (the first transfinite cardinal) is the cardinality of the integers, and 2[sup][sup]Aleph[sub]0[/sub][/sup][/sup] is the cardinality of the set of all subsets of the integers (and the cardinality of the set of real numbers), but it isn’t necessarily the case that 2[sup][sup]Aleph[sub]0[/sub][/sup][/sup] = Aleph[sub]1[/sub] (the second transfinite cardinal). 2[sup][sup]Aleph[sub]0[/sub][/sup][/sup] is at least as big as Aleph[sub]1[/sub], but it could be much larger. The continuum hypothesis says that they are, in fact, equal, but the continuum hypothesis is independent of the standard set theory axioms. (The generalized continuum hypthesis says that 2[sup][sup]Aleph[sub]n[/sub][/sup][/sup] = Aleph[sub]n+1[/sub] for all n).
Some set theorists have been working toward a “natural” extension of the standard axioms which would decide the truth/falsehood of the continuum hypothesis. The tendency has been to reject the continuum hypothesis; in fact, a new axiom has been proposed (Woodin’s axiom, which I know nothing about) which implies that the continuum hypothesis is false–the cardinality of the reals is strictly greater than Aleph[sub]1[/sub].
I’ll put two cents also, though my knowledge of foundations is VERY rusty:
I’d like you to tell me what aleph[sub]aleph[/sub] means. I can’t attach a meaning to it, as alephs with integer subscripts are defined orders of cardinality, aleph not being a symbol with meaning in its own right.
Chronos, you are accepting something called the continuum hypothesis when you say that aleph[sub]0[/sub] = cardinality of the real numbers. Like the axiom of choice, this is something you can be consistent both with and without. Most people accept “with”.
The generalized continuum hypothesis suggests that you get all orders of cardinality by marching through “power set of power set of …” to obtain the sequence of aleph[sub]n[/sub]'s.
I poked around for a background document. I’m not happy with this, as it’s a bit heavy going for the casual reader, but anyway:
And I saw your use of !!. Thanks.
Ok, I’ll defer to the mathematicians on that one… I must have mis-learned the continuum hypothesis. The way I was taught, was that it was proven using the standard axioms that the power set of the integers had the same cardinality as the reals. What was presented in the (admittedly introductory-level) class I had as the Continuum Hypothesis was that there were other transfinite cardinals in between aleph[sub]0[/sub] and aleph[sub]1[/sub]. Apparently, I was taught wrong, or at least, learned wrong.
I see my reply crossed Cabbage’s. Cabbage is probably more up to date than I am when he suggests current trends to reject the continuum hypothesis. Believe him.
yabob:
I read it as meaning aleph[sub]aleph[sub]0[/sub][/sub], i.e., the smallest transfinite cardinal that is greater than infinitely many transfinite cardinals (aleph[sub]0[/sub], aleph[sub]1[/sub], aleph[sub]2[/sub], aleph[sub]3[/sub],…(running through all the natural numbers),…,aleph[sub]aleph[sub]0[/sub][/sub]).
Actually, however, I think it would be more proper to index the aleph’s with ordinals rather than cardinals, so instead of aleph[sub]aleph[sub]0[/sub][/sub], I would write it as aleph[sub]omega[/sub] (omega is the first infinite ordinal, just as aleph[sub]0[/sub] is the first infinite cardinal).