There is no largest natural number. If n’s a natural number, then so is n +1, and that’s bigger. Aleph-0, on the other hand, is not a natural number. But as DrMatrix said, you can construct larger numbers through the power set operation. So that’s why we know there’s no largest cardinal: we can make bigger ones.
ultrafilter - not wanting to harp on about this, but I truly am curious about is, speculative as it is (and I feel it’s relevant to the spirit of the OP) and I don’t think I explained what I meant quite right.
OK, I 100% agree that there can be no largest cardinal, in the same way that (even post-Cantor) there is no “largest” natural number. However, isn’t it possible that there’s a whole bunch of mathematics out there beyond “cardinals” - that there is some entity which lies beyond all the alephs, not a cardinal itself but something stranger, in the same way aleph-null stands just beyond all the natural numbers?
i.e.
0,1,2,3,4… aleph-null
aleph-null, aleph-one, aleph-two…aleph-aleph… <something>
I haven’t read Cantor’s work (probably should), only summaries of it, but I believe this is something he referred to as “The Absolute” or “Omega”. It seems to me that by denying that there is anything at the “limit” of all the cardinals, we are guilty of the same type of thinking that rejected Cantor’s “actual infinities”. To me the history of mathematics seems to be a broadening of the entities we consider - and at each point, as soon as they were introduced, new concepts were widely rejected.
How many of these concepts were considered preposterous when first discovered?
Irrational numbers
“Zero”
Negative numbers
Transcendental numbers
“Imaginary” numbers
Transfinite numbers
Infinitesimals
(my ordering is probably not quite right, chronologically)
Isn’t it just possible that some day, someone as bright as Cantor will come up with a theory of what lies “beyond the cardinals” ? (At that point the major problem will be which alphabet to use, since Roman, Greek and Hebrew already seem to work full time in mathematics 
)
You should realize by this point that it doesn’t cut it to say “infinite”, but I’ll assume every time you say “infinite” you mean “countable”. (“Countable” refers to sets with cardinality Aleph[sub]0[/sub], for reasons that escape me.)
Now, let’s assume CH for the duration of this post. That is, let’s assume that 2[sup]Aleph[sub]0[/sub][/sup] = Aleph[sub]1[/sub]. Then of course #3 is correct. But #4 is not, I believe, and here’s why…
If there are Aleph[sub]0[/sub] dice in one game, and we assume that each of the dice is distinct, then one die roll has 6[sup]Aleph[sub]0[/sub][/sup] = Aleph[sub]1[/sub] possible outcomes. Even if you rolled the dice Aleph[sub]0[/sub] times in one game, that’s Aleph[sub]1[/sub] × Aleph[sub]0[/sub] = Aleph[sub]1[/sub] possibilities. Now, if there are Aleph[sub]1[/sub] possible combinations of Yahtzee games going on, then that’s Aleph[sub]1[/sub] × Aleph[sub]1[/sub] = Aleph[sub]1[/sub] combinations of dice rolls in combinations of games. This should give you some respect for just how big of a number Aleph[sub]2[/sub] is. 
There are cardinal numbers that are above the alehps. Off the top of my head, I don’t know what they’re called, but they do exist. However, they’re still cardinals–and if you have something that’s not a cardinal, there’s no natural way to say that it’s bigger than the cardinals. The definition of the order relation on the cardinal numbers depends on the fact that each of them is the cardinality of a set.
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[symbol]w[/symbol] is the first transfinite ordinal, but I don’t think that that’s what Cantor had in mind. The “absolute” was Cantor’s term for the universal set, I believe, and that doesn’t exist in any modern set theory.
Frankly speaking i don’t what in the infinite world is all this you ppl are talking abt even after reading through but i got question : what would infinity/(infinity+1) be? 1? but is that logical?
damn my english when i’m sleepy :smack:
Nishroch Order: first of all, for infinite cardinal numbers like aleph[sub]0[/sub] it happens that aleph[sub]0[/sub]+1=aleph[sub]0[/sub]. So “infinity/(infinity+1)” is the same as “infinity/infinity”.
But “infinity/infinity” is undefined anyway, for the same reason that 0/0 is undefined: there’s no unique solution.
Nishroch Order
It depends on what infinity you’re talking about. In calculus, infinity is not a number, but a formal symbol. Infinity + 1 equals infinity. So your expression is infinity/infinity. This is not a number, but rather a description of some function whose limit you’re trying to determine. Depending on the function, the value could be any finite value or be infinite (undefined).
If you are using cardinal arithmetic, you need to specify which infinity you mean. Let’s take infinity to mean aleph-null. Aleph-null + 1 is aleph null. Division is defined in terms of multiplication. So aleph-null/aleph-null is the number x that satisfies: x * aleph-null = aleph-null. This is satisfied by any finite cardinal as well as aleph-null. So your expression is not well-defined.
You’re expression is also not well-defined for trans-finite ordinals. Omega-null + 1 is not the same as omega-null. But you still have a problem with division. There is no number that satisfies: x * (omega-null + 1) = omega-null.
According to Achernar’s TI calculator: infinity + 1 = infinity and infinity/infinity = undef. So there’s your answer according to TI.
In John Conway’s system of numbers, (taking infinity to be aleph-null) infinity/(infinity+1) is a number that’s infinitesimally less than one.
[hijack]Has anyone else read John Conway’s book On Numbers and Games? Or the fictional work inspired by Conway’s results: Surreal Numbers: How two ex-students turned on to pure mathematics and found total happiness by Donald E. Knuth?[/hijack]
They’re both on the list. I’m also looking for references on supernatural numbers as discussed in Hofstadter’s Gödel, Escher, Bach, but it seems to be hard to find information on those.
Dr. Matrix: I’ve read both of the books you mention, but it was a couple of years ago.
Here’s the one thing I don’t understand: if the continuum hypothesis is false, how do you construct aleph-1?
OK, Why then does this above not prove the continuum axiom that aleph[sub]reals[/sub] = aleph[sub]1[/sub] ?
2 to the power aleph[sub]n[/sub] = aleph[sub]n+1[/sub]
Does the set of aleph[sub]0[/sub] length binary
fractions not form a continuum on [0,1)
Interested to know why the above is not sufficient to answer the continuum puzzle.
Cheers, Keithy
OK, Why then does this above not prove the continuum axiom that aleph[sub]reals[/sub] = aleph[sub]1[/sub] ?
2 to the power aleph[sub]n[/sub] = aleph[sub]n+1[/sub]
Does the set of aleph[sub]0[/sub] length binary
fractions not form a continuum on [0,1)
Interested to know why the above is not sufficient to answer the continuum puzzle.
Cheers, Keithy
Sorry about the double post.
b.t.w. is there a correct word for the “decimal” point in a binary?
Cheers, Keithy
Radix point.
Duh…Aleph-1 is the cardinality of the reals, isn’t it? Never mind…
But that still doesn’t tell me how higher alephs are constructed if the continuum hypothesis is false.
No. 2[sup]aleph-n[/sup] = 2[sup]aleph-n+1[/sup] is a generalization of the Continuum Hypothesis called, what else, the Generalized Continuum Hypothesis. CH is undecidable. GCH is undecidable even if you assume CH. You are assuming GCH. If you assume GCH, then CH follows trivially as a special case.
I’m not sure what you mean here. Are you asking if the set of binary representations forms a continuum? If so, then yes, since the reals form a continuum and the reals map to the binary repesentations. (Glossing over . . .) It follows that the binary representations form a continuum.
You’re not going to like the answer. There are two types of existence proofs. One type, the constructive, says there exists X that satisfies P(X), and constructs an X and shows that X satisfies P(X). The other type, the non-constructive proof, just proves that there must exist some X that satisfies P(X) without giving any clues as to how one might construct such an X. If you assume CH is false, you still know that aleph-one exists and you can show that is a subset of the reals with cardinality aleph-one. You cannot construct the subset, because if you could, that would be a constructive proof of CH. You know that it exists, but you can never construct it because CH is undecidable.
OK, I need to do some more reading. Would the two books you mentioned earlier address this, or should I look elsewhere?
Ugh, I’m somewhat confused now. I was taught… no, nevermind what I was taught. It will just confuse things more. Here are the facts as I understand them. Someone please tell me if any of them are wrong:[ul][li]Aleph-0 is defined to be the cardinality of the integers.[/li][li]Aleph-0 can be shown to be the smallest cardinal number. Every subset of the integers is either finite or has cardinality Aleph-0.[/li][li]Continuum © is defined to be the cardinality of the reals.[/li][li]It can be shown that 2[sup]Aleph-0[/sup] = C and that C > Aleph-0. That is, the cardinality of the power set of the integers is C.[/li][li]Aleph-1 is defined to be the smallest cardinal number greater than Aleph-0. (Someone please tell me, is Aleph-1 known to exist? That is, can it be shown that there is a smallest cardinal number greater than Aleph-0?)[/li][li]Aleph-2 is defined to be the smallest cardinal number greater than Aleph-1. Etc. (Same question for Aleph-2 as Aleph-1.)The Continuum Hypothesis (CH) states that C = Aleph-1.[/li][li]CH can not be shown to be true or false. Either one is consistent. This is sort of reminscent of Euclid’s Fifth Postulate.[/li][li]If we assume that CH is false, then there exists some subset of the reals with cardinality Aleph-1, but it is impossible to construct this subset.[/li][li]If we assume that CH is false, then C = Aleph-N for some integer N > 1. In fact, we can assume that N = any integer and still be consistent. (Right?)[/li][/ul]Well, that’s my entire understanding of cardinalities. Anything else?
Achernar
You are spot on except for one small thing in the last bullet. It should read:[ul][li]If we assume that CH is false, then C = Aleph-N for some ordinal N > 1. In fact, we can assume that N = any integer and still be consistent. We can even assume that N is trans-finite; it could be omega-null or larger.[/ul][/li]
ultrafilter,
It’s been about 30 years since I read Surreal Numbers and I lost my copy of On Numbers and Games some time ago. So I’m going from memory here. If they address CH or GCH at all, they assume it. I am sure that they don’t deal with non-Cantorian set theory. (Where Cantorian set theory assumes CH; non-Cantorian set theory denies CH.)
Do you mean that, for every aleph, there is a cardinal greater than that aleph? Or do you mean that there is a cardinal so large that such that every aleph is less than that cardinal?
If you mean the second: given a cardinal [symbol]k[/symbol], since the cardinals are a subset of the ordinals, don’t we have aleph[sub][symbol]k[/symbol][/sub], an aleph greater than [symbol]k[/symbol]?