What is the obvious way? Obviously the inductive definition (n!=n*(n-1)! and 0!=1) won’t work. I would have defined aleph_0!=X[sub]1[/sub] x X[sub]2[/sub] x … x X[sub]aleph_0[/sub] (where x[sub]n[/n] is a set of size n) which looks at least as large as X[sub]2[/sub] x X[sub]2[/sub] … = aleph_1 (iirc).
The obvious way:
aleph[sub]0[/sub]! = aleph[sub]0[/sub]
aleph[sub]n+1[/sub]! = aleph[sub]n+1[/sub]*aleph[sub]n[/sub]!
btw, [symbol]w[/symbol] + 1 != [symbol]w[/symbol]. Ordinals are very weird.
Why is it obvious that aleph[sub]0[/sub]!=aleph[sub]0[/sub] ? I can see it might make sense, but the other feels more natural to me. My way has the advantage that it fits the ‘n!=ways of ordering n objects’ definition (I think). Or have I boobed somewhere?
What about the (w+1)!=w ? I believe you - no one could make that up. But when we were taught ordinals we didn’t get round to factorials.
Using ultrafilter’s definition and cardinal multiplication rules gives:
aleph[sub]n[/sub]! = aleph[sub]n[/sub] for all n. Well, at least for finite n.
Remember that the produce to two cardinals is the larger of the two when at least one of them is transfinite.
Shade:
You’re assuming the continuum hypothesis again. Aleph-1 is the smallest cardinal greater than aleph-0. The cardinality of the power set of the naturals is c (same as the cardinality of the reals). December didn’t show that aleph-1 > aleph_0, he showed that c > aleph_0, and hence c >= aleph-1.
ultrafilter: I’m not familiar with any standard definition of factorial for ordinals, either. Your definition:
isn’t complete, either–what would aleph-omega! be? You could extend your definition by defining (for limit ordinals a):
aleph-a! = sup{aleph-b!:b < a}
I don’t think this would be of any interest, however, for the reason DrMatrix points out.
Also:
I don’t think anyone claimed otherwise, maybe you misread Shade’s comment? He said
w + 1 = {1,2,3,…,w}, while
1 + w = w
(I assume you wrote that to mean w + 1 is not w, rather than (w+1)factorial = w.)
Golly! When DrMatrix and Ultrafilter get roped in like this it makes me want to be a rodeo cowboy!
I thought I had made an OK joke about a silly question. I know we’re fightin’ ignorance ‘n’ all but shucks, can’t we have a laugh from time to time?
How’s about the power of infinity to the power of the size of the rock that God can’t move? Smoke in your pipe and put that in!
You’d ha’ thunk from Hebe’s reply early on that it was a little funnin’. C’mon. Get with the program.
:smack: continuum hypothesis.
:smack: w+1 != 1 ! != factorial
Right, I was using the logical != operator. No factorials there.
The transfinite cardinal factorial definition is something I just pulled out of thin air. It seemed obvious to me, but maybe that’s just me.