How long is the song 'aleph-1 green bottles, hanging on the wall'?

[jack]

Poll #14435 Aleph-1 green bottles

Open to: Registered Users, detailed results viewable to: All, participants: 12

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How long is the song 'aleph-1 green bottles, hanging on the wall'

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omega-0, songs are inherently sequential
1 (8.3%)

omega-1, one verse per bottle
3 (25.0%)

omega-3, because... fish?
1 (8.3%)

Something else
4 (33.3%)

What?
3 (25.0%)

Comments

[fhtagn.livejournal.com]

That's of a length with a Grateful Dead song, isn't it?

[khoth.livejournal.com]

I'm not sure how the song goes exactly. Could you jog my memory by writing out the second verse, please?

[chess]

Slightly shorter than most Green Bottles incidents, because it gets really boring and/or complicated to sing really quickly, so people give up earlier.

[fluffymormegil]

aleph-1, because lengths are cardinal, not ordinal.

[jack]

Maybe "length" was the wrong term, but surely it's only a song if it's sequential? :)

[ptc24]

Perhaps you're looking for the ordinal of the last verse?

[jack]

Pretty much, but it seems to me like "length" can mean either "size of set" or "length of sequence"...

[azurelunatic]

Always the length of the attention span (or artificial equivalent) of those singing.

[naath]

Very short indeed, because you have to stop to argue about what aleph-1 minus 1 is.

[jack]

We did indeed argue that :)

My original assumption was that removing one bottle from aleph-1 bottles would still leave omega-1 bottles, so the song was naturally omega-0 long (but could equally well be omega-1, omega-2, etc). But other people's assumptions was that it referred to a _particular_ set of bottles, so even if at each stage, the number of bottles remaining was aleph-1, the limit would leave nothing.

[ptc24]

Does the answer depend on the continuum hypothesis? Does the question assume the inverse of the continuum hypothesis? Or am I barking up the wrong tree entirely?

[jack]

I'm not sure why -- every possible interpretation seems to assume the bottles are discrete. Am I missing something?

[ptc24]

I think if you're trying to get one verse per bottle, then your song might have to be a continuum in order to sing it all in time. FSVO "sing" and "time", of course.

You could invent a language where you use a precisely-timed note to convey an infinite amount of information, that lets you get aleph-null verses out of the way in finite time. Possibly you could even invent a really freakish sort of sound wave, where the amplitude at each instant represents a real (and thus a block of aleph-null verses), and you don't have to bother about boring things such as finite frequency or anything like that.

[jack]

Oh, I see! Right, that makes sense. (Assuming you're not picky about the verses coming sequentially.)

I'd assumed that if anything, you'd imagine a "more infinite" timeline :)

But does CH make a difference? Isn't aleph-c always at least as big as aleph-1, just CH says not bigger?

[ptc24]

I was thinking that if aleph-1 < c then aleph-1 might be a bit more tractable, possibly you wouldn't need a continuum. Of course my ideas about such scenarios are deeply hazy.