Subject: Re: [lojban] Sets and classes

From: [email-masked]
Date: Sun, 22 Sep 2002 15:47:52 EDT

In a message dated 9/21/2002 7:11:57 PM Central Daylight Time, 
[email-masked] writes:

<<
> In mathematical usage, "class" is {klesi} (or maybe {zilkle} since x2 is 
> always sets) and "set" appears to be {selcmi}. But what of the empty set? 
> {} 
> selcmi noda, and {selcmi noda} is equivalent to {na selcmi}. So what is the 
> 
> word for "set"?
>>
Well, for 1, Lojban seems to have gone opposite to one major mathematical 
usage: in theories that allow both sets and classes, sets are usually the 
smaller ones, the classes that can be members as well as have them (there is 
a class of all sets, but a set of all sets would be self-contradictory). 
For 2, it is not clear how blotting x2 helps, since most classes are subsets 
of other classes and it is often important to know which classes those are -- 
although this can be built into the defining characteristic (x3) if you want 
always to start from scratch.
3) {cmima} seems to be, when used mathematically, just the member epsilon, so 
its x2 is anything that goes on the open side of epsilon, including the empty 
set, of which it is true that nothing is a member. Now, of course, {selcmo} 
does not exactly mean "set" but is just the inversion of epsilon, {na selcmi} 
does not mean "is not a set" but just "is not a set with the usual {zo'e} 
member" -- as it should in the case of the null-set.

xorxes:
<<
One could ask, does {lo selcmi be no da} belong to {lo'i selcmi}?
I don't see how it could.
I wonder what {lo'i} really means -- "set" or "class" (I suspect "class" -- 
in which case there is no other problem with the null set being a member; if 
"set" then there is a problem with there being such a set, let alone any 
particular set being a member of it). But I suppose the issue is whether {lo 
selcmi be no da} can be a member of the set of things satisfying {selcmi 
zo'e/da}. It can't, of course; so the conclusion is that "selcmi" does not 
mean "set," any more than {y: Ex x e y} can be the set of sets -- or even 
the class of them.

<<
{zilselcmi} should cover all sets though, including the empty one.
Yes, "is a thing just like a set but for consideration of what its members 
are," that is, all things that are meaningfully referred to by the 
expression on RHS of epsilon, including the one which never gives a true 
sentence.

<<
> > One could ask, does {lo selcmi be no da} belong to {lo'i selcmi}?
> > I don't see how it could.
>
>I don't see how it couldn't.

Then a bicycle, which is {lo selcmi be noda}, is a member of
{lo'i selcmi} too? Is there anything that is not a member?
I think this is a bit unfair, since the bicycle cannot be meaningfully 
referred to by and expression on RHS of epsilon. On the other hand, Lojban 
gives not specific way of distinguishing this case, so it is not ahpelessly 
bad point.

<<
> > {zilselcmi} should cover all sets though, including the empty one.
>
>I think selcmi should also.

Only if it can be interpreted as {selcmi be zi'o}, which may very
well end up being what happens.
Lord, I hope not. That goes along way toward making {zi'o} a sumti, first of 
all, and secondly a sumti whose referent is nothing. Fredegesus aside, the 
first would miss the role of {zi'o} (compare {se}) of making new predicates 
out of old -- not of referring to something , the second would give a name to 
what is not and so is unnameable. Neither desirable moves.

jordan:
<<
Saying that containing 0 things is the same as not being a container
would be pretty broken, though. We shouldn't just deny that 0 is a
valid number.
Not what is said, since {vasru} is also a relation between container and 
content, which keeps an empty container from being a vasru tout court (see an 
earlier go around about bottles that don't have caps). This may nt be the 
way that your (nor I) would have done it, but it is the Lojban way, leaving 
us to figure out how to deal wih empty not-quite-containers and (I suppose) 
contents running free.

<<
su'o da selcmi node ==
su'o da selcmi naku de ==
su'o da naku de zo'u da selcmi de ==
naku roda de zo'u da selcmi de
It is false that, for all X there is a Y such that X is a set
containing Y.

i.e., that says exactly what you'd expect from the the first one:
su'o da selcmi node
there is at least one set which contains nothing.
It actually says "there is at least one thing which is not a set containing 
anything," and, given Lojban's level ontology, a bicycle or the concept of 
green will do.

<<
A bicycle can't go into x1 of selcmi.
Well, a reference to a bicycle can go into the x1 of {selcmi}, but it always 
gives a false atomic sentence, whatever happens after that. Unless you want 
to cry {na'i} at this point, which might be hard to justify.

<<
Sure it isn't containing, but ja'a it is a container. Lojban's
brivla places claim more than just the relationship to the other
places. For example, as we were discussing earlier, putting something
in x1 of carce claims that it has wheels, even though there is no
place for the wheels.
A non-containing container is a container, it just isn't a vasru. As for the 
{carce} case, something that IS a carce has wheels (we pretty much agree), 
but that doesn't mean we can't put sumti into x1 of {carce} that refer to 
things without wheels. they just yield false atomic sentences, is all.

<<
It's just plain unfair to 0 to say that it's not on-par with the other
numbers here. ;P

> >We shouldn't just deny that 0 is a
> >valid number.
> 
> Nobody is denying that.

If you say that there's a special provision that if a selcmi contains
0 things it isn't a selcmi, then you are treating 0 special.
Yes, it would be unfair to say that a selcmi with 0 members is not a selcmi, 
but what is said here is that ANYTHING with zero members is not a selcmi -- 
totally even handed and on a par (as xorxes notes) with denying that a father 
with no children (ever) is not a father, i.e., that the original description 
is vacuous.

<<
Yes it does [say x1 is a set]. It is in x1 of selcmi. Of course, the 
assertion *can*
be a false one (as you would likely contend). But my point is that
da selcmi node
isn't the same as
da na selcmi
Right, it is the same as {da naku selcmi de}, your first is true, your second 
false, your third true again.

<<
Also (and back to the original thing), what about "lu'i no da" for
empty set?
Mebbe, but {le/lo/tu'o nomei} is safest.