From pycyn@aol.com Sun Sep 22 12:47:57 2002 Return-Path: X-Sender: Pycyn@aol.com X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-8_1_1_3); 22 Sep 2002 19:47:56 -0000 Received: (qmail 64682 invoked from network); 22 Sep 2002 19:47:56 -0000 Received: from unknown (66.218.66.216) by m9.grp.scd.yahoo.com with QMQP; 22 Sep 2002 19:47:56 -0000 Received: from unknown (HELO imo-r06.mx.aol.com) (152.163.225.102) by mta1.grp.scd.yahoo.com with SMTP; 22 Sep 2002 19:47:56 -0000 Received: from Pycyn@aol.com by imo-r06.mx.aol.com (mail_out_v34.10.) id r.188.e62d7e4 (4402) for ; Sun, 22 Sep 2002 15:47:52 -0400 (EDT) Message-ID: <188.e62d7e4.2abf7868@aol.com> Date: Sun, 22 Sep 2002 15:47:52 EDT Subject: Re: [lojban] Sets and classes To: lojban@yahoogroups.com MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="part1_188.e62d7e4.2abf7868_boundary" X-Mailer: AOL 7.0 for Windows US sub 10509 From: pycyn@aol.com X-Yahoo-Group-Post: member; u=2455001 X-Yahoo-Profile: kaliputra --part1_188.e62d7e4.2abf7868_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit In a message dated 9/21/2002 7:11:57 PM Central Daylight Time, phma@webjockey.net 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. --part1_188.e62d7e4.2abf7868_boundary Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: 7bit In a message dated 9/21/2002 7:11:57 PM Central Daylight Time, phma@webjockey.net 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. --part1_188.e62d7e4.2abf7868_boundary--