Re: Questioning the defintions of set and element.

On May 1, 1:00 pm, "porky_pig...@xxxxxxxxxxx" <porky_pig...@my-
deja.com> wrote:
On May 1, 3:42 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On May 1, 12:41 pm, "Mark" <u...@xxxxxxxx> wrote:

http://en.wikipedia.org/wiki/Collection_%28mathematics%29

You will immediately improve your mathematical vocabulary by ceasing
to rely on Wikipedia for it.

MoeBlee

But Mark is correct: both Wolfram and Wikipedia state [...]

It may be that those two sites corroborate one another in this
instance, and, of course, those sites are often correct on many
matters. My remark was general...In my opinion a good mathematical
vocabulary is not built by grabbing definitions ad hoc from such
sites, especially Wikipedia, which is nothing but ad hoc defintions
and formulations, written article-by-article (and edited by no single
responsible authority), without a systematic presentation among the
various articles on even one single branch of mathematics.

that the
'collection' is a common synonym for multiset. I looked at the
'multiset' definition on both Wolfram and Wiki sites but missed the
Wolfram definition of 'collection' as well as the fact that Wiki
redirects 'collection' to multiset. OK, mea culpa.

Though other people may have different experiences, it is my
impression that 'multiset' is a special notion in a way that
'collection' is not. It seems to me that usually when a writer refers
to a collection (itself, a quite loose rubric), he means (or at least
is ordinarily in a context of) an object still subject to the
principle of extensionality, whereas as a multiset is a some kind of
object exempt from extensionality by allowing multiple "instances" of
certain members (though, as I gave a formulation, even in an
extensional theory, we can define something close enough to the notion
of a multiset, which I would take to be formalization of the notion
unless it is formalized otherwise in some non-extensional formal
theory). In sum, given the literature I happen to have come across, I
would not use 'multiset' as synonymous for 'collection'.

For sure, such terminological matters are subject to the differences
among whatever texts and courses different people have happened to
read and take, so, understandably, my impression of this particular
terminology might not be consistent with someone elses. But still, I
don't find it a good policy to take sites such as (and especially)
Wikipedia to be definitive in such matters or even a good starting
point.

And yet, once again, the fact that 'collection' is a common synonym
for multiset does not mean more than saying that 'family' is a common
synonym for the aggregation of sets. Synonym is not a definition.

Simple stipulation of synonymy may suffice as a definition. For
example:

x is a natural number <-> x is a finite ordinal

(where 'finite ordinal' has been previously defined).

Neither sets not multisets are defined. OK, the multiset is
generalization of a set,

Not as far as I have seen the word 'multiset' in use, especially in
books on combinatorics and computability (I don't want to scurry to
look them up; if I'm doubted on this point, then so be it, it's not
worth a "my textbook can beat up your textbook" battle).

but that does not make multiset more defined
than set, ditto for 'collection' understood as a synonym of multiset.

In fact, the only time I saw the word 'collection' is as a reference
to the 'second level aggregation' of sets, just like family.

And in ordinary set theory that pretty much reduces both 'collection'
and 'family' to 'set' and, in class theory, to 'class', though, of
course, the emphasis (in the way the "picture is being drawn") toward
explanation is different (especially in such locutions as "an indexed
family").

But then
I never read anything on multisets, so may be this is a common
convention. *Still* does not make it a definition.

If Wolfram and Wikipedia take 'multiset' as synonymous with
'collection' then I can't deny that there is probably a strong trend,
a convention, in which they are used synonymously. It's just that I've
also seen a strong enough trend in which they are not synonymous, thus
conflicting conventions, and my original point was just that it's
better to get a sense of such things from math books than from,
especially Wikipedia.

MoeBlee

.

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