Re: Specifying Sets

Chris Menzel wrote:
On Wed, 12 Nov 2008 19:55:51 +0000, John Jones <jonescardiff@xxxxxxx> said:
Chris Menzel wrote:
On Tue, 11 Nov 2008 20:39:37 +0000, John Jones <jonescardiff@xxxxxxx>
said:
Chris Menzel wrote:
an empirically verifiable fact
about the usage of the term in mathematical contexts.
So, according to maths naming convention, a 'shopping list' is a set
I'm not sure "shopping list" is a precise enough notion to say
exactly what one is. For my part, I tend to think of a shopping list
as a piece of paper or the like on which I've scribbled some words
and phrases in random order as reminders, e.g.,

Pound pastrami
can kraut
six bagels--bring home for Emma

and not a list.
A shopping list is indeed not a mathematical list, if that is your
question. It is quite clear what a mathematical list is. Not so clear
-- as a matter of ontology -- what a shopping list is. But we do
generally know one when we see one -- at least, pre-apocalypse -- and
that's plenty good enough to get the errands done.

BTW, if this convention flies in the face of common usage don't make
sour faces by privileging one use over the other.
No one who is clearheaded about such matters would dream of it. It is
common practice in science and mathematics to co-opt terms from ordinary
language and invest them with a precise technical meaning that might
ultimately depart drastically from their ordinary meaning.

So, an unordered collection is not called a list but is called a set.
An ordered collection is called a list.
Well, ordered in the proper sort of way.

Why?
Aspects of their vague English meanings were reasonable approximations
to the corresponding defined or axiomatized mathematical notions. But,
ultimately, that is neither here nor there. The important thing is to
have a tag to which one can affix a clear definition or a rigorous set
of axioms. It can be heuristically useful to start with suggestive
terms out of ordinary language, but we could start calling mathematical
sets "lists" and vice versa and it wouldn't affect the mathematics one
bit.

Which different words am I using? The set of all x reduces to all x.

So how would you reduce "The set of all solar planets has nine
members" to a statment about "all solar planets"?
There are 9 solar planets. If we insist on employing the term
'members' then "The set of all solar planets has nine members" reduces
to 'set of nine objects',
How could it possibly, since the former is a sentence and the latter is
not?

which reduces further to '9'.
*boggle*

The point you seem to be missing is that sets are mathematical
objects.
A 'mathematical set' is the sign surrounding glyph {}.
Well, see now, mate, there's where things go all Merleau-Ponty for you.
You've not studied a lick of mathematics, your are abysmally ignorant of
the subject, yet you have the cheek to pontificate about the meaning of
mathematical discourse. The result, of course, is clueless prattle like
the above, etched into the repository of the web for all time and for
all to see. Think on these things.

Reference to them therefore requires locutions like the former. "all
x" does not pick out a set, it simply quantifies over the x's and
hence will not do the job.
If 'all' blondes picks out all blondes, what does the 'set' of all
blondes pick out?
Well, I should think the set of all blondes, wouldn't you agree? Not
that the ordinary meaning of either "set" or "pick out" is all that
clear here.

If you can answer that it will be a surprise, for everything I've seen
so far indicates that the term set is used indiscriminately.
Indeed, in ordinary discourse, "set" is entirely ambiguous. This is
hardly news, and it is not of the least significance for mathematics.
"1) The method of enumeration lists the contents or objects of the set
{The President of the United States, Donald Rumsfeld, 2005}
2) The method of description states what condition must be satisified for something to be a member of the set. {x: x is blond}: the set of all x such that x is blond, or the set of all blonds."

If you reread that, it is plain that the intention was to read 'list' in much the same way as we read a shopping list, ie. unordered. I got that quote from a logical encyclopaedia.

You may well have -- though I'd be curious what "logical encyclopedia"
this is from. Be that as it may, all you've done is illustrate that
point that "set" and "list" don't have precise meanings in ordinary
language -- you do recognize that the above is just an ordinary language
use of "list", right, and not a mathematical definition of the term?

The term list used in the quote is different from the list you claim is used by mathematics. What is the mathematical term for the list as it is used in the quote? Just use that.

And that, moreover, it is not even talking about lists, just about two
different informal ways of picking out a set?

The list in the quote doesn't pick out a set, informally or formally, or anything that could be considered being called a set. The trap is to think that where the word 'set' is used, we have a new property, or something substantive, but we don't. A list and 'all' didn't pick out a 'set' in the quoted material, despite the fact that the author thought that they did.

It's the sort of thing
one might see in the introductory chapter of an elementary set theory
text where the author provides an intuitive introduction to the subject
before actually getting into the mathematics proper.

The greatest battles are fought before the first elementary textbook can make an appearance.
.

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