Re: A simple question?

David Marcus wrote:

Would you prefer if I said that "S is a well ordered set" means "S =
(T,R) where R is an well ordering of T"? Now, both T and R are new and
unbound.

Why should I prefer that when I'm arguing AGAINST dragging in UNbound
variables. I already said what I would like "S is well ordered" to be
regarded as: "ER R is a well ordering of S."

So, you want the name of the thing to be "well ordered structure"
instead of "well ordered set". I doubt you will convince people to
switch.

I have no serious intention of changing anyone's way of speaking. I'm
just explaining to you my reasons for preferring a more "clean cut"
style, and to explain why I took the original question in this thread
the way I did.

But, why argue with the name?

Because I find the whole set of phrasing informally conflicting.
Remember, I said I'm only talking about these informal ways of writing.
Of course, there's no substantive mathematical disagreement between us
in this matter.

"Well ordered set" seems at least
as good a name as "well ordered structure".

Actually, I said "well order structure", which I'm not happy about
either. And, as I mentioned, I eschew 'set' for a set theory of which
every object is a set. I would just say "S is well ordered" (which may
use a bound existential 'R') or "S is well ordered by R", which is a
DIFFERENT statement (and uses an unbound 'R') or something like that,
and if I need to mention isomorphisms and things like that then I can
mention "<S R>".

After all, it is a set with
a well ordering. Where did you get the word "structure" from that you
used as part of the name?

'structure' is used often the way I used it. Unfortunately, it has
another, very related, yet, technically different sense too, viz. a
structure (model) for a language. (By the way, there's another
terminological glitch there, but that's another can of worms probably
best not to open right now.)

But again, I reiterate, in any given system of precise definitions,
these are not problems; Whatever problems there are occur when we use
looser conversational language, as, of course, we will, since we're not
robots.

Is the reason that you don't like "well ordered set" because you think
that something named "well ordered set" should be a set (not an ordered
pair)?

Yes, that's the basic idea (though of course I do recognize that an
ordered pair is a set too).

First, 'set' is superfluous in a theory of which there are only sets.
(Oh, by the way, there's another terminological glitch. I used to say
"there are only sets in the theory", but actually a theory is a set of
sentences closed under entailment, when what I really mean is that "Ax
x is a set" is a theorem of the theory, and even though, complicating
even further, in my view sentences themselves are sets (since sentences
are sequences of symbols and a sequence is a function and a function is
a set; for that matter, symbols themselves are sets, since the
formal-metatheory I use for formal languages is a set theory, of which
every object is a set).)

Oh boy, I hate that I'm saying all of this about informal terminology
glitches, since it is not so mathematically important but is really a
matter of writing and communication style, which should be a much lower
priority for me while my primary priority is to learn more math.

Second, yes, I think of S as what is well ordered and R as the (or one
of the) well ordering(s) of S. For <S R>, where <S R> is a well order
structure (notice, my neologism, if it is a neologism is not 'well
ordered structure' but rather is 'well order structure') and S is well
ordered and R is a well ordering of S.

'well order structure' is clunky, so, call it whatever you like, say,
'wo-structure' or whatever. Then my definitions are, starting with
Suppes's definition:

R is a well ordering of S <->
R is connected in S & Ab((~b= 0 & b subsetof S) -> Em(meb & Az(zeb -> ~
<z m>eb))).

R is a well ordering <->
ES R is a well ordering of S

S is well ordered <->
ER R is a well ordering of S

<S R> is a wo-structure <->
R is a well ordering of S

Then:

Well ordering theorem: AS S is well ordered.

And I never allow the redundcany of a predicate symbol rendered as 'is
a set' in a formula of Z set theory except when proving and mentioning
theorems about all objects being sets and related concerns.

So, since I eschew 'is a set', I don't like writing:

ASr(<S r> is well ordered -> ~ <S r> is similar to an initial segment
of <S r>)

which seems to me the bind Halmos's phrasing puts us in if we take him
too literally and also eschew the redundancy of the phrase 'is a set'.

Here you've lost me. What is wrong with what you wrote?

Because I don't like to say "<S R> is well ordered", since I would
rather say "S is well orderded by R". Actually, <S R> is well ordered
(or as others would say 'is well orderable' or 'capable of being well
ordered'), even without the well ordering theorem, since <S R> = {{S}
{S R}}, which is finite and obviously well ordered by a trivial well
ordering such as {<{S} {S R}>}. That is incredibly pedantic and beside
the point, except that it points out that, if we are literal, saying
"<S R> is well ordered" (given that I hold that "S is well ordered"
means "ER R is a well ordering of S", and I disagree that "S is well
ordered" can be rendered with an UNbound 'R') makes sense in a way I
would NOT intend.

And that is kind of the OTHER side of the coin of "G is a group". To
me, G is not a group. <G f> is a group. G is the carrier set for the
group <G f>. And f is a function on the carrier set G. But the group
itself is <G f>. At least that's the way I would write it. Other people
write as they see fit.

Do you see what I'm saying?

Sure. But, I don't think you will convince people to always say "Let (G,
+) be an abelian group"

Often people do say that. But, yes, far be it from me to convince
people who don't. I'm just giving my reasons for my own preferences.

or "Let (M,F) be a differentiable manifold".
People will say this if they need to make clear that they are using "+"
or "F", but otherwise they will probably just say "Let G be an abelian
group" or "Let M be a differentiable manifold".

Sure, but all I'm saying is that, personally, I prefer to say, "Let G
be the carrier set for an abelian group."

The book "A First Undergraduate Course in Abstract Algebra" by Hillman &
Alexanderson defines a group G as a set \hat{G} with an operation. They
keep this up for two pages, then say, "A group is more than a set in
that a group is a set with an operation satisfying the group axioms.
Although it is important that one appreciate the distinction between a
group G and its set of elements \hat{G}, consistently stressing the
difference results in notational awkwardness. Hence we shall frequently
write G for \hat{G}. For example, we shall refer to 'an elemet of the
set \hat{G} of the group G' as 'an element of the group G'. When this is
done, the context will indicate whether G represents a group or a set."

Right, that is the usual kind of understandable informality and liberty
a writer may take. Personally, I prefer to keep more precise (or at
least try), even at the cost of pedanticism and occasional awkardness.

MoeBlee

.

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