Re: A simple question?

David Marcus wrote:
MoeBlee wrote:
David Marcus wrote:
MoeBlee wrote:
David Marcus wrote:
To me, if someone says a set is well-ordered, they mean whatever
ordering relation the set has

First, I stress that this is a matter of how we use informal language
about set theory. So what is most important is for us to agree on our
use so we understand one another, while the question of what is the
"correct" (or at least the most common) informal use is secondary
though important too.

In accord with Suppes (here, 'e' for the epsilon membership symbol):

R well orders S <-> R is connected in S & Ab(b is a nonempty subset of
S -> Ex(xeb & Az(zeb -> ~ <z x>eR)))

Then, personally, my use:

R is a well ordering of S <-> R well orders S <-> S is well ordered by
R

I'll go along with that.

That's a two-place predicate.

So:

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

That's a one-place predicate.

I don't think I've seen that use before.

I'm just saying that when I say "S is well ordered", I mean "There
exists an R that well orders S."

Yes, I understood that.

But what does your phrase, "whatever ordering relation the set has"
mean? There can be many orderings of a set. If there is a well ordering
of the set, then one of the orderings that the set "has" is a well
ordering.

I mean whichever ordering relation has been specified on the set earlier
in the exposition by the author or, if the author hasn't specified one,
whichever ordering relation it would be natural to assume. For example,
if I define a certain subset of the reals, I would expect the reader to
assume the ordering inherited from the reals, unless I said otherwise.

Okay, I don't know, maybe that is how most people understand this. But,
personally, I would rather not presume, in the context of the
expressions "S is well ordered" or "S is not well ordered", that a
particular but unmentioned ordering is in question.

(or whatever ordering relation we are
talking about) is a well-ordering.

Okay, but then we have to have mentioned a particular ordering.

Exactly! Or, there has to be an obvious one from the context.

If
someone says "S well ordered", without mentioning a particular
ordering, then I take that in the sense of "There is a a well ordering
of S".

If someone says "S is well ordered" and it isn't clear to me which
ordering of S he is saying is a well ordering, then I ask!

Yes, if we want to know which ordering in particular the speaker has in
mind. But I can also take the statement "S is well ordered" to stand by
itself as the assertion that there exists a well ordering of S.

We should not presume that the statement means "The standard
ordering of S a well ordering", especially in a general context since,
sets having standard orderings is an exception not a generality. And
especially for the power set of omega, there is no notion whatsoever of
a standard ordering anyway.

It doesn't mean that I can construct
a new ordering relation that is a well-ordering.

What is a "new" ordering?

By "new", I meant one that I haven't mentioned before in the book or
article or newsgroup post that I'm writing and one that isn't the
natural ordering on the set which I might expect the reader to assume if
I don't specify an ordering.

All orderings of a set exist
"simultaneously". And even a standard ordering of set is not something
that exists in some kind of temporal or ontological priority over other
orderings.

The temporal ordering is that of reading the book or article or
newsgroup post.

Standard orderings may be more naturally of concern or even
more natural for us to construct as to our human interest in them, but
they are "older" than other orderings only in the sense that we as
humans may have studied them prior to other orderings. In the actual
theory, they're not somehow "older" or more "primitive" than other
orderings.

Do you see somewhere in Enderton where he uses "well-ordered" to mean a
well-ordering exists?

I don't know that the book has that, since I'm not looking for every
mention in the book, but I do find right away:

Page 196:

"Well Ordering Theorem: For any set A, there is a well ordering of A.
This theorem is often stated more informally: Any set can be well
ordered."

So, I take that to indicate that 'there is a well ordering of S' and 'S
can be well ordered' are equivalent. And I take 'S can be well ordered'
to be equivalent to 'S is well ordered', since surely "can" should not
be taken to refer to human agency but rather I take it is as
anthropomorphic only in a figurative sense. What "can" be be ordered IS
ordered, whatever our human knowledge of the particular orderings may
be.

That's why Enderton said it is informal: because the "can" is referring
to the fact that I can say, "Let's use an ordering on A that is a well
ordering". And, the Well Ordering Theorem says I am justified in saying
this, if I want to.

To me, "S is well ordered" is different from "S can be well ordered". If
someone says to me that "S is well ordered", and it isn't obvious to me
which ordering they are using on S, then I would ask them what ordering
they are referring to.

Okay, we do take this particular informal speaking differently, then.

Page 191:

"A set is well ordered by epsilon iff [...]"

That mentions not only the well ordering but the relation that is the
well ordering. And I regard 'S is well ordered by R' to entail 'S is
well ordered'.

It seems to me from your examples and my looking at the book that
Enderton doesn't say "S is well ordered" without also saying what
ordering he is referring to.

No, I haven't looked through the book to see whether he uses the plain
expression "S is well ordered" without mentioning a particular well
ordering and, for all I know, he might agree with your way of
understanding such informal locutions.

But here's Suppes:

233:

"[...] either two well-ordered sets are similar or one is similar to an
intitial segment of another".

I take that to mean that if you have two sets A1 and A2 and an ordering
R1 on A1 and an ordering R2 on A2 and R1 and R2 are well orderings, then
A1 and A2 are similar or one is similar to an initial segment of the
other.

Right. And there is some relation that is the ordering. But he doesn't
mention any specific ordering.

Yet, I'm starting to feel that your notion is closer to the usual way
the words are used and that my way of using the words is not ordinary.

First, I missed in Halmos that he mentioned that a partially ordered
set is a set WITH a partial ordering, so presumably, that would carry
over to a well ordered set so that when he talks about a well ordered
set, he's talking about a set with a well ordering (whereas, I was
(against the grain of ordinary use it is starting to seem to me)
advocating that we can talk about a well ordered set and just suppose
there is some well ordering but not supposing that we have a particular
well ordering in mind).

Also, I notice that Bernays speaks of 'well orderable', which is
another point against my argument.

Also, I didn't mention that many writers refer to not just the set S
but the pair <S R> where 'a well ordered set' is meant by these writers
to mean a pair <S R> where R is a well ordering of S.

So, I wish not to persist to argue that my way of speaking is in accord
with an ordinary way of speaking. (Nevertheless, of course, if I make
explicit definitions such as "S is well ordered <-> ER R is a well
ordering of S", then that is my prerogative, though I could not
reasonably claim that my definition is ordinary if it is indeed not
ordinary.)

And he mentions on that page:

"[...] this theorem is an analogue for well-ordered sets [...]"

236:

"[...] the fundamental theorem for well ordered sets [...]"

And my take on "can" is in accord with Suppes:

242:

"Every set can be well ordered; that is, for every set A there is a
relation R such that R well-orders A."

Yes, I agreed with how you, Enderton, and Suppes interpret "can".

And I really do NOT want to distinguish between the loose speaking
"can" and the more ontologically exact "is".

But, I think Enderton and Suppes don't think "can" is the same as your
meaning for "is".

I don't see why not. There's no modality of "can" in the formal theory.
"can" is a loose way of speaking, which is okay, but I don't know of
anything in IN the theory to distinguish 'can' from 'is'.

Halmos:

70:

"[...] on any well ordered set W [...]"

He speaks of a set W being well ordered with no mention whatsoever of a
particular ordering.

I take that to mean that whatever he is about to say is true for any set
W with an ordering R as long as R is a well ordering.

Sure, but he's not mentioning that R any more than as I mention it when
I use it as a bound variable for an existential quantifier in "S is
well ordered <-> ER R is a well ordering of S" or "S is a well ordered
set <-> ER R is a well ordering of S" so that R drops out COMPLETELY
after it's done its job as a bound variable in a definition.


70:

"If W is a well ordered set [...]"

I.e., if W is a set and R is an ordering on W and R is a well ordering,
then ...

Even though I am conceding the general point to you, in fairness to
myself, in this example, you are blatantly injecting "R" when it is
simply not mentioned.

72:

"It is easily possible for a well ordered set to be similar to a proper
subset [...]"

Again, speaking of a well ordered set without mentioning any particular
well ordering.

73:

"[...] a well ordered set is never similar to one of its initial
segments [...]"

There he even mentions segments without saying anything about the
ordering of which these are segments.

I take it to be the same ordering that is a well ordering of the entire
set.

Again, you assume "THE" well ordering [all caps added]. There usually
is not THE well ordering of the set, since there may be many well
orderings of the set. There is "the" well ordering only as soon as we
specify WHICH well ordering we are talking about so as to define "THE
well ordering such that [...]" So what he says applies to ANY well
ordering of the set.

"[...] comparability theorem for well ordered sets [...]"

"[...] if X and Y are well ordered sets [...]"

And more on that page mentioning "well ordered sets" with no context
whatsoever as to what the well ordering is.

It is the ordering that is also a well ordering.

Again, you inject "the".

And, in a theory in which every object is a set, such as ZF, I really
do not see any point in distinguishing between "S is a well ordered
set" and "S is well ordered".

Both of these phrases mean to me that we are talking about a set S, an
ordering R on S, and R is a well ordering.

But why can't we talk in greatest generality too? I can just say that
there exists SOME well ordering of the set, without mentioning a
particular well ordering. Then I can talk about such sets that have
some well ordering on them.

Anyway, as to the way people do in fact informally speak ordinarily, I
think you are right. I guess I'm just saying now that it, if the use is
made clear in any given exposition, it would be just fine to say "S is
a well ordered set" to mean "There exists a well ordering of S."

MoeBlee

.

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