Re: A simple question?

David Marcus wrote:
MoeBlee wrote:
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'.

I think we can give a formal meaning (and I think Enderton would agree
with this): If we say "S is a well ordered set", then we really mean
"(S,R) is an ordered pair where S is a set, R is an ordering on S, and R
is a well ordering".

You left 'R' free in that formulation. But, other than 'S', there is
no free variable in 'S is a well ordered set'. So for your formulation
to work, it has to be, "There EXISTS an R such that <S R> is a well
ordered set" or "There EXISTS and R such that <S R> is structure in
which R is a well ordering of S."

If we say "S can be well ordered", then we mean "S
is a set and there exists R such that R is an ordering on S and R is a
well ordering". So, with "is", R is unbound, but with "can" it is bound.

But you see that, since we do have to bind 'R', as I mentioned above,
it turns out that "S is well ordered" is equivalent to "S can be well
ordered".

You can't take "S is well ordered" and treat it as if it has an UNbound
'R' in it; you can only treat it as if it has a BOUND 'R'. And when you
bind 'R', it turns out that "S is well ordered" is tantamount to "S can
be well ordered".

That's why people say that the axiom of choice means "Any set can be
well ordered" rather than "Any set is well ordered".

And what I'm saying is that I recognize that that is widespread
conversational style, but I think the distinction is one of nuance of
emphasis but does not have a formal counterpart. (Like the distinction
between 'but' and 'and'; conversationally, there is a difference in the
informal emphasis; but in the formalization, 'but' and 'and' get washed
over and the distinction is, as DESIRED, lost).

If I give you a set
S, then it doesn't make sense to say S is an ordered pair (S,R), but it
does make sense to say that I can find R such that (S,R) etc.

Exactly. That is my point. "S can be well ordered" stands for "There
exists an R such that <S R> is a structure in which R is a well
ordering of S." Or more succinct, "S can be well ordered" stands for
"There exists an R that is a well ordering of S." And "S is well
ordered" stands for the same thing, EXCEPT for the nuance that "S is
well ordered" suggests that we have a particular well ordering in mind,
whereas, "S can be well ordered" suggests we're less concerned with
what any of the particular well orderings might be.

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.

That's because whatever Halmos says has an implicit "for all R" wrapped
around the outside of it, and the resulting statement is probably
equivalent to your statement where the R is bound by your "there
exists" quantifier.

Well, as I mentioned to qualify that Halmos is actually less on my side
here than I first claimed, Halmos does say that a partially ordered set
is a set with a partial ordering, which I take to mean that when he
mentions a partially ordered set, he's mentioning an implict R (bound
by an existential quantifier) such that <S R> is a structure where R is
a partial ordering of S. So that would carry over to wel ordering too.

What I'm saying is that I don't like that way of speaking. I'd rather
say, "S is well ordered" to mean "There exists a well ordering of S" or
less succinctly "There exists an R such that R is a well ordering of S"
so that R is bound, and then to say "<S R> is a well order structure"
or more simply "R is a well ordering of S" or "S is well ordered by R"
when I want to mention R (UNbound) specifically.

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.

Yes. I believe it is part of the definition of the phrase "well ordered
set".

Yes, a BOUND 'R'.

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.

But, Halmos's conclusion refers to the ordering, so you can't translate
what he has as "If S is a set and ER(R is a well ordering of S), then S
is never similar to one of its initial segments" because you need R to
give meaning to "similar" and to "segment". You have to do something
like "If (S,R) is a well ordered set, then it is never similar to one of
its initial segments."

I can take it as "If there exists an R such that R is a well ordering
of S, then S is never similar to one of its R-segments", which is
pretty much the way Suppes would put it. But I have conceded that
Halmos implicitly brings in a structure, not just a set that has a well
ordering, in his discussions.

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.

But, isn't that what I'm doing? If I say I have a well ordered set
(meaning I have a set and an ordering that is a well ordering) and then
what I say about it doesn't use anything specific about the ordering
(other than that it is a well ordering), then I am being general.

I don't think that is what you're doing when you don't existentially
bind 'R'.

Anyway, I do concede the original point to you as to how people
actually do speak and regard this infomal terminology. Now we are both
splitting minute hairs about nuances of informal mathematical
discourse.

MoeBlee

.

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