Re: A simple question?

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

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.

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.

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

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

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'.

MoeBlee

.

Relevant Pages

  • Re: A simple question?
    ... I mean whichever ordering relation has been specified on the set earlier ... And even a standard ordering of set is not something ... "Well Ordering Theorem: For any set A, there is a well ordering of A. ... That's why Enderton said it is informal: because the "can" is referring ...
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  • Re: A simple question?
    ... assumes the axiom of choice. ... that a set need not be well ordered, even if a well-ordering of the set ... The standard ordering is not a well ordering, ... ordering relation the set has (or whatever ordering relation we are ...
    (sci.math)