Re: A simple question?

MoeBlee wrote:
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'.

Well, just because you don't see it, doesn't mean it isn't (implicitly)
there. If S can be free, then why can't the (invisible) R?

Of course, any unbound variables will be bound by putting "for all"
around the entire statement.

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."

I'd do "For all S, for all R, if <S,R> is a well ordered set, then ..."

How is it different from saying "G is an abelian group"?

"[...] 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.

How would you write that more formally?

AS ER [<S,R> is a well ordering -> <S,R> is not similar to one of its
initial segments]

Seems to me I can make this true by picking R so <S,R> is not a well
ordering. I think Halmos means

AS AR [<S,R> is a well ordering -> <S,R> is not similar to one of its
initial segments]

But I have conceded that
Halmos implicitly brings in a structure, not just a set that has a well
ordering, in his discussions.

--
David Marcus
.

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