Re: A simple question?

David Marcus wrote:
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.

'S' is implicity bound by a universal quantifier. But it doesn't work
for 'R' that way. Or, tell me what formulation you have in mind where
'R' is bound by a universal quantifier to translate, specifically, "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."

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"?

First, just as a personal matter, I just don't like "<S R> is a well
ordered set." Yes, <S R> is a set, but, to me, what is well ordered is
S, not <S R> (well, actually, quite literally, <S R> is well ordered
since it is {{S} {S R}}, which is a finite set, hence it has a well
ordering; but that is not what we have in mind). So, I say S is well
ordered by R and S is a well ordered set and <S R> is a well order
structure.

Second, I don't see the relevence of your analogy. We were talking
about formalizing "S is a well ordered set" not some other larger
statement "If S is a well ordered set, then [...]"

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?

That was a mistake. I made a correcting post in which I said what I
actually meant, and it does work to the advantage of your point of
view, I admit.

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]

Yeah, that existential version is my goofup. I didn't really mean that.
And, yes, I agree that the above is correct, as I mentioned in my
correcting post.

MoeBlee

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