Re: Aleph_Aleph_1

On Jun 26, 5:32 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:

Talking about Aleph_1 + 1 while specifying we mean ordinal
addition is simply perverse.

Ideally (in principle), there are different symbols for ordinal and
cardinal addition. I wouldn't use '+' just be itself without
specifying which symbol is meant (or have it clear by context).
(Actually, in my own notes, I either subscript or use different fonts
for different '+'s).

And, of course, it is well understood that it is ordinarily better
style not to use the ordinal '+' symbol with alephs.

There are many places in mathematics where we know what
we mean to define, and we concoct an official definition that
has all the properties we need, even though the official
definition really doesn't "look" at all like what we had in
mind. A real number is a set of rationals such that... right.

When we do that it's a very bad idea to use notation that
refers to the explicit "implementation" of the concept,
instead of referring to concepts at a higher level, which
have to do with what it "really means".

I basically agree. Indeed, I prefer definitions and treatments at
greatest generality. In addition to the example you give below, an
example that occurs to me is the notion of continuousness - one finds
out about continuous real functions, then, oh wait, it's more general,
for metric spaces, then, oh wait, even more general, for topologies. I
prefer to work the oppositie direction - from topologies, to certain
T_n-axioms, to a metric space as a certain kind of T_n, to the reals
as a metric space. Then I usually work to understand in both
directions - from general to specific and from specific to general.

But it's no always so easy to first define from general to specific,
since textbooks don't always work that way, so I find myself making
mental notes to re-organize my notes some time later.

If x is a real number
and one talks about the elements of x (_except_ in the
context of a discussion of the definition of real numbers
as Dedekind cuts) one is doing a very bad thing. It
would be a very bad thing even if there were only
one standard version of the definition. In fact talking
about the elements of a real number is _very_ bad,
because the reader may be thinking of the definition
as an equivalence class of Cauchy sequences.
Anything one says about the real number x should
make sense assuming only that we have fixed a
particular complete ordered field, called it R,
and we are assuming that x is an element of R.
If we need to talk about the elements of a real
number then (except in special contexts) we
shouldn't be calling it a real number.

Yes, I basically agree.

The situation here is not _quite_ so bad, since
my impression is that the definition of cardinals
as initial ordinals is pretty standard.

Exactly. And I made that clear from my first remark.

But "a cardinal is an initial ordinal" is _not_
a true fact about ZFC, it is a _convention_
in common _expositions_ of the theory of
ZFC. Saying "Aleph_Aleph_1 properly
refers in ZFC" needs a little qualitification
to become correct;

Sure, and I made that qualification.

much less qualification
than "Aleph_w_1 properly refers in ZFC",
since as far as I know it's true of _every_
existing exposition. It makes sense if we just
think about what things "really mean", instead
of requiring certain "implementation details".

Sure, but my remark was in the context, as I even mentioned it, of
cardinals as certain kinds of ordinals, which is a quite ordinary
context.

MoeBlee
.

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