Re: An uncountable countable set

On 19 Oct 2006 12:16:24 -0700, "MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

Lester Zick wrote:
to definition. Your definition for card(x) assumes we know what x is
to begin with.

x is a variable.

So what kind of variable is x?

It's just ridiculous that you don't understand the
basic form of a mathematical definition.

Oh I understand the basic form of mathematical definition quite well.
People have been letting mathematkers get away with it long enough
because of the arcane terminology involved. The fact is that you don't
define the v(x) you define the v.That's what definition is.You explain
a term in terms independent of objects to which it applies. You define
"cardinality" in terms which don't rely on what is defined in terms of
"cardinality". You define "infinity" in terms which don't rely on what
is defined as "infinite". In the case of "cardinality" "cardinality"
is the subject and not things such as x to which cardinality applies.

If you want it in intuitive
terms, 'x' is a pronoun, such as 'it' or an expression such as 'that
thing'. Then the definition is understood as saying that if we have a
given thing, call it 'x', then the cardinality of that thing (called
'x') is the least oridinal equinumerous with that thing (called 'x').

I'm interested in "cardinality" which is what you're supposed to be
defining and not the things to which it applies.

We're not defining the variable.

Yeah? Then why don't you try defining the variable with the subliminal
"equinumerosity" clause and see what pops up whose ratio might not be
3.14159 . . .

We're USING the variable as a "place
holder" that you plug in any object of the domain of discourse so that
the defintion tells you what the cardinality of that object is.

Well excuse the *** outta me. So why do you need a placeholder
exactly when the object to be defined is "cardinality" and not x?

That's longwinded for simply:

card(x) = the least ordinal equinumerous with x

So why don't you just say "cardinality = least ordinality"? If you
insist on the "equinumerous with x" qualification then you indeed have
a circular definition with the properties of x in both subject and
predicate where you can't explain the cardinality of x without the
equinumerosity of x which relies on the cardinality of x.

It's exactly analogous to such mathematical defintions as:

square(n) = n times n

I don't doubt that at all. However that doesn't define the square. It
just shows how to take the square of x assuming x is understood to be
a cardinal to begin with. I rather doubt "square(n)=n x n" works quite
as well for ordinals or people.

Certainly there's no evidence in your definition for
what x may be that wouldn't affect the definition of cardinality.

'x' is a variable that ranges over the entire domain of disourse.

Yeah well this "domain of discourse" would seem to be pretty vague and
problematic unless understood to begin with and conform to whatever
assumptions people want to apply to circular definitions.

x
could be instantiated to ANY object of the theory.

Sure maybe in the land of OZ where ordinals and cardinals merge.

So
you wind up with no definition for "card" that doesn't rely on x.

Of course it relies on x when you instantiate 'x' to any object of the
theory.

Not if you insist on the "equinumerous" clause in the predicate. Then
you're relying on an unstated assumption. Circular definitions are not
instantiations of any theory which can be justified as true. They are
however certainly instantiations of any theory which can be justified
as mechanically ambiguous.

This is ridiculous that you don't understand what a variable is in
mathematics.

Oh I understand variables quite well thank you very much. I just don't
understand why you use them to avoid simple abstract definitions in
terms of subject and predicate. Why not just say "cardinality is . .
.." or "infinity is . . .". This is grade school stuff, Moe. We don't
need x to tell us what cardinality is or infinity means. Maybe you do.
If x fits the bill then it fits the bill regardless of whether it is
explicitly specified. The only reason I can think of why you would
need to specify x is to smuggle in considerations not explicitly
evident in an otherwise ambiguous definition.

You
go on to claim that this means you're defining "card" but I don't see
any definition for "card". Maybe it's a counter intuitive definition?

The definition is there. Give me an object, ANY object, which is
instantitated from the variable 'x',

Why would I give you an object? You're the one doing the definition.
We expect you to explain the concept "cardinality" to us. I have no
idea what objects you're talking about. If your definition for a math
concept is so parochial as to require limitation to particular items I
suggest you specify them as part of the definition. For example, if
asked to define "infinity" and I reply "the number of infinitesimals"
I've specified the objects I'm concerned with. Or if asked to define
"cardinality" and I reply "equal differences" I've specified that the
concept of cardinality only applies to objects with equal differences.
Just don't expect someone else to do your explaining for you.

then the cardinality of that
object is the least ordinal equinumerous with that object. Again, it's
a simple as:

card(x) = the least ordinal equinumerous with x

And what if I don't want to know about x? You can't tell me what the
cardinality of x is without further specification of x such that your
phrase "equinumerous with x" has some mechanical significance
independent of the cardinality of x. Because that's exactly where your
definition for card(x) becomes circular if "equinumerous with x"
relies on the cardinality of x for its mechanical significance. And
your definition for card(x) doesn't show that it doesn't.

So what is it exactly that "set" theory allows us to do in mathematics
that we couldn't already do without it?

Any axiomatization (such set theories have axioms) allows you to show
proofs of theorems from those axioms. And definitions of operation
symbols added to the language of the theory are from primitives of the
theory as long as the axioms prove the existence of a unique object
apropros the definition. For example, ZFC proves that for any object,
there exists a unique least ordinal equinumerous with that object.
(Conditional definitions are the subject of another lessson.)

So this means we couldn't do exactly what without set theory?

Answering each of your objections without you knowing even the most
basic things about the subject is not an efficient way for me to teach
you about basic mathematical logic. You just need to do some of this
basic work for yourself with a good textbook.

Which of course is a blatant evasion. We seem to be able to do quite a
good many mathematical things without set theory (by which I assume
you mean ZFC). So set theory proves things about things. The question
I ask once more is why does anyone need set theory to do mathematics?

You seem
to be of a psychological frame of reference prevalent among modern
mathematikers that arithmetic in the form of set theory represents
some kind of TOE.

What in the world are you talking about? I never said anything like
that set theory "represents a theory of everything" (let alone that
arithmetic in the form of set theory does that). On the contrary, I've
posted that I do not make such a claim. Please do not put words in my
mouth.

I don't mean to put words in your mouth. You put plenty of words in
your own mouth and I'm just trying to decrypt their significance. You
say you set theory texts define "cardinality" in a certain way which
is pretty much circular if relying on cardinality for equinumerosity.

~v~~
.

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