Re: An uncountable countable set

In article <44fdcf67@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

MoeBlee wrote:
Tony Orlow wrote:
The difference between = and <-> disappears when logical truth values
are quantities from 0 through 1, so I don't see that as any better, but
equivalent.

You say, in the absence of having specified a syntax for a language in
which this all happens.

'equality' would be a better word than 'equivalence' here, I think.
I suppose, though the same applies to "equivalence classes" doesn't it?
No matter.

No, that is the point. There is a difference between members of an
equivalence class and the equivalence class itself.

I didn't say that all objects within a class are EQUAL, but given some
criterion for distinguishing objects, one can form CLASSES where a given
property is the same for all members of any given class, ignoring all
other properties.

That raises the issue of which classes are also sets and which are
"proper" classes.

In ZF, ZFC and NBG, one can avoid the issue of equivalence classes
entirely by having a unique representative for each of what would
otherwise require an equivalence class, and comparing arbitrary objects
to the representative objects.

This is the principle used in NBG using the von Neumann naturals as
representatives of both finite cardinality and finite ordinality.
So that there no equivalence classes need be posited.


All I was saying is that if the set of property values IS the object,
then the object IS the set of property values. Is that so difficult to
understand?

Since that requires that classes of properties must all be sets, it
raises all sorts of problems, as well as being a breeding ground for
Russell's paradox and other anomalies.

A theory is a set of sentences closed under entailment. Theories are
not made subjective for our reasons for interest in them. The
subjectivity is in our deciding to study one theory and not another,
but as a set of sentences closed under entailment, the theory itself is
not affected by whether we are interested in it or not or by our
reasons for interest or disinterest in it.

You must need another cup of tea. I am not talking about psychological
subjectivity, but the fact that any normal theory only addresses certain
properties of the objects is discusses, and therefore may not have
distinctions that are available in other theories.

That presumes that the 'properties' in one theory are even available in
another theory, or that, even if they are, that the objects they apply
to are the same in both. TO's list of axiomatic (unverifiable)
assumptions is growing by leaps and bounds.


Two objects are equal only if there exists no way to distinguish them.

See, that is what is subjective (or epistemological). We don't define
equality by "way to distinguish" but rather by FORMULAS.

Actually "equality" in any theory is by definition.
In ZF, ZFC, NBG, equality of sets is determined exclusively by what
objects are members of the sets being compared. Any other "properties"
are irrelevant.

If there are only finitely many primitive predicate symbols, then there
are only finitely many properties being addressed by the theory. For
instance, set theory only uses 'e' and '=', and misses most properties
of sets.

Which properties are mostly, if not entirely, irrelevant to set theory.


An OBJECT is a set of its defining properties, and a set is a collection
of objects which share one or more properties.

Not in ZF, ZFC or NBG, nor any related set theory. And there is no
coherent set theory in which things "are" their properties.

Yes, If I just think hard enough, in a blaze of enlightenment I'll see
that you and you alone have the answers.

There's lots of people working on answers in the face of this kind of
dismissal. :)

And a lot of people issuing this kind of dismissal to the garbage those
people are proposing.

Others have been there and done that and learned the hard way that it
doesn't work very well, if at all.

First thing you know, TO will reinvent a theory of types, or some such
anachronism.
.

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