Re: An uncountable countable set

Tony Orlow 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,

I didn't say that you said that all objects in a class are equal. In
fact that is why it is imporatant to keep clear the difference between
equality and equivalence.

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's pretty much what we do in set theory. Except we don't "ignore"
other properties; rather, we just have only a finite number of
primitives. And we don't need to refer to classes; rather sets x and y
are the same set.

..> >> Yes, it's that simple. If the object IS the unique set of logical
values
applied to all properties, then each unique set of logical values for
each statement about an object IS a unique object. :)

Whatever that means, I doubt it is the principle of the symmetry of
identity, which is that a=b <-> b=a, which makes no mentions whatsoever
of "logical values" or "properties".

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?

I understand that it is your postulate. But it is not the same
statement as the symmetry of identity. And apparently that IS difficult
for you to understand.

and the
inability to discern two objects by their properties makes them equal,
at least until some property is discovered which can discriminate
between the two.
That's going to make the theory subjective - depending on discoveries.
Why don't you look at how different mathematical theories handle
identity?
Ummm.... Isn't each isolated theory "subjective" in terms of the
properties that it explores?

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.

Sorry for not reading your mind when you mentioned, in your own scare
quotes, "subjective". As to your point, I don't want to comment at this
time since I see some philosophical complexities here I couldn't
address properly within only a paragraph. Anyway, my original point is
that a theory such as set theory avoids having the identity of objects
depend upon DISCOVERY (which is epistemological) of properties.

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

Formulas are a fine way to distinguish objects. For instance, I
distinguish a vastly greater number of different infinities than
cardinality simply by ordering formulas on a unit infinity. Good suggestion.

As usual, you seem not to recognize my point.

No, we may do better than that in theories in which there are only
finitely many primitive predicate symbols, such as set theory. I told
you all about that already.

If there are only finitely many primitive predicate symbols, then there
are only finitely many properties being addressed by the theory.

No, because properties are not just primitive but are also compound.

For
instance, set theory only uses 'e' and '=', and misses most properties
of sets.

The reason is that we are interested in those properties we need to
formulate mathematics.

And you might not be confused over the nature of objects and properties,
over inductive logic vs. inductive proof,

I may be confused about ontology, but not because I have taken some
firm stance about it while doing everything I can to not understand the
many philosophies that have been formed.

As to induction, I've never shown any confusion between inductive logic
and inductive proof. If at some time I did not realize which of the two
subjects you had in mind, then that doesn't entail that I dont'
understand the difference between the subjects but rather only that I
failed to discern of which you were addressing. I'm surprised that even
you would resort to such intellectual dishonesty as to take insinuate
that I don't know the difference in the subjects themselves only
because I may have been mistaken at some point as to which of the
subjects you were addressing.

I thought a 'bit' is a 0 or 1. In that case, in set theory, it is not
the case that each object is a bit.

Sorry, a bit position.

Now, if I were you, I'd say you don't know the difference merely from
the fact that you misspoke. But I'm not you, so I don't resort to that
kind of cheap tactic.

Number the objects in the universe starting from
0. Every unique set is therefore a unique bit string representing which
elements are members.

If that's your axiom, fine. But it contradicts set theory. So, again,
we need to be clear what theory we're talking about. But you don't have
a theory, so it's nugatory anyway. You just keep announcing disparate
axioms without stating a logistic system, primitives, or definitions.

The set y IS the set of
objects which are members of y, no more, and no less.

That's correct regarding set theory, and it conflicts with your notion
that a set is the set of its defining PROPERTIES. A set is the set of
its MEMBERS, and it is not the set of defining PROPERTIES.

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

You may define words however you like. However, you should at least be
aware when your definitions and meanings depart from those of most of
the other people in a converstation.

In second order, it would be a=b <-> AP(P(a) <-> P(b)), and P=Q <->
Aa(P(a) <-> P(a)).
If you prefer, you may use <-> instead of =.

But those don't ential that, for example, b = {P | P is a defining
property of b}.
Really, they do.

Because you say so. But you couldn't demonstrate that entailment in a
system.

I'll have to think about that. What makes you so sure?

Because YOU present yourself as so sure that you don't need to bother
with understanding what is involved in such things. But, of course, as
we recognize, appearences could be deceiving and it's possible you've
already digested an entire graduate library of mathematical logic, set
theory, and mathematics but are only pretending to know virtually
nothing about them.

Please just read a book on logic.
Please just think hard. :)

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. :)

What dismissal? YOU won't read even a SINGLE book from the ENTIRE
history of HUMAN THOUGHT on the subject.

MoeBlee

.

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