Re: An uncountable countable set

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.

So, it's not that a->b -> b->a, but that a=b <->
b=a.

That part seems messed up. a=b <-> b=a is just the symmetry of
identity.

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".

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.

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.

How do we know if this is the case? By enumerating all possible
properties of each. Can we do that? No. We can only say that, given the
set of properties under discussion in any given theory, the two are not
distinguishable, within that theory. We cannot say that they are
absolutely the same object.

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.

It depends on the specific theory. In a first order theory with
infinitely many primitive predicate symbols, we have no theorem schema
for doing what you suggest. But set theory has only two primitive
predicates (one if you take equality as defined) so we can state such a
theorem schema. However, we don't need to do that since the axiom of
extensionality allows us to prove x=y merely by proving Az(zex <->
zey).
And what is z besides one of the set of properties which defines the
sets x and y?

That's a confused view of the axiom of extensionality and the role of
variables.

The perception of confusion would appear to be a subjective and rather
relative phenomenon.

True. You might not really be confused about the axiom of
extensionality and the role of variables, but rather only pretending to
be.

The distinction between elements and properties is rather
tenuous. Is it not a property of y that z e y?

For any PARTICULAR z, it's a property of y that z is or is not a member
of y. That doesn't entail that y IS the set of properties that y has.

Consider each object in the universe to be a bit. Does each unique set
correspond to a unique bit string, where each object's bit posiiton is a
1 if the object is a member, and 0 if it is not?

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.

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.

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.

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.

MoeBlee

.

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