Re: An uncountable countable set

MoeBlee wrote:
Tony Orlow wrote:
MoeBlee wrote:
Tony Orlow wrote:
MoeBlee wrote:
Tony Orlow wrote:
No, it all rests on the notions of identity and equality. As Leibniz
pointed out, when the properties of two objects are all exactly the
same, then they are the same object. So, when we say two numbers are
equal, that means all properties of the two are equal.
Ha! The fallacy of reversing implication right there! An example of
just about the most basic fallacy.
When you say "a=b", you say "A a A b P(a)=P(b)". The two are equivlent
statements, and therefore imply each other.
I explained to you a long time ago that in general, in first order
logic we cannot state the identity of indiscernibles, even as a schema,
let alone have it implied from something else in first order logic.
However, as exception to the generalization just mentioned, in a
language with only finitely many non-logical primitive symbols, we can
state the identity of indiscernibles as a schema. And in very general
terms not tied to any specific kind of system, we may say that the
indiscernibility of identicals implies the identity of indiscernibles
only in the sense that the identity of indiscernibles is a logical
priniciple (or at least taken by many people to be a logical
principle), thus a given.

But what was incorrect in your original statement was the word 'so' in
the sense that you were RELYING on one principle to infer the other. In
that sense, you committed the fallacy of inferring B -> A from A -> B.
Looking at what I wrote now (I've beeen tied up for a bit) it's not
correct. I should have said a=b = A P P(a)=P(b).

Better would be a=b <-> AP(P(a) <-> P(b)).

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.


Since one cannot
quantify over sets, or the properties that define them, in first order
logic, this is not a first order statement, but second order.

Right.

In any
case, I agree with Leibniz that the unique identity of an object is
defined by its unique set of properties, and that equivalence between
two objects is the SAME as equivalence between the entire set of
properties of each.

'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.


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. :)


The object IS the set of properties which defines it,

That's going well beyond the principles of the identity of
indiscernibles and the indiscernibility of identicals. I think you're
going to run into some big problems if you try to have an object equal
to its set of defining properties, beginning with defining a 'defining
property' and then vicious circles that I suspect will arise from your
postulate.

I am aware that there are difficulties defining what constitutes a valid property in this sense, as Russell's Paradox demonstrates, but I think the kind of statement that produce such issues can be identified. That would be an interesting discussion....


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? If there is no universal system of cohesive mathematics, then this is surely the case. In the example of the staircase in the limit vs. the diagonal line, point-set topology cannot discern the two objects because it looks only at proximity of corresponding points. When one defines the objects using a segment-sequence topology, as I suggested, there is a very discernible difference between the two objects, namely, as I intuited from the beginning of that discussion, in direction of the curve. Thus, from the "subjective" perspective of sets of points, they are equal, but from the "subjective" perspective of sequences of segments, they are not. So, are they equal?

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


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.


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


Anyway, I don't disagree with the principle of the identity of
indiscernibles; my point is that we don't infer it (or if we do, then a
demonstration is required) from the principle of the indiscernibility
of identicals (except in the trivial sense that we can infer a logical
principle from anything at all).
No, but we can take as axiomatic that a=b = A P P(a)=P(b), and that P=Q
= A a P(a)=Q(a).

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.


Thus properties are defined by the objects to which
they pertain, and objects are defined by the properties which pertain to
them. Despite the fact that this statement is not first-order, I see no
problem arising from it.

Please just read a book on logic.

Please just think hard. :)


MoeBlee

.

Relevant Pages

  • Re: An uncountable countable set
    ... let alone have it implied from something else in first order logic. ... state the identity of indiscernibles as a schema. ... the sense that you were RELYING on one principle to infer the other. ... to its set of defining properties, ...
    (sci.math)
  • Re: An uncountable countable set
    ... let alone have it implied from something else in first order logic. ... state the identity of indiscernibles as a schema. ... the sense that you were RELYING on one principle to infer the other. ... we have no theorem schema ...
    (sci.math)