Re: An uncountable countable set

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)).

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.

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.

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.

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?

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 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.

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)).

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

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.

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. ... 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. ... 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)
  • Re: An uncountable countable set
    ... 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)
  • Re: Uniqueness of physical objects in the universe.
    ... >> the identity of indesceribles. ... "His Principle of the "Identity of Indiscernibles" states that no two ... I have to congratulate you - it's the best rebuttal I've heard thus far. ...
    (sci.math)