Re: An uncountable countable set

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.

No, the indiscernibility of identicals does NOT imply the identity of
indiscernibles. You need both implications; you can't derive one from
the other. And, in first order logic, one direction can be posited only
in the semantics not in the axioms.

You prove two quantities equal by showing there is no difference, do you
not?

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).

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).

MoeBlee

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