Re: An uncountable countable set

In article <44fdc460@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

MoeBlee wrote:


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.

"0 through 1"?

Does TO expect to find any truth values strictly between 0 and 1?

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....

In ZF, predicate definition of sets is limited to defining subsets of
sets which are otherwise known to exist, so that Russell's paradox is
vanquished.

Absent some mechanism of similar effectiveness, TO's system will crash.


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.

But in TO's case, the objects are not mere sets of points but much more
complicated structures, about which TO asserts many properties he has
not proved which do not hold for mere sets of points.



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?

The issues are not subjective, but matters of definition. In standard
mathematics representing such objects by sets of points is standard, and
as TO wants them represented is not.


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

If they are sets, they are indistinguishable if and only if every
member of either is a member of both and every non-member of either is a
non-member of both.

If they are TO's constructs and they are anything other than sets, it is
up to him to define what he means by equality.

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.

That is the trouble with any such property definitions, you never can
tell whether you have considered all properties.

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 only subject being confused by it is TO

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

But as every "object" in TO's universe is a bit, every bitstring is a
bit also and one does not need any strings of bits separately.



Please just read a book on logic.

Please just think hard. :)

TO rekes not his own rede.
.

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