Re: ZFC means?

David C. Ullrich wrote:
> On 9 Jan 2006 08:50:52 -0800 Charlie-Boo wrote:
> >Daryl McCullough wrote & Charlie-Boo says & Barb Knox wrote:

> >For what values of . . . is {x|. . .} a set (examples, general rules to
> >decide and create)?

> This is hilarious.
>
> There is no axiom of set theory that says that {x:...} is a set.

That has no real relevance to the question. It is only stating that
the answer is not the trivial "It is always a set due to axiom
such-and-such." There can be no such axiom as that would imply that
{x|~(xex)} is a set. The question remains, "When is {x|. . .} a
set?"

> There is an axiom that says that _if_ S is a set and P is a
> predicate then {x in S : P(x)} is a set.

Yes, you have a sufficient condition. But you mean P is a formula (or
an expressible predicate.) See e.g.
http://en.wikipedia.org/wiki/Zermelo_set_theory
: "In the modern ZFC system, the 'propositional function'
referred to in the axiom of separation is interpreted as 'any
property definable by a first order formula with parameters' "

(Once again the expressible/representable distinction eludes you! You
refer to a predicate when you mean a formula, which defines an
expressible predicate. That's at least 4 times you have confused
(misused) the notions of expressible and representable: (1) with
respect to expressing a relation for program synthesis which you
implied is representable, (2) the distinction in the Smullyan/CB
metatheorem, (3) unawareness of the terminology (writing, "Given a
wff, what are the sets 'expressed by' and 'represented by' the
wff, and in particular how do they differ (how do the definitions
differ)?") or, moreover, unawareness of the very concept of there
being multiple methods of defining a set with a wff (writing, "I
can't guess at reasonable definitions that come out different."), and
(4) now the misuse of terminology.

Yet the distinction between expressible and representable is what Godel
proved in 1931 and it is discussed routinely. And you claim to be a
judge of those who misuse terminology!: "No, I don't harp on that. I
harp on people _using_ standard terms _incorrectly_."

"Doctor, heal thyself."

(You also need to get rid of the unnecessary kludge of {x in S : P(x)}
using set notation in the variable declaration. Furthermore, written
as {x|x is S ^ P(x)} you then in fact have an instance of the original
question concerning {x|. . .} which is not so of your clumsy {x in S :
P(x)}.)

> (And hence the informal notation {x : P(x)} is a set _if_ P
> has the property that there is a set S such that every x
> satisfying P is an element of x.)

Huh??

1. "every x satisfying P is an element of x" means P contains only
non-well-founded sets. The axiom of separation doesn't refer to sets
of non-well-founded sets.

2. If you change it to "every x satisfying P is an element of S"
then you are saying that any subset of any set is a set. But that is
not what the axiom of separation is saying! It is saying that you must
have a formula that effects that separation. The condition must be
that x is in the set and a particular formula holds for x.

Now what were you saying about people who misuse terminology?
Something about them being "stupid"?

Anyway, so you now have a sufficient condition! And if you show that
it is also necessary, then you have answered the question! What's no
hilarious about finding a sufficient condition? (ZF is full of them,
actually. You just quoted (tried to quote) from one axiom.)

>"He who lives by the sword dies by the sword." would be apt if you'd caught me _using_ a term that I didn't know the definition of.

Consider the above for starters.

C-B

.

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