Re: ZFC? countable?uncountable?

On Feb 21, 1:24 pm, "zuhair" <zaljo...@xxxxxxxxx> wrote:
You say that 'Ay' means every y that exists.

What I was saying is that 'Ay' means 'for all y'. We don't need to say
'that exists'.

And you mentioned that there are two contexts of 'exists'
1) y exists no matter what since it is refering to something.
so here existance is used in the sense of referrability to something.

I wouldn't put it that way, but I'll play along...

2)y exists in a sense that y is proved to be a UNIQUE set fulfilling a
formula Q, and the prove comes from a rigorouselly defined axiomatic
set theory.

No, here it is not y that is questioned or considered to exist, but
rather whether there is a y that has the property described by 'Q'.

Although you are objecting to what you called it redundant term of
existing but informally it sometimes help(that's my opinion).

So I will name the two senses of existance you've mentioned

1) Existance by reference.
2) Existance by uniqueness.

That really doesn't capture what I was saying.

Here 1) refers to a more vivid and more shadawy form of existance,
which is related to first order logic itself and not to any theory in
it. I personally don't agree with it.

I think you're making this into something different from what I was
saying.

I'll put it more somewhat rigorously:

The two different sense I mean are:

(1) For any term 'T' in the language, for a given model (structure)
for the language, 'T' refers to some member of the universe of the
model. And if 'T' is an open term (has free variables), then we'll
need not just a model but an assigment for the variables.

(2) For any description such as "the unique object such that Q" ('Q'
is some formula for "defining" a property), we ask whether there
exists such a unique object (and instead of 'object' we use 'y'). So
ask, do we have this as a theorem:

E!yQ.

see that: Robin hood is a good man. So I am refering to
Robin hood here, Thus Robin hood exists?Strange?

That's a natural language problem. In a mathematical language, using
'Robin Hood' as an analogy, we have this:

With a Fregan approach, the name 'Robin Hood' is not allowed NOT to
refer to SOMETHING. Unlike, perhaps natural languages, with the
Fregean approach, whenever we have a term (which is like a "name" if
it is a closed term), that term will refer to something given an
interpretation of the language.

Now, will it refer to something that has the properties of being a
bandit in Sherwood forest who robs from the rich and gives to the poor
(or whatever properties you take to be those of Robin Hood?*). Not if
there is no such bandit. And so what do we do with the name
(description) 'the bandit in Sherwood forest who robs from the rich
and gives to the poor' especially as that may be our definition of the
name 'Robin Hood'? Well, that is the part I would tell you about
perhaps later.

Actually
this existance by reference you've mentioned is better terms
as 'essence' of reference.

No, PLEASE, 'essence' is a related but VASTLY more complicated
philosophical question. Please do not drag the notion of essence into
this. And, generally, there are philosophical issues to follow up with
regarding this subject, but just for right now, to get the
mathematical method in place, let's concentrate on the mathematics,
which is as I described in points (1) and (2) above.

while 2) refers to a more rigorouselly defined concept where
the vivid existance of x becomes a Proved uniqueness of x having a
specific formula ,

We need the proof in order to know that our definition will provide
for a properly referring term (i.e., that there is a unique object
that fulfills the definition). But the more fundamental point is that
there is or there is not such a unique object, regardless of which, if
either, of those disjuncts we can prove.

and the prove comes from a rigorouselly defined
axiomatic set theory. which is a sense
of existance that I like. Thought philosophically speaking
this is not enought to define existance, but yet it is acceptable that
the form of existance in logical language means actually proved
uniqueness of fulfillment of a specific formula.

Am I right?

Not quite. See my paragraph just above.

Now my question is simply the following:

when we say 'Ay' it means every y that exists.
my question is in what sense y exists, is it 1) or 2).

'Ay' means 'for all y'. And y exists, since 'y', being a variable, is
a kind of "pronoun" that does refer.

For example 'Ay y is ugly' means "For any object, call it 'it', it is
ugly." So instead of 'it' we use 'y'. "For any object, call it 'y', y
is ugly."

I always thought it is in the sense of 2)i.e Existance by provable
uniquess of x having Q .

No.

So just suppose you are working in a set theory that doesn't
prove the Uniqueness of y were y={0},but prove the uniqueness
of 0,{1} and {0,1}.

Proves or does not prove EXISTENCE and uniqueness.

And you got it wrong still. We're not concerned with proving the
uniqueness of y = {0}. We're concerned with proving:

E!yAx(xey <-> x = 0)

which is to say, we're concerned with proving that {0} is a PROPERLY
referring term.

Now suppose this theory has the power axiom.
then for z={0,1} , what would be P(z)?
would it be P(z)={0,{1},{0,1}}?
Or it would be P(z)={0,{1},{0},{0,1}}?

I already answered that a few posts ago.

And I should give you an even MORE technical answer, but it won't help
because there are too many basic confusions of yours and too much
basic knowledge that you don't have in the predicate calculus.

This is frustrating because I can't give you very helpful answers
UNTIL you learn the predicate calculus.

The second question.
In ZFC, and for N={0,1,2,3,4,...}
Define: x is a subset of N <-> Ay(yex->yeN)
Does schema of separation prove the uniqueness
Of every subset of N?
The purpose of this question is obvious.
Since the number of subsets of N is uncountablly infinite.
While we have countablly infinite number of formulas Q that
are used by the countablly infinite schema of separation.
Then the logical answer is that schema of separation alone cannot
prove the uniqueness of every subset of N.

Am I right?

I don't know. I didn't even read that paragraph you just wrote,
because I'm getting tired and frustrated trying to explain this to you
while you don't have the needed basic knowledge in predicate calculus
that is required for understanding the explanations I'm giving.

*By the way, coincidentally, last night, I watched a video of the
movie "Cartouche" with Jean Paul Belmondo and Claudia Cardinale, which
is about a character much like a Robin Hood. Interesting and
entertaining slapstick, farce, and satire, with Belmondo filling the
legend superbly and Cardinale at her most beautiful and enjoyable to
watch in the role of the "queen of the bandits" who is extraordinarily
beautifuly but earthy and pistol shootin' worthy herself.

MoeBlee

.

Relevant Pages

  • Re: ZFC? countable?uncountable?
    ... Existance by uniqueness. ... For any term 'T' in the language, ... Robin hood here, Thus Robin hood exists?Strange? ...
    (sci.math)
  • Re: ZFC? countable?uncountable?
    ... from a cetain theorum or axiom in a specified set theory. ... E!xAy ~yex. ... First prove the existence and uniqueness clause, which, in this case ... so here existance is used in the sense of referrability to something. ...
    (sci.math)
  • Re: Discussion about transformation TSP to UniqueTSP
    ... UP is a complexity class, so it is not a set of problem instances, but a set ... It depends if we assume that uniqueness of solution is ... For example let us consider language: ... transformations you're talking about. ...
    (comp.theory)
  • Re: Who did not have the least losses in 2006?
    ... If we need the uniqueness indicator, ... Joe Blogg came complaining that at 3,000 bucks he got less bonus paid ... in the language per se, as far as applied logic goes, i can't fault ...
    (rec.autos.sport.f1)
  • Re: Who did not have the least losses in 2006?
    ... If we need the uniqueness indicator, ... Note isMost will still evaluate to True. ... Joe Blogg came complaining that at 3,000 bucks he got less bonus paid ... in the language per se, as far as applied logic goes, i can't fault ...
    (rec.autos.sport.f1)
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