Re: Existence and Non -Existence

On Jun 28, 3:23 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 27, 1:58 pm, apoorv <sudhir...@xxxxxxxxxxx> wrote:



On Jun 20, 2:21 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 19, 11:50 am, apoorv <sudhir...@xxxxxxxxxxx> wrote:

On Jun 17, 11:43 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 17, 4:09 am, apoorv <sudhir...@xxxxxxxxxxx> wrote:

For ease ,I will use 'in' for 'in_1' and 'e' for 'in_2'.

When we say 'There exists X' (at the object level),

We DON'T say that (unless as an abbreviation for something else). We
say:

There exists an x such that P.

We don't just say:

There exists an x.

The above is not well formed in our mathematical language.

'Let x be a set' ---is this well formed?

Expressions such as 'let x be a set' or 'let x be prime', etc., are
not ordinarily thought of as formal. Usually, such an expression
occurs in an informal proof or discussion. Usually it can be thought
of as an infomal way of conveying (formal or informal, as the case may
be) application of universal instantiation or existential
instantiation (as the case may be).

To say that 'x is a set' is not well formed appears equivalent to
saying '-is a set' is not an admissible predicate.

No, I didn't say 'x is a set' is not well formed. I said that such
expressions are usually informal; but I didn't say that there cannot
be a formalization.
I take it that you agree that 'x is a set' is well formed?
Further,
' x is a variable' is a part of the definition of the syntax of FOL.
To me, this appears equivalent to ' ' x is in the domain.' or 'x
exists'.

Whatever "appears" to you, the simple fact is that

Ex

standing alone as you used it or as a conjunct, is not well formed.

In the language of set theory, this would be equivalent to
'x is a set'.

No.

Ex

standing alone as you used it or as a conjunct, is simply not well
formed, not syntactical, for it to be equivlent with ANYTHING.
Well, what is the meaning of 'exists'? It just means 'is in the
domain'.
Like the square root of 2 does not exist in the domain of rational
numbers.
So, x exists <--> x is in the domain <--> x is a set.
If the last is well formed , the other too are also.


Why don't you just get a book where you can read how the syntax of the
language is specified?



There exists an x such that P

Is this to be read as ' There exists an x ( in the domain over which
the variable ranges)such that P'

As it literally reads off, it says "There exists an x such that P".
THEN, if we care to followup on that in the sense of our semantics, it
says, "In ANY model with a domain chosen for the variables to range
over, there exists an object in that domain that has the property
specified by the formula P".

What is the difference between 'There exists an x such that P'
and 'There exists an x and P'?

The second one makes no sense in ordinary predicate logic. It would be
formalized as

Ex & P

But the first supposed conjunct there - Ex - does not qualify to be a
conjunct since it is not a formula.

I am not so sure. How would you interpret
'There exists a flying dog such that its name is brute,?

You are "not so sure". Then why don't you get a book already in which
you can find out for yourself how the syntax of first order predicate
languages is given?

How would you interpret
'There exists a flying dog such that its name is brute,?

You mean how would I TRANSLATE to a first order sentence.

Bear in mind, that we can translate to various degrees of detail. Here
is one translation:

Let Fx stand for 'x flys'. Let Dx stand for 'x is a dog'. Let 'Nxy'
stand for 'x is the name of y'. Let b stand for the name 'brute'.

Ex(Fx & Dx & Nxb)

That is a formal formula, but the interpretation part (the 'Let'
clauses) is informal.



It would appear that the use of 'in' is implicit at the
object level itself.

What does that mean? The 'e' (epsilon membership symbol) is a
primitive symbol of set theory, if that's what you're getting at.

No. I am saying that the statements at the object level implicitly
use 'in' ,a relation between objects in the Domain, and the Domain.

If you want to call it 'implicit', okay, in the sense that, yes, our
method of models is set theoretical.

Since we are assuming a relation 'in' between the objects and the
domain,
the domain exists and is not a set.

No, it is simply a non sequitur that a domain of discourse is not a
set. BY DEFINITION, a domain of discourse is a set. Also, remember,
for a given formula, set of formulas, or theory, there is not "THE"
domain of discourse. Rather, there are many domains of discourse - as
different models for the language may have different domains of
discourse (each model has one domain of discourse, but there are
different models for any given language).

You STILL have not gotten a good book on introductory symbolic logic?

I admit that I am quite baffled.If in the axioms of ZFC, the variables
could range over
all sets,

You didn't answer the question. Why don't you get a good book?

And I never say "the variables range over all sets". Rather, PER a
given model for the language, the variables range over the members of
the set that is the universe for that model.

then how could the 'domain of discourse' be a set?

Why couldn't it? Given the remark I just made, Again, PER a given
model for the language, the variables range over the set that is the
universe (domain of discourse) for that model.

Alternately, if the
'domain of discourse' is a set, would every set have a power set in
the domain?

Depending on the domain of discourse, of course power sets may be
members.

Could 0
possibly contain elements not in the domain (which elements should
exist}?

There is no object that stands in the membership relation to the empty
set.

The SYMBOL '0' is defined by:

x=0 <-> Ay ~yex

So any model of the axioms maps '0' to some object O such that there
is no object (in the universe for the model) that bears to O whatever
relation the model maps from 'e'. But it is possible that '0' gets
mapped to an object that has many members but that are not in the
universe. Nevertheless, it is always a THEOREM that ~Ey ye0. And that
just reinforces by example the point I made above: The variables
range, PER a model, over the members of the universe for THAT model.
Well, the 'e' relation or any other relation
is between members 'of' the universe. The universe is not
its own member.However, there is a relationship 'in' between the
members
and the universe; a relation between members of the universe and an
entity
not in the universe.


You're getting nowhere trying to figure this stuff on your own without
some good books. It's been about two or three(?) YEARS now that I've
seen you floundering in such discussions as this one, as you still
haven't even gotten to a level of understanding of the first two pages
of an introductory textbook in symbolic logic. Three years (more
even?) and all you have to show for it are your own self-confusions.
I'm sorry, but that is a stupid methodology. Get the books, and study
them like any other reasonable person would, then you'll learn
something.


Let me address your last remarks (which are more in the nature of
distractions).
Maybe you read too much and think too little. I hope you do realize
that it is
only set theorists and priests who speak of the actual infinite and
you do give
the impression of a priest always swearing by the scriptures.the good
thing is
that you engage , so there is some hope still.
As for floundering in discussions, I guess that some of the points
that
emerged in the discussions have just passed you by--
That set theory does not prove the existence of more than one
infinity;
it just assumes it .For, in any universe of discourse with no largest
member, the assumption of one infinite member is equivalent to
assuming the existence of innumerable infinities.

That though set theory does not deal with the notion of motion,
in some intuitive sense, it is incompatible with the concept of
motion.

And it would be quite interesting to see this definition of the
empty set in your books "the empty set is the set containing
the set that contains all sets"

..--apoorv
.

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