Science and Technology news... :: View topic

John Jones · Guest

Recent posts have brought to my attention the problem of defining set
membership. A primary problem, it seems to me, is the definition of an
object or element of a set. There are at least two possibile definitions
- objects (elements) have set-defined PROPERTIES, or objects have TAGS.
Here we go then:

PROPERTIES
If objects are identified by their properties, then these properties
ought to be fixed for each object. Otherwise, if the list of properties
for an object were not fixed, but variable, then we would not be able to
consistently identify the same object.

Now, if an object>s properties are determined by set membership, and if
we still wish to be able to reidentify an object, then the number of set
memberships that that object is 'in' must also be fixed. If an object>s
set membership was not fixed, but variable, then the properties of an
object would also be variable. In which case, an object would not be
reidentifiable.

An advantage of the idea of properties, however, is that we can form
subsets. For even though the property-defined object is not
reidentifiable (!) it can have as many properties as we wish, and these
properties can be arranged as subsets, arising as subsets of an original
set of properties. But then what counts as an 'original', or source, set
of properties?

TAGS
Unlike an object which is defined by its properties, an object that is
defined through its tags is reidentifiable. It is reidentifiable because
an object has its own tag-independent properties (like price-tags -
these do not affect the objects actual properties).

A disadvantage of tags is that objects that are reidentifiable by their
tags cannot be arranged into, and identified by, their subsets. Not even
the tags can be organised into sets, for an object>s tags are appended
to the object and not to other tags.

CONCLUDING
Property-defined objects can be organised into set membership, and are
not reidentifiable; tag-defined objects are not organised into
heirarchical sets but are reidentifiable. Neither properties nor tags
satisfy the conditions of set membership - both reidentifiability and
heirarchical organization.

Q. What other definitions of an element are available to us, that aren>t
explicitly or implicitly, properties or tags, or their unholy compromise?

JP · Guest

On Jul 20, 10:59 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Recent posts have brought to my attention the problem of defining set
membership. A primary problem, it seems to me, is the definition of an
object or element of a set. There are at least two possibile definitions
- objects (elements) have set-defined PROPERTIES, or objects have TAGS.
Here we go then:

PROPERTIES
If objects are identified by their properties, then these properties
ought to be fixed for each object. Otherwise, if the list of properties
for an object were not fixed, but variable, then we would not be able to
consistently identify the same object.
[/quote]
In NL the list of properties is variable from one speaker to another
and that is why we have problems communicating using NL.
You may try to label every possible combination of properties in order
to have them fixed to these labels and not to an "object".
JP
[quote]
Now, if an object>s properties are determined by set membership, and if
we still wish to be able to reidentify an object, then the number of set
memberships that that object is 'in' must also be fixed. If an object>s
set membership was not fixed, but variable, then the properties of an
object would also be variable. In which case, an object would not be
reidentifiable.

An advantage of the idea of properties, however, is that we can form
subsets. For even though the property-defined object is not
reidentifiable (!) it can have as many properties as we wish, and these
properties can be arranged as subsets, arising as subsets of an original
set of properties. But then what counts as an 'original', or source, set
of properties?

TAGS
Unlike an object which is defined by its properties, an object that is
defined through its tags is reidentifiable. It is reidentifiable because
an object has its own tag-independent properties (like price-tags -
these do not affect the objects actual properties).

A disadvantage of tags is that objects that are reidentifiable by their
tags cannot be arranged into, and identified by, their subsets. Not even
the tags can be organised into sets, for an object>s tags are appended
to the object and not to other tags.

CONCLUDING
Property-defined objects can be organised into set membership, and are
not reidentifiable; tag-defined objects are not organised into
heirarchical sets but are reidentifiable. Neither properties nor tags
satisfy the conditions of set membership - both reidentifiability and
heirarchical organization.

Q. What other definitions of an element are available to us, that aren>t
explicitly or implicitly, properties or tags, or their unholy compromise?[/quote]

David C. Ullrich · Guest

On Sun, 20 Jul 2008 20:59:00 +0100, John Jones <jonescardiff@aol.com>
wrote:

[quote]Recent posts have brought to my attention the problem of defining set
membership. A primary problem, it seems to me, is the definition of an
object or element of a set.
[/quote]
Of course that seems like a problem to you - this is just another
example of your complete ignorance of the subject you>re
talking about.

In axiomatic set theory there is _no_ definition of elementhood;
"x is an element of A" is taken as an _undefined_ primitive.

No doubt you>re shocked by this. If so you>re overlooking the
fact that regardless of what system of definitions we adopt it>s
impossible to define _everything_ using only previously
defined terms. The terms in the first definition must necessarily
be undefined. So _something_ in the foundations of mathematics
_must_ be undefined - the standard choice is to take that something
to be "is an element of". We assume this relation has various
properties and off we go.

In particular, when we "define" AuB to be the set of all x
such that x is an element of A or x is an element of B we
do need to show from the axioms that there _is_ such
a set. Nobody>s been worrying about explaining this to
you since you>ve been stuck on silly objections indicating
you don>t understand simple English. Here>s the proof.

First, there are two standard axioms, the axiom of pairs
and the axioms of union. The axiom of pairs says informally
that for any two sets A and B there is a set {A, B}. Formally:

Note first that everything below (x, A, B, etc) is a set.

Axiom of Pairs: For every A and B there exists C such that
for every x, x e C if and only if x = A or x = B.

The set C given by the axiom is usually denoted {A,B}.
(Note I>m using "x e A" as an abbreviation for "x is an
element of A".)

The aziom of unions says informally that given a set
A there exists a set which is the union of all the elements
of A. Formally:

Axiom of Unions: For every A there exists B such
that for every x we have x e B if and only if
there exists y e A with x e y.

The set B given by the axiom is uusually denoted UA.

Now given two sets A and B the axiom of pairs says
that there is a set {A,B}, and the axiom of unions
says that then there is a set U{A,B}. We define

AuB = U{A,B}

and it>s easy to verify from the definitions that
x e AuB if and only if x e A or x e B:

Since y e {A,B} if and only if y = A or y = B, we
have x e AuB if and only if there exists y in {A,B}
with xey, if and only if xeA or xeB. QED.

[quote]There are at least two possibile definitions
- objects (elements) have set-defined PROPERTIES, or objects have TAGS.
Here we go then:

PROPERTIES
If objects are identified by their properties, then these properties
ought to be fixed for each object. Otherwise, if the list of properties
for an object were not fixed, but variable, then we would not be able to
consistently identify the same object.

Now, if an object>s properties are determined by set membership, and if
we still wish to be able to reidentify an object, then the number of set
memberships that that object is 'in' must also be fixed. If an object>s
set membership was not fixed, but variable, then the properties of an
object would also be variable. In which case, an object would not be
reidentifiable.

An advantage of the idea of properties, however, is that we can form
subsets. For even though the property-defined object is not
reidentifiable (!) it can have as many properties as we wish, and these
properties can be arranged as subsets, arising as subsets of an original
set of properties. But then what counts as an 'original', or source, set
of properties?

TAGS
Unlike an object which is defined by its properties, an object that is
defined through its tags is reidentifiable. It is reidentifiable because
an object has its own tag-independent properties (like price-tags -
these do not affect the objects actual properties).

A disadvantage of tags is that objects that are reidentifiable by their
tags cannot be arranged into, and identified by, their subsets. Not even
the tags can be organised into sets, for an object>s tags are appended
to the object and not to other tags.

CONCLUDING
Property-defined objects can be organised into set membership, and are
not reidentifiable; tag-defined objects are not organised into
heirarchical sets but are reidentifiable. Neither properties nor tags
satisfy the conditions of set membership - both reidentifiability and
heirarchical organization.

Q. What other definitions of an element are available to us, that aren>t
explicitly or implicitly, properties or tags, or their unholy compromise?
[/quote]
David C. Ullrich

"Understanding Godel isn>t about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)

Balthasar · Guest

On Mon, 21 Jul 2008 05:59:48 -0500, David C. Ullrich
<dullrich@sprynet.com> wrote:

[quote]
In particular, when we _define_ AuB to be the set of all x
such that x is an element of A or x is an element of B we
do need to show from the axioms that there _is_ such
a set.

Right. Though we might define the _predicate_[/quote]

union(c, a, b) :<-> Ax(x e c <-> x e a v x e b)

"c is an union of a and b."

without any reference to the axioms of set theory.

This would allow to check for three given sets A, B and C, if C is an
union of A and B.

For example.

Let A = {1} and B = {2, 3}, then C = {1, 2, 3} is
an union of A and B.

Indeed: for any x: x e {1, 2, 3} <-> x e {1} v x e {2, 3})

(...where for any x: x e {r, s, ..., z}" iff x = r or x = s or x = t or
.... or x = z.)

B.

JP · Guest

On Jul 21, 9:27 pm, John Jones <jonescard...@aol.com> wrote:
[quote]JP wrote:
On Jul 20, 10:59 pm, John Jones <jonescard...@aol.com> wrote:
PROPERTIES
If objects are identified by their properties, then these properties
ought to be fixed for each object. Otherwise, if the list of properties
for an object were not fixed, but variable, then we would not be able to
consistently identify the same object.

In NL the list of properties is variable from one speaker to another
and that is why we have problems communicating using NL.
You may try to label every possible combination of properties in order
to have them fixed to these labels and not to an "object".
JP

If you try to label an object with every possible combination of
properties in order to give it a fixed identity, then it would seem that
all objects will be the same, and hence unidentifiable.
[/quote]
I did not say to try to label an object with every possible
combination of properties but that every possible combination can be
categorized as an "object" or not.
If it is not, you can combine the previous non "object" labels into
new ones and so on until you get to the "obhect" level.
But that is not how we do it in NL.
JP.

Mitch · Guest

On Jul 21, 2:27 pm, John Jones <jonescard...@aol.com> wrote:
[quote]JP wrote:
On Jul 20, 10:59 pm, John Jones <jonescard...@aol.com> wrote:
PROPERTIES
If objects are identified by their properties, then these properties
ought to be fixed for each object. Otherwise, if the list of properties
for an object were not fixed, but variable, then we would not be able to
consistently identify the same object.

In NL the list of properties is variable from one speaker to another
and that is why we have problems communicating using NL.
You may try to label every possible combination of properties in order
to have them fixed to these labels and not to an "object".
JP

If you try to label an object with every possible combination of
properties in order to give it a fixed identity, then it would seem that
all objects will be the same, and hence unidentifiable.
[/quote]
Then don>t try to do that. Just label one object with the properties
it has, and another object with the properties it should have (maybe
some of the same as the first one, but maybe some that are not).

Mitch

george · Guest

On Jul 21, 6:59 am, David C. Ullrich <dullr...@sprynet.com> wrote:
[quote]In axiomatic set theory there is _no_ definition of elementhood;
"x is an element of A" is taken as an _undefined_ primitive.
[/quote]
Indeed, and precisely for this reason, THE CONJUNCTION OF ALL
the axioms *is* the (proposed) definition of the primitive.
Other proposals would be other axiomatizations.

george · Guest

On Jul 20, 3:59 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Q. What other definitions of an element are available to us, that aren>t
explicitly or implicitly, properties or tags, or their unholy compromise?
[/quote]
You simply are not in the spirit of the game here.
You don>t GET to just SAY what properties and tags are.
You MUST Give Some Axioms DEFINING what properties and
tags are. The axiomatic set theorists have ALREADY given some
axioms defining what sets and elements are. So right now, THEY>RE
WINNING
and YOU>RE LOSING.

Chris Menzel · Guest

On Mon, 21 Jul 2008 15:41:24 -0700 (PDT), george <greeneg@cs.unc.edu>
said:
[quote]On Jul 21, 6:59 am, David C. Ullrich <dullr...@sprynet.com> wrote:
In axiomatic set theory there is _no_ definition of elementhood; "x
is an element of A" is taken as an _undefined_ primitive.

Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.
[/quote]
There is no such thing in first-order logic, as there are infinitely
many axioms. And even if it were, it still wouldn>t be a definition in
any formally correct sense.

John Jones · Guest

JP wrote:
[quote]On Jul 20, 10:59 pm, John Jones <jonescard...@aol.com> wrote:

PROPERTIES
If objects are identified by their properties, then these properties
ought to be fixed for each object. Otherwise, if the list of properties
for an object were not fixed, but variable, then we would not be able to
consistently identify the same object.

In NL the list of properties is variable from one speaker to another
and that is why we have problems communicating using NL.
You may try to label every possible combination of properties in order
to have them fixed to these labels and not to an "object".
JP
[/quote]
If you try to label an object with every possible combination of
properties in order to give it a fixed identity, then it would seem that
all objects will be the same, and hence unidentifiable.

Jesse F. Hughes · Guest

george <greeneg@cs.unc.edu> writes:

[quote]On Jul 21, 6:59 am, David C. Ullrich <dullr...@sprynet.com> wrote:
In axiomatic set theory there is _no_ definition of elementhood;
"x is an element of A" is taken as an _undefined_ primitive.

Indeed, and precisely for this reason, THE CONJUNCTION OF ALL
the axioms *is* the (proposed) definition of the primitive.
Other proposals would be other axiomatizations.
[/quote]
That>s not a definition, silly person.

--
Jesse F. Hughes
"Time and again, history has shown that people who think their beliefs
trump reality lose, and lose badly. Luckily, I don>t have to listen
to you." -- James Harris on reality avoidance

Nam Nguyen · Guest

Chris Menzel wrote:
[quote]On Mon, 21 Jul 2008 15:41:24 -0700 (PDT), george <greeneg@cs.unc.edu
said:
On Jul 21, 6:59 am, David C. Ullrich <dullr...@sprynet.com> wrote:
In axiomatic set theory there is _no_ definition of elementhood; "x
is an element of A" is taken as an _undefined_ primitive.

Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.

There is no such thing in first-order logic, as there are infinitely
many axioms. And even if it were, it still wouldn>t be a definition in
any formally correct sense.

[/quote]
I could be wrong but I don>t think he meant that way literally: because
he did say "Indeed", "proposed"; and because "(proposed) definition of
the primitive" is never part of FOL, as I think you>ve mentioned. I mean
basically there>s a binary predicate symbol "epsilon", and formally that>s
it! So I think he meant something like:

All the axioms collectively "*is* the (proposed) definition of the primitive".

Imho.

herbzet · Guest

Chris Menzel wrote:
[quote]george said:
David C. Ullrich wrote:

In axiomatic set theory there is _no_ definition of elementhood; "x
is an element of A" is taken as an _undefined_ primitive.

Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.

There is no such thing in first-order logic, as there are infinitely
many axioms. And even if it were, it still wouldn>t be a definition in
any formally correct sense.
[/quote]
It could be said that the axioms implicitly define the primitive(s)
of the theory, or at least delimit the possible meanings they could
bear.

"Implicit definition" is not a formal notion, but makes sense.

--
hz

JP · Guest

On Jul 22, 12:04 am, Mitch <maha...@gmail.com> wrote:
[quote]On Jul 21, 2:27 pm, John Jones <jonescard...@aol.com> wrote:

JP wrote:
On Jul 20, 10:59 pm, John Jones <jonescard...@aol.com> wrote:
PROPERTIES
If objects are identified by their properties, then these properties
ought to be fixed for each object. Otherwise, if the list of properties
for an object were not fixed, but variable, then we would not be able to
consistently identify the same object.

In NL the list of properties is variable from one speaker to another
and that is why we have problems communicating using NL.
You may try to label every possible combination of properties in order
to have them fixed to these labels and not to an "object".
JP

If you try to label an object with every possible combination of
properties in order to give it a fixed identity, then it would seem that
all objects will be the same, and hence unidentifiable.

Then don>t try to do that. Just label one object with the properties
it has, and another object with the properties it should have (maybe
some of the same as the first one, but maybe some that are not).

Mitch- Hide quoted text -

- Show quoted text -
Personally, this is the way I think that NL operates.[/quote]
We create a definition of an "object" or a concept as a set of
properties and as a result we encounter the following 3 distinct
problems in communication:
1. The speaker>s "object" or concept has one or more properties but
not all of the definition set, IOW it is a subset.
2. The speaker>s "object" or concept has some or all of the properties
that are included in the definition set, but it has properties that
are not included, IOW we have an intersection.
3. The speaker>s "object" or concept has none of the properties of
the definition set, IOW they are different, they do not intersect.
JP.

Balthasar · Guest

On Tue, 22 Jul 2008 00:04:51 -0400, herbzet <herbzet@gmail.com> wrote:

[quote]
In axiomatic set theory there is _no_ definition of elementhood; "x
is an element of A" is taken as an _undefined_ primitive.

Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.

There is no such thing in first-order logic, as there are infinitely
many axioms.

Well, ok. So let>s consider the set of all axioms of ZFC instead. :-)[/quote]

[quote]
And even if it were, it still wouldn>t be a definition in
any formally correct sense.

Right. Though...[/quote]

[quote]
It could be said that the axioms implicitly define the primitive(s)
of the theory ...

"Implicit definition" is not a formal notion, but makes sense.

It was a common notion in German „math“ (some time ago).[/quote]

B.


www.GetXFactor.com Leading Technology, Science, Agriculture News and information	Part of the Identityscape.com network... getxfactor.com jmoodmusic.com smartbusinesschoices.com mintdepot.com lowfaresalways.com evangelicalview.com shoppingpodder.com soproudlywehail.com webnews.ws currenthumor.com

Science and Technology news... Forum Index -> Logic Forum	Goto page 1, 2, 3, 4 Next
Page 1 of 4