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Mitch · Guest

On Jul 22, 8:56 am, John Jones <jonescard...@aol.com> wrote:
[quote]JP wrote:
On Jul 22, 12:04 am, Mitch <maha...@gmail.com> wrote:
On Jul 21, 2:27 pm, John Jones <jonescard...@aol.com> wrote:

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).

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.

Yes, and a consequence of that is that objects must have variable
properties. In which case, they cannot be reidentified.
[/quote]
You>re unnecessarily confusing yourself with the 'variable properties'
idea. Like in physics, let>s just ignore friction for the moment.
Consider the 'objects' 2 and 17. Both have them have all sorts of
properties.

Do they share any properties (are they both members of some sets?)?
Yes (e.g. both are < 18, both are prime, etc.).

Do either have a property that the other has (is one in a set that the
other is not?) ? Yes. (2 is even but 17 is not, 17 is > 16 but 2 is
not).

You might think that there are still variable properties for 2 and 17
(there are so many more possible properties/sets that could be
tested), but do you really think that 2 and 17 are indistinguishable?

Of course you are trying to -find- a 'theory' of set membership that
accounts for the fact that 2 and 17 really should be considered as
distinct. What you>ve presented so far about properties and tags
really don>t say much about how the usual theory works.

[quote]How can there be an intersection of
set-determined properties with properties that are not set-determined?
There is no foundation for such a hybrid of association.
[/quote]
Sure there is. Hm.. maybe you>re still thinking in your head about
using your conception of set membership. I don>t have a good hold on
what you>re thinking, but there are perfectly reasonable, usable,
efficient conceptions of set membership that already exist. google for
them, even wikipedia. please. Pretty much any book that mentions 'set
theory' in the title should help.

[quote]the relationship between sets and elements has still not
been identified.
[/quote]
No one here has given that to you because it is considered a primitive
notion, that is, 'x is a member of S' has been culturally rendered
(has been found to be most expedient) to be an undefined notion, whose
meaning is created/implied/described by set theory axioms. Oh yeah,
and first order logic. So you might want to get some background
reading/practice with that (things like and/or/xor/not, implication,
rules of inference, etc).

That>s sort of a formal relationship, so maybe you>d be happier with
an informal one (that ends up describing what the axioms eventually
do). If you>re unhappy with the informal style (you find too many
contradictions, terms that don>t match what you think them to be) then
you can always fall back on the formalization.

Informally, a 'set' is a collection of things that are unordered and
have no repeats. This is in distinction to a 'list' (ordered, with
repeats), a 'bag' or 'multiset' (unordered, but with repeats), and
'permutation' or 'map' (ordered, no repeats). These are just names,
with many things undefined (what exactly is a 'collection'...something
that has... 'members' !).

There -are- problems with this in formal notion, but they>re not the
problems that you>ve been describing. Things like self-reference,
infinity, existence.

Mitch

David C. Ullrich · Guest

On Mon, 21 Jul 2008 15:41:24 -0700 (PDT), george <greeneg@cs.unc.edu>
wrote:

[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]
Supposing that there were such a thing as the conjunction of all
the axioms, it wouldn>t be a valid "definition", since it does not
characterize "is an element of".

You did have a valid point, of course, but you didn>t state
it very well.

[quote]Other proposals would be other axiomatizations.
[/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.)

Chris Menzel · Guest

On Tue, 22 Jul 2008 09:43:25 +0200, Balthasar <nomail@invalid> said:
[quote]On Tue, 22 Jul 2008 00:04:51 -0400, herbzet <herbzet@gmail.com> wrote:
In axiomatic set theory there is _no_ definition of elementhood;
"x is an element of A" is taken as an _undefined_ primitive.
[Someone said:]
Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.
[Menzel said]
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]
Well, I *guess* the smiley is there because it was obvious that set
theory was the context for "axioms" here.

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

Right. Though...

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

"Implicit definition" is not a formal notion...
[/quote]
It is, or at least there is a formal notion in the neighborhood. Evert
Beth defined a completely rigorous, model theoretic notion of implicit
definability which he proved equivalent to explicit definability in his
well-known Definability Theorem.

[quote]It was a common notion in German "math" (some time ago).
[/quote]
Hilbert in fact said at one point that the axioms of set theory define
membership "implicitly", though in an intuitive sense, not Beth>s.

John Jones · Guest

JP wrote:
[quote]On Jul 22, 12:04 am, Mitch <maha...@gmail.com> wrote:
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.
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.
[/quote]
Yes, and a consequence of that is that objects must have variable
properties. In which case, they cannot be reidentified.

[quote]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.
[/quote]
But is this an 'intersection'? How can there be an intersection of
set-determined properties with properties that are not set-determined?
There is no foundation for such a hybrid of association.

[quote]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.
[/quote]
That looks a little like my definition of a tag, but I think only
notionally, for the relationship between sets and elements has still not
been identified.

John Jones · Guest

george wrote:
[quote]On Jul 20, 3:59 pm, John Jones <jonescard...@aol.com> wrote:
Q. What other definitions of an element are available to us, that aren>t
explicitly or implicitly, properties or tags, or their unholy compromise?

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.
[/quote]
You can pull them out of my account. eg tags are appended to or
associated with their objects, and not with other tags. Set properties
on the other hand are appended to other properties.

It would not be hard to present this in neat, bulletted format.

But I don>t think that properties or tags do the job, for the reasons I
gave.

The axiomatic set theorists have ALREADY given some
[quote]axioms defining what sets and elements are.
[/quote]
Are you sure? I haven>t seen any. Have I missed something?

So right now, THEY>RE
[quote]WINNING
and YOU>RE LOSING.[/quote]

John Jones · Guest

Mitch wrote:
[quote]On Jul 22, 8:56 am, John Jones <jonescard...@aol.com> wrote:
JP wrote:

You>re unnecessarily confusing yourself with the 'variable properties'
idea. Like in physics, let>s just ignore friction for the moment.
Consider the 'objects' 2 and 17. Both have them have all sorts of
properties.
Do they share any properties (are they both members of some sets?)?
Yes (e.g. both are < 18, both are prime, etc.).

Do either have a property that the other has (is one in a set that the
other is not?) ? Yes. (2 is even but 17 is not, 17 is > 16 but 2 is
not).

You might think that there are still variable properties for 2 and 17
(there are so many more possible properties/sets that could be
tested), but do you really think that 2 and 17 are indistinguishable?

Of course you are trying to -find- a 'theory' of set membership that
accounts for the fact that 2 and 17 really should be considered as
distinct. What you>ve presented so far about properties and tags
really don>t say much about how the usual theory works.
[/quote]
If I want to distinguish one symbol from another, like 2 from 17, then I
could say either 1) there must be some minimal, unique set of properties
associated with each object, or 2) there is no object 2 or 17, but there
is only a minimal, unique set of actions or functions.

What I am saying is that, from the way you have described it, it looks
like 2). We don>t actually need the numbers, elements, or objects,
called 2 and 17. They become empty signs that serve only to handily
collate properties. We don>t need an object in this case.

We also need to get rid of the idea of the object if properties are
determined by set membership. Because if we can have a set of anything,
then there are no limits to set formation. If there are no limits to set
formation then we can>t claim that set-determined objects are
reidentifiable. So by getting rid of the idea of the object, and the
terminology of 'properties' we might be able to accomodate your view.

[quote]How can there be an intersection of
set-determined properties with properties that are not set-determined?
There is no foundation for such a hybrid of association.

Sure there is. Hm.. maybe you>re still thinking in your head about
using your conception of set membership. I don>t have a good hold on
what you>re thinking, but there are perfectly reasonable, usable,
efficient conceptions of set membership that already exist. google for
them, even wikipedia. please. Pretty much any book that mentions 'set
theory' in the title should help.
[/quote]
I mean only this. How can there be an intersection of A and not-A; or an
intersection of one thing and another thing; or an intersection of
set-determined properties and non-set-determined properties?

[quote]the relationship between sets and elements has still not
been identified.

No one here has given that to you because it is considered a primitive
notion, that is, 'x is a member of S' has been culturally rendered
(has been found to be most expedient) to be an undefined notion, whose
meaning is created/implied/described by set theory axioms.
[/quote]
'Culturally rendered' might mean 'learned by rote'. Which is the case I
think. Sets are taught to youngsters through traditional diagrams. This
visualisation practice has become taken for granted. When challenged it
is declared a 'primitive notion'. It isn>t a primitive notion, its just
that no-one has looked at it.

[quote]Oh yeah,
and first order logic. So you might want to get some background
reading/practice with that (things like and/or/xor/not, implication,
rules of inference, etc).

That>s sort of a formal relationship, so maybe you>d be happier with
an informal one (that ends up describing what the axioms eventually
do). If you>re unhappy with the informal style (you find too many
contradictions, terms that don>t match what you think them to be) then
you can always fall back on the formalization.

Informally, a 'set' is a collection of things that are unordered and
have no repeats. This is in distinction to a 'list' (ordered, with
repeats), a 'bag' or 'multiset' (unordered, but with repeats), and
'permutation' or 'map' (ordered, no repeats). These are just names,
with many things undefined (what exactly is a 'collection'...something
that has... 'members' !).

There -are- problems with this in formal notion, but they>re not the
problems that you>ve been describing. Things like self-reference,
infinity, existence.

Mitch
[/quote]
The problems associated with set-membership cannot be resolved by a
desription of the behaviours of objects in different types of
association, I believe. Rather, I offer the idea that a set is not
determined by its elements at all, and that the notion of a
'set-element' is neither coherent nor required.

Mitch · Guest

On Jul 22, 4:36 pm, John Jones <jonescard...@aol.com> wrote:
[quote]Mitch wrote:
On Jul 22, 8:56 am, John Jones <jonescard...@aol.com> wrote:
JP wrote:
You>re unnecessarily confusing yourself with the 'variable properties'
idea. Like in physics, let>s just ignore friction for the moment.
Consider the 'objects' 2 and 17. Both have them have all sorts of
properties.
Do they share any properties (are they both members of some sets?)?
Yes (e.g. both are < 18, both are prime, etc.).

Do either have a property that the other has (is one in a set that the
other is not?) ? Yes. (2 is even but 17 is not, 17 is > 16 but 2 is
not).

You might think that there are still variable properties for 2 and 17
(there are so many more possible properties/sets that could be
tested), but do you really think that 2 and 17 are indistinguishable?

Of course you are trying to -find- a 'theory' of set membership that
accounts for the fact that 2 and 17 really should be considered as
distinct. What you>ve presented so far about properties and tags
really don>t say much about how the usual theory works.

If I want to distinguish one symbol from another, like 2 from 17, then I
could say either 1) there must be some minimal, unique set of properties
associated with each object, or 2) there is no object 2 or 17, but there
is only a minimal, unique set of actions or functions.
[/quote]
Both seem reasonable in an informal sense (of course note the
difficulty with explaining 'set' as those objects that have the same
property in their set of properties).

As to 'there are no objects just sets of properties', sure, if you
must, but informally thinking of an object as a set of properties is
enough (I think it>s unnecessarily radical to say 'there are no
objects'. Why deny objecthood, just think of properties if there>s
ever a problem)

[quote]What I am saying is that, from the way you have described it, it looks
like 2). We don>t actually need the numbers, elements, or objects,
called 2 and 17. They become empty signs that serve only to handily
collate properties. We don>t need an object in this case.
[/quote]
Hm..yeah, you don>t -need- an object. It sure is useful to think like
that, but one can also think of sets of properties.

[quote]We also need to get rid of the idea of the object if properties are
determined by set membership. Because if we can have a set of anything,
then there are no limits to set formation.
[/quote]
That inference is pretty tenuous. What does 'anything' mean? What do
you mean by limits? This sounds like a debating tactic. (formally, you
-can>t- have a set of everything... see Russell>s paradox.).

[quote]If there are no limits to set
formation then we can>t claim that set-determined objects are
reidentifiable.
[/quote]
That doesn>t follow either. Even if there aren>t limits (there are as
above, but take the hypothetical), it still depends on -how- you form
a set which will determine if it is identifiable (or reidentifiable).

[quote]So by getting rid of the idea of the object, and the
terminology of 'properties' we might be able to accomodate your view.
[/quote]
no need to get rid of it.

[quote]How can there be an intersection of
set-determined properties with properties that are not set-determined?
There is no foundation for such a hybrid of association.

Sure there is. Hm.. maybe you>re still thinking in your head about
using your conception of set membership. I don>t have a good hold on
what you>re thinking, but there are perfectly reasonable, usable,
efficient conceptions of set membership that already exist. google for
them, even wikipedia. please. Pretty much any book that mentions 'set
theory' in the title should help.

I mean only this. How can there be an intersection of A and not-A;
[/quote]
Easily. The intersection of A and not A (or maybe this is what is
throwing you...the intersection of A and complement A) is the set
normally referred to as the 'empty set', the set that has no members.
You might have problems with that kind of set, which of course is
philosophically OK; for many centuries, 'zero' was not considered a
legitimate number. People eventually got over it.

[quote]or an
intersection of one thing and another thing;
[/quote]
easy. check which elements are common to both sets. YOu -are- talking
about the intersection of two sets, right?

[quote]or an intersection of
set-determined properties and non-set-determined properties?
[/quote]
That I don>t know. I wouldn>t try to apply the intersection operator
in such an instance.

[quote]the relationship between sets and elements has still not
been identified.

No one here has given that to you because it is considered a primitive
notion, that is, 'x is a member of S' has been culturally rendered
(has been found to be most expedient) to be an undefined notion, whose
meaning is created/implied/described by set theory axioms.

'Culturally rendered' might mean 'learned by rote'.
[/quote]
Sure, but it>s not totally arbitrary. The math behind the things is
pretty non-contingent, it>s the language it>s expressed in, the
particular choice

[quote]Which is the case I
think. Sets are taught to youngsters through traditional diagrams. This
visualisation practice has become taken for granted.
[/quote]
It>s a learning aid. Sure, one can complain about the vagueness of the
learning aid, that doesn>t necessarily say anything about the
underlying thing being learned. You complained about set notation
using curly braces; venn diagrams makes it less formal but a bit more
palatable.

[quote]When challenged it
is declared a 'primitive notion'. It isn>t a primitive notion, its just
that no-one has looked at it.
[/quote]
- Everything can be challenged/questioned.
- the informal concept of set membership has been questioned, and the
current accepted notion is to consider it as a primitive that is
defined by a set of axioms.
- if you can determine more primitive notions that then allow
definition of set-membership, then great.

[quote]Oh yeah,
and first order logic. So you might want to get some background
reading/practice with that (things like and/or/xor/not, implication,
rules of inference, etc).

That>s sort of a formal relationship, so maybe you>d be happier with
an informal one (that ends up describing what the axioms eventually
do). If you>re unhappy with the informal style (you find too many
contradictions, terms that don>t match what you think them to be) then
you can always fall back on the formalization.

Informally, a 'set' is a collection of things that are unordered and
have no repeats. This is in distinction to a 'list' (ordered, with
repeats), a 'bag' or 'multiset' (unordered, but with repeats), and
'permutation' or 'map' (ordered, no repeats). These are just names,
with many things undefined (what exactly is a 'collection'...something
that has... 'members' !).

There -are- problems with this in formal notion, but they>re not the
problems that you>ve been describing. Things like self-reference,
infinity, existence.

The problems associated with set-membership cannot be resolved by a
desription of the behaviours of objects in different types of
association, I believe. Rather, I offer the idea that a set is not
determined by its elements at all, and that the notion of a
'set-element' is neither coherent nor required.
[/quote]
OK. That>s fine but then do you have any ideas about what determines a
set? Can you make the notion of 'set-element' more coherent or give an
alternative that is coherent?

Mitch

Mitch · Guest

On Jul 22, 9:00 am, John Jones <jonescard...@aol.com> wrote:
[quote]george wrote:
The axiomatic set theorists have ALREADY given some
axioms defining what sets and elements are.

Are you sure? I haven>t seen any. Have I missed something?
[/quote]
Not that wikipedia is definitive, but it sure is convenient...

http://en.wikipedia.org/wiki/Set_theory

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

You probably want to look up some basic logic too since you need that
to understand how the axioms are processed.

Mitch

David C. Ullrich · Guest

On Tue, 22 Jul 2008 09:43:25 +0200, Balthasar <nomail@invalid> wrote:

[quote]On Tue, 22 Jul 2008 00:04:51 -0400, herbzet <herbzet@gmail.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.

Well, ok. So let>s consider the set of all axioms of ZFC instead. :-)
[/quote]
Huh? That>s what he meant. ZFC has infinitely many axioms.

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

Right. Though...

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]
Regardless, the axioms of ZFC do _not_ suffice as a definition
of set membership. Because they do not _characterize_ set
membership.

[quote]
B.
[/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 Wed, 23 Jul 2008 05:43:58 -0500, David C. Ullrich
<dullrich@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.

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. :-)

Huh? That>s what he meant. ZFC has infinitely many axioms.

Yes, I know that this is what HE "meant". Still you can>t form a[/quote]
CONJUNCTION of infinitely many axioms in FOPL. --- Which was the claim I
replied to.

And that>s why I "proposed" to consider the "set" of axioms instead.

B.

--

"For every line of Cantor>s list it is true that this line does not
contain the diagonal number. Nevertheless the diagonal number may
be in the infinite list." (WM, sci.logic)

Chris Menzel · Guest

On Wed, 23 Jul 2008 13:14:46 +0200, Balthasar <nomail@invalid> said:
[quote]On Wed, 23 Jul 2008 05:43:58 -0500, David C. Ullrich
dullrich@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.

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

Huh? That>s what he meant. ZFC has infinitely many axioms.

Yes, I know that this is what HE "meant". Still you can>t form a
CONJUNCTION of infinitely many axioms in FOPL. --- Which was the claim I
replied to.

And that>s why I "proposed" to consider the "set" of axioms instead.
[/quote]
How is that supposed to help? Definitions are sentences, at least in
the standard theory of definitions stemming from Padoa.

george · Guest

[quote]george <gree...@cs.unc.edu> writes:
Indeed, and precisely for this reason, THE CONJUNCTION OF ALL
the axioms *is* the (proposed) definition of the primitive.
[/quote]
Jesse Hughes replied:
[quote]That>s not a definition, silly person.
[/quote]
You *can>t* ARGUE with a DEFINITION!
"Definition" in the first-order sense is not sufficiently
universally defined to get THIS barred.
When YOU get some axioms determining what is vs. isn>t a
definition, WELL, PUBLISH them.

Jesse F. Hughes · Guest

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

[quote]george <gree...@cs.unc.edu> writes:
Indeed, and precisely for this reason, THE CONJUNCTION OF ALL
the axioms *is* the (proposed) definition of the primitive.

Jesse Hughes replied:
That>s not a definition, silly person.

You *can>t* ARGUE with a DEFINITION!
"Definition" in the first-order sense is not sufficiently
universally defined to get THIS barred.
[/quote]
Can you show a single source that defines the term "definition" so
broadly that the axioms of ZFC count as a definition for set
membership?

If not, it seems silly to say that the (infinite) conjunction of the
axioms is a definition. I might as well say that the conjunction is a
peach pit and you *can>t* ARGUE with *my* DEFINITION.

[quote]When YOU get some axioms determining what is vs. isn>t a definition,
WELL, PUBLISH them.
[/quote]
Brilliant, insightful, decisive rejoinder with very keen
capitalization, too. As usual.

--
"By initially making it virtually impossible to maintain a heterogenous
environment of Word 95 and Word 97 systems, Microsoft offered its customers
that most eloquent of arguments for upgrading: the delicate sound of a
revolver being cocked somewhere just out of sight." --Dan Martinez

Jesse F. Hughes · Guest

"Jesse F. Hughes" <jesse@phiwumbda.org> writes:

[quote]george <greeneg@cs.unc.edu> writes:

george <gree...@cs.unc.edu> writes:
Indeed, and precisely for this reason, THE CONJUNCTION OF ALL
the axioms *is* the (proposed) definition of the primitive.

Jesse Hughes replied:
That>s not a definition, silly person.

You *can>t* ARGUE with a DEFINITION!
"Definition" in the first-order sense is not sufficiently
universally defined to get THIS barred.

Can you show a single source that defines the term "definition" so
broadly that the axioms of ZFC count as a definition for set
membership?
[/quote]
Alternatively, what meaning of "definition" did you have in mind?

Must>ve had one, since you claim I was arguing with it.

--
Jesse F. Hughes

"[Iota]'s the smallest infinitesimal, Russell, there are smaller
infinitesimals." -- Ross Finlayson

herbzet · Guest

Chris Menzel wrote:
[quote]On Tue, 22 Jul 2008 09:43:25 +0200, Balthasar <nomail@invalid> said:
On Tue, 22 Jul 2008 00:04:51 -0400, herbzet <herbzet@gmail.com> wrote:

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

"Implicit definition" is not a formal notion...

It is, or at least there is a formal notion in the neighborhood. Evert
Beth defined a completely rigorous, model theoretic notion of implicit
definability which he proved equivalent to explicit definability in his
well-known Definability Theorem.

It was a common notion in German "math" (some time ago).

Hilbert in fact said at one point that the axioms of set theory define
membership "implicitly", though in an intuitive sense, not Beth>s.
[/quote]
I>m not familiar with Beth>s definability theorem, nor with the
formal notions of implicit and explicit definability.

A cursory google search only turns up stuff couched in technical jargon,
so I>m not sure if we>re in agreement or not!

But thanks for the interesting response!

--
hz


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