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george · Guest

On Jul 23, 9:32 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
[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?
[/quote]
Appeal to authority IS A FALLACY, DIPSHIT.
Can you show a single source that PURPORTS to "define"
"set membership"??
Or to "define" "addition"?
Or to "define" number?

Just what IS a set (or a number) ANYway?

The WHOLE POINT of this whole enterprise is that
ANY OLD *model* will DO!

[quote]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]
That is not a valid analogy and you don>t even yourself believe that
it is;
you are just proposing it because it is an absurdity. You basically
just
HAVE said that a conjunction is a peach pit, and the fact that you>ve
said that is YOUR problem, NOT anybody else>s. "Conjunction" (in our
context) HAS a definition, although you would (again, in our context)
BE RIDICULOUSLY HARD-PRESSED TO *STATE* it without waxing
CIRCULAR. Seriously, you just wind up defining "conjunction" as "and"
and "and" as "/\" and "/\" BY ITS TRUTH-TABLE, except that you THEN
run into the problem that "a definition is NOT a truth-table"!

You really haven>t thought this through at all.

[quote]When YOU get some axioms determining what is vs. isn>t a definition,
WELL, PUBLISH them.

Brilliant, insightful, decisive rejoinder
[/quote]
As usual.

My point is, YOU DON>T HAVE A SOURCE, EITHER.
Or you have conflicting sources.
Everybody who purports to define "definition" has to acknowledge that
different
things are going to be appropriate in different contexts, ESPECIALLY
for things
that DON>T EVEN HAVE a first-order definition.

george · Guest

On Jul 23, 9:45 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
[quote]Alternatively, what meaning of "definition" did you have in mind?

Must>ve had one, since you claim I was arguing with it.
[/quote]
Obviously, terms can be used in more than one way in different
contexts.
I know perfectly well what meaning[s] of definition YOU had in mind,
SINCE I HAVE a philosophy degree.

Maybe you should stop playing dumb (or dipshit) and engage
intellectually
and appropriately instead of in a 4th-grade pissing contest.

george · Guest

On Jul 22, 1:02 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
[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.
[/quote]
You KNEW this and you gave THAT response TO ME
*anyway*??
Aren>t you getting a little OLD to be performing acts assholic enough
to land you in hell??

george · Guest

[quote]On Mon, 21 Jul 2008 15:41:24 -0700 (PDT), george <gree...@cs.unc.edu
said:
Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.
[/quote]
On Jul 21, 7:19 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
[quote]There is no such thing in first-order logic, as there are infinitely
many axioms.
[/quote]
BITCH, *PLEASE* -- YOU>RE the professor. YOU>RE the one who is going
to
be held to the higher standard. There are PLENTY of infinite
collections
in first-order logic, SINCE EVERY first-order language with at least a
0-ary
functor, a 1-ary functor, and 1-ary predicate IS INFINITE.
The fact that terms or atomic applications of predicates-to-lists-of-
terms
are never (in that context) themselves infinite does NOT stop all of
them
(and all subcollections of them) from EXISTING and it does not stop
anyone from
MODELING those infinite subcollections; indeed, given a recursive
INFINITE set of axioms, the question of whether it does or doesn>t
have a model is arguably WHAT THE WHOLE FIELD IS ABOUT, SO IT WOULD BE
STUPID to allege that their conjunction "doesn>t exist". Infinitary
logic exists. I realize that infinitary logic is not first-order
logic but we are talking about the simultnaeous truth of an infinite
collection OF FIRST-ORDER SENTENCES, so to say that that does not
exist "in" first-order logic is an unreasonably restrictive sense of
"in", AND YOU KNEW THAT.
Besides, there are finite axiomatizations of set theory, AND YOU KNEW
THAT,
TOO.

At some point you really do HAVE to just STOP BEING AN ASSHOLE
and make an effort to be intellectually co-operative.

[quote]And even if it were, it still wouldn>t be a definition in
any formally correct sense.
[/quote]
THAT is "formally correct"'s problem, NOT anybody else>s.

george · Guest

[quote]On Mon, 21 Jul 2008 15:41:24 -0700 (PDT), george <gree...@cs.unc.edu
said:

Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.
[/quote]
slipping back down into sophomoric undergrad mode,
On Jul 21, 7:19 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
[quote]There is no such thing in first-order logic, as there are infinitely
many axioms.
[/quote]
This is insufferable, frankly.
I HAVE a philosophy degree.
I KNOW that standard classical first-order logic does not have
infinitary
conjunctions. I>VE BEEN POSTING HERE for over 20 years.
Everybody who has ALSO been here that whole time, INCLUDING
PROF.MENZEL,
*ALREADY KNOWS* that I already know that the standard doesn>t have
infinite
conjunctions. SO WHY is it being belabored?? Could it be....
SATAN ??
Could it be that people are just getting curmudgeonly in their old
age ??

[quote]And even if it were, it still wouldn>t be a definition in
any formally correct sense.
[/quote]
That is JUST STUPID.
There are DIFFERENT senses of "definition" AND ALL of them are
"formally correct" IF or when one BOTHERS to formalize them!
ANYthing can be formally correct IF it is well-defined and consistent
enough TO GET consistently axiomatized! Whether that consistent
axiomatization has or has not achieved any collegiate popularity IS
NOT even IMPORTANT as far as its "correctness" is concerned!!

AFTER you decide to be in the context of first-order logic, you can
have a USUAL
treatment of "definition" under that framework that looks something
like this:

[quote]Gloss of "Definition"

Definition - an explicit eliminable definition
[/quote]
Please note the circularity.
If I were talking to halfway reasonable people INSTEAD OF ASSHOLES
then I COULD JUST STOP here, the point being that EVEN YOUR stupidly
narrow sense ALREADY concedes that there CAN be OTHER KINDS of
definitions.

[quote]of a mathematical operator or concept ;
In the sense of a definition of a mathematical operator or concept,
we mean an eliminable definition that does not have any further
epistemic content. The "meaning" of an expression should remain
unchanged when an a definiendum is replaced by it definiens.
In a program source this is meant to be like a macro; in a mathematical
discourse, the intention is that the truth value or provability of any assertion
is preserved by definition elimination or introduction, although the proof
itself may require modification with regard, for example, to whether the
definiendum is mentioned in inference specifications.

While the creation of a definition provides a new, though semantically
shallow, resource for possible use in expression or argument,
no substantial epistemic content is implied by the definition;
this is in contrast to the addition of axioms or primitive rules
of inference. The same goes for theorems and derived rules of inference.
[/quote]
This is (OBVIOUSLY) ASSUMING A CONTEXT in which the default is to
start
with the language and then worry about what you can or cannot ADD, by
way
of new abbreviations or sentences to some axiom-list. THAT IS NOT
EVEN
THE CONTEXT WE WERE IN in talking about "definitions" of "set"!!
THERE, OBVIOUSLY, the context was WE ALREADY INTUIT what numbers
are and what sets are, and are trying to come up with axioms
CONSISTENT WITH AND CONGRUENT TO the *definitions* of these concepts
*ALREADY IN OUR
NATURAL*-language UNDERSTANDING. "Set" and "number" HAVE definitions
IN THE DICTIONARY, DESPITE not having eliminable definitions! One
could say something next about when inductive definitions are
eliminable, but that is just COMPLETELY MISSING THE POINT.

The point IS that you>re NOT educating anybody by responding with
something canned from a sophomore seminar.

Chris Menzel · Guest

On Thu, 24 Jul 2008 06:36:11 -0700 (PDT), george <greeneg@cs.unc.edu> said:
[quote]On Jul 22, 1:02 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
Hilbert in fact said at one point that the axioms of set theory define
membership "implicitly", though in an intuitive sense, not Beth>s.

You KNEW this and you gave THAT response TO ME *anyway*??
[/quote]
Er, well, I wasn>t talking to you. I was providing a small confirmation
of a remark of Balthasar>s, who noted that "implicit definition" used to
be a "common notion" in German mathematics.

Such a bizarre response to an utterly innocuous post.

Chris Menzel · Guest

On Thu, 24 Jul 2008 06:44:17 -0700 (PDT), george <greeneg@cs.unc.edu>
said:
[quote]
On Mon, 21 Jul 2008 15:41:24 -0700 (PDT), george <gree...@cs.unc.edu
said:
Indeed, and precisely for this reason, THE CONJUNCTION OF ALL the
axioms *is* the (proposed) definition of the primitive.

On Jul 21, 7:19 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
There is no such thing in first-order logic, as there are infinitely
many axioms.

BITCH, *PLEASE* -- YOU>RE the professor.
[/quote]
Careful now. You got unplonked from my newsreader for some reason, but
back you>ll go if you *still* insist upon responding like an angry 13
year old.

[quote]YOU>RE the one who is going to be held to the higher standard. There
are PLENTY of infinite collections in first-order logic, SINCE EVERY
first-order language with at least a 0-ary functor, a 1-ary functor,
and 1-ary predicate IS INFINITE.
[/quote]
Well, of course. However, there are no infinite conjunctions in
first-order logic, which was the context of the thread. And since the
formal notion of definition in first-order logic is given in terms of
sentences, not infinite sets of sentences, your suggestion doesn>t fly.
Of course, if you>d like to provide an alternative notion of definition,
that>s fine, but all you did was assert (well, shout) that the
(infinite) conjunction of all the axioms of ZF constitutes a definition.
Since you didn>t say you were using "definition" in a sense other than
the only formal notion in the neighborhood, I responded accordingly and
pointed out that what you said was not feasible.

[quote]The fact that terms or atomic applications of predicates-to-lists-of-
terms are never (in that context) themselves infinite does NOT stop
all of them (and all subcollections of them) from EXISTING and it does
not stop anyone from MODELING those infinite subcollections; indeed,
given a recursive INFINITE set of axioms, the question of whether it
does or doesn>t have a model is arguably WHAT THE WHOLE FIELD IS
ABOUT, SO IT WOULD BE STUPID to allege that their conjunction "doesn>t
exist".
[/quote]
Actually, it wouldn>t. It would be exactly right. In the context of
first-order logic (which was the context of the thread), definitions, in
the formal sense introduced by Padoa, are are given in terms of
sentences. So to propose that an infinite set of sentences can serve as
a definition -- in the only formal sense in the neighborhood -- is not
tenable. Again, if you>d suggested that you were proposing an
alternative notion of "definition", I>d likely have responded
differently. But you didn>t.

[quote]Infinitary logic exists.
[/quote]
Indeed. But that was not the logic at issue in the thread.

[quote]I realize that infinitary logic is not first-order logic but we are
talking about the simultnaeous truth of an infinite collection OF
FIRST-ORDER SENTENCES, so to say that that does not exist "in"
first-order logic is an unreasonably restrictive sense of "in", AND
YOU KNEW THAT. Besides, there are finite axiomatizations of set
theory, AND YOU KNEW THAT, TOO.
[/quote]
You are changing the subject. I was responding to *your* claim about
an "infinite conjunction" serving as a definition.

[quote]At some point you really do HAVE to just STOP BEING AN ASSHOLE and
make an effort to be intellectually co-operative.
[/quote]
It is terribly odd that you consider someone an asshole for simply
giving a response that you do not feel adequately addresses what you had
in mind, especially in a slap-dash forum like sci.logic. If you were
genuinely interested in intellectual cooperation, you would have
responded with something along the lines of: "Well, let me be more
specific about what I had in mind." A civil conversation could then
ensue. But you seem far more interested in abuse and bombast.

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

THAT is "formally correct"'s problem, NOT anybody else>s.
[/quote]
No, actually. As you yourself have emphasized in this very forum, you
don>t *get* to use widely used, well-understood scientific terms in ways
that depart from their conventional definitions, at least, not without
providing equally rigorous alternatives. "Definition" is one of those
terms. Until you provide a rigorous alternative to its current
definition in mathematical logic, you have no grounds for talking about
what is or isn>t a definition in anything other than its conventional
sense.

Jesse F. Hughes · Guest

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

[quote]On Jul 23, 9:32 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
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?

Appeal to authority IS A FALLACY, DIPSHIT.
[/quote]
I didn>t mean to do so, but I want to know what meaning of
"definition" you had in mind. If you had no such meaning, then how
could I be arguing with a non-existent definition?

[...]

--
Jesse F. Hughes
"Radicals are interesting because they were considered 'radical' by
the people who played with them who wrote a lot of math work that
modern mathematics depends on." --Another JSH history lesson

Jesse F. Hughes · Guest

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

[quote]On Jul 23, 9:45 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
Alternatively, what meaning of "definition" did you have in mind?

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

Obviously, terms can be used in more than one way in different
contexts.
I know perfectly well what meaning[s] of definition YOU had in mind,
SINCE I HAVE a philosophy degree.
[/quote]
Wow. Impressive!

[quote]Maybe you should stop playing dumb (or dipshit) and engage
intellectually
and appropriately instead of in a 4th-grade pissing contest.
[/quote]
--
"That>s all the legacy I ever wanted, to have people remember me like
a shooting star streaking across their Life sky, illuminating, for
just one moment, unparalleled beauty unique to itself."
-- Weblogs are a particularly humble medium, unique to themselves.

Gc · Guest

On 25 heinä, 14:53, David C. Ullrich <dullr...@sprynet.com> wrote:
[quote]On Thu, 24 Jul 2008 06:32:47 -0700 (PDT), george <gree...@cs.unc.edu
wrote:

The WHOLE POINT of this whole enterprise is that
ANY OLD *model* will DO!

Really? Of course I don>t have a degree in philosophy,
just a few in mathematics. But it seems to me that a
definition of set membership should (at least in principle)
allow us to answer the question "is x an element of y?"
for any x and y. Now, given x and y there exists a model
of set theory in which x is an element of y, and there
exists a model of set theory in which x is not a model
of y. So it>s hard to see how this gives a "definition".
[/quote]
I would say that in FOL we can define the set of real numbers, even if
we can not characterize them. Otherwise is there anything interesting
in FOl what we can actually define?

David C. Ullrich · Guest

On Thu, 24 Jul 2008 06:34:16 -0700 (PDT), george <greeneg@cs.unc.edu>
wrote:

[quote]On Jul 23, 9:45 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
Alternatively, what meaning of "definition" did you have in mind?

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

Obviously, terms can be used in more than one way in different
contexts.
I know perfectly well what meaning[s] of definition YOU had in mind,
SINCE I HAVE a philosophy degree.

Maybe you should stop playing dumb (or dipshit) and engage
intellectually
and appropriately instead of in a 4th-grade pissing contest.
[/quote]
Wow. Jesse gives an awesomely charitable reply - instead
of just pointing out that what you said isn>t right according
to the standard meaning of the terms, he actually _asks_
you what notion of "definition" you _did_ have in mind.
And you don>t bother to even hint at an answer to the
question.

Good to know that you HAVE a philosophy degree. That>s
where you learned that an appropriate intellectually
engaging reply to "what do you mean by this term?" is "Maybe
you should stop playing dumb (or dipshit)".

I guess there>s been a lot of progress in philosophy since
I was in school - the current version seems a lot easier.

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.)

David C. Ullrich · Guest

On Thu, 24 Jul 2008 06:32:47 -0700 (PDT), george <greeneg@cs.unc.edu>
wrote:

[quote]On Jul 23, 9:32 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
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?

Appeal to authority IS A FALLACY, DIPSHIT.
[/quote]
Regarding matters of fact, yes. Not regarding matters of _definition_.

[quote]Can you show a single source that PURPORTS to "define"
"set membership"??
[/quote]
I don>t know whether he can, but I certainly can>t. In standard
set theory set membership is undefined.

[quote]Or to "define" "addition"?
[/quote]
What? Of course we can give references for the definition
of addition.

[quote]Or to "define" number?

Just what IS a set (or a number) ANYway?

The WHOLE POINT of this whole enterprise is that
ANY OLD *model* will DO!
[/quote]
Really? Of course I don>t have a degree in philosophy,
just a few in mathematics. But it seems to me that a
definition of set membership should (at least in principle)
allow us to answer the question "is x an element of y?"
for any x and y. Now, given x and y there exists a model
of set theory in which x is an element of y, and there
exists a model of set theory in which x is not a model
of y. So it>s hard to see how this gives a "definition".

[quote]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.

That is not a valid analogy and you don>t even yourself believe that
it is;
you are just proposing it because it is an absurdity. You basically
just
HAVE said that a conjunction is a peach pit,
[/quote]
No, he hasn>t.

[quote]and the fact that you>ve
said that is YOUR problem, NOT anybody else>s. "Conjunction" (in our
context) HAS a definition,
[/quote]
Yes, it does. And by that definition there is no such thing as an
infinite conjunction.

[quote]although you would (again, in our context)
BE RIDICULOUSLY HARD-PRESSED TO *STATE* it without waxing
CIRCULAR. Seriously, you just wind up defining "conjunction" as "and"
and "and" as "/\" and "/\" BY ITS TRUTH-TABLE, except that you THEN
run into the problem that "a definition is NOT a truth-table"!

You really haven>t thought this through at all.

When YOU get some axioms determining what is vs. isn>t a definition,
WELL, PUBLISH them.

Brilliant, insightful, decisive rejoinder

As usual.

My point is, YOU DON>T HAVE A SOURCE, EITHER.
Or you have conflicting sources.
Everybody who purports to define "definition" has to acknowledge that
different
things are going to be appropriate in different contexts, ESPECIALLY
for things
that DON>T EVEN HAVE a first-order definition.
[/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.)

Gc · Guest

On 25 heinä, 17:35, Frederick Williams <frederick.willia...@tesco.net>
wrote:
[quote]Gc wrote:

On 25 heinä, 14:53, David C. Ullrich <dullr...@sprynet.com> wrote:
On Thu, 24 Jul 2008 06:32:47 -0700 (PDT), george <gree...@cs.unc.edu
wrote:

The WHOLE POINT of this whole enterprise is that
ANY OLD *model* will DO!

Really? Of course I don>t have a degree in philosophy,
just a few in mathematics. But it seems to me that a
definition of set membership should (at least in principle)
allow us to answer the question "is x an element of y?"
for any x and y. Now, given x and y there exists a model
of set theory in which x is an element of y, and there
exists a model of set theory in which x is not a model
of y. So it>s hard to see how this gives a "definition".

I would say that in FOL we can define the set of real numbers,

How so?
[/quote]
I may have understanded this discussion way wrong, particulary if some
relevant information was cutted from the tread before I posted
anything. What I meant is that a definition doensn`t have to
characterize what is defined. The definition lives in a theory and to
bring in models is, in my opinion, unrelevant when speaking about
definitions. We don`t need anything "metamathetical" to understand
definitions. I agree with George that the axioms of the set theory is
some kind of "work-definition" about set membership, because that`s
all we know about set membership, and I guess we must have some
picture about set membership in our mind. To say that epsilon is
undefined is technically right, but that`s not my point.

[quote]even if
we can not characterize them. Otherwise is there anything interesting
in FOl what we can actually define?

--
He is not here; but far away
The noise of life begins again
And ghastly thro' the drizzling rain
On the bald street breaks the blank day.[/quote]

Frederick Williams · Guest

Gc wrote:
[quote]
On 25 heinä, 14:53, David C. Ullrich <dullr...@sprynet.com> wrote:
On Thu, 24 Jul 2008 06:32:47 -0700 (PDT), george <gree...@cs.unc.edu
wrote:

The WHOLE POINT of this whole enterprise is that
ANY OLD *model* will DO!

Really? Of course I don>t have a degree in philosophy,
just a few in mathematics. But it seems to me that a
definition of set membership should (at least in principle)
allow us to answer the question "is x an element of y?"
for any x and y. Now, given x and y there exists a model
of set theory in which x is an element of y, and there
exists a model of set theory in which x is not a model
of y. So it>s hard to see how this gives a "definition".

I would say that in FOL we can define the set of real numbers,
[/quote]
How so?

[quote]even if
we can not characterize them. Otherwise is there anything interesting
in FOl what we can actually define?
[/quote]

--
He is not here; but far away
The noise of life begins again
And ghastly thro' the drizzling rain
On the bald street breaks the blank day.

Gc · Guest

On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:
[quote]On Fri, 25 Jul 2008 05:13:44 -0700 (PDT), Gc <Gcut...@hotmail.com
wrote:

On 25 heinä, 14:53, David C. Ullrich <dullr...@sprynet.com> wrote:
On Thu, 24 Jul 2008 06:32:47 -0700 (PDT), george <gree...@cs.unc.edu
wrote:

The WHOLE POINT of this whole enterprise is that
ANY OLD *model* will DO!

Really? Of course I don>t have a degree in philosophy,
just a few in mathematics. But it seems to me that a
definition of set membership should (at least in principle)
allow us to answer the question "is x an element of y?"
for any x and y. Now, given x and y there exists a model
of set theory in which x is an element of y, and there
exists a model of set theory in which x is not a model
of y. So it>s hard to see how this gives a "definition".

I would say that in FOL we can define the set of real numbers, even if
we can not characterize them.

Not that I see what your point is here or below, but of course
we can characterize the real numbers (up to isomorphism):
the reals are a complete ordered field.
[/quote]
Not in FOl. We can characterize the real closed fields by the Tarski
axioms.

[quote]Otherwise is there anything interesting
in FOl what we can actually define?

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.)[/quote]


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