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Gc · Guest

On 26 heinä, 15:33, Gc <Gcut...@hotmail.com> wrote:
[quote]On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:

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.

Not in FOl. We can characterize the real closed fields by the Tarski
axioms.
[/quote]
David Ullrich I respect you a lot, but remember that we can`t express
the least upper bound property in FOl.

[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]

Gc · Guest

On 26 heinä, 15:34, Gc <Gcut...@hotmail.com> wrote:
[quote]On 26 heinä, 15:33, Gc <Gcut...@hotmail.com> wrote:

On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:

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.

Not in FOl. We can characterize the real closed fields by the Tarski
axioms.

David Ullrich I respect you a lot, but remember that we can`t express
the least upper bound property in FOl.
[/quote]
I mean there is no way to characterize the reals in FOl up to
isomorphism, there always could be subsets of the integers (each
subset of the integers can be seen as a real number) which are not
definable by the language we are using in our theory.

[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]

Gc · Guest

On 26 heinä, 16:01, Gc <Gcut...@hotmail.com> wrote:
[quote]On 26 heinä, 15:34, Gc <Gcut...@hotmail.com> wrote:

On 26 heinä, 15:33, Gc <Gcut...@hotmail.com> wrote:

On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:

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.

Not in FOl. We can characterize the real closed fields by the Tarski
axioms.

David Ullrich I respect you a lot, but remember that we can`t express
the least upper bound property in FOl.

I mean there is no way to characterize the reals in FOl up to
isomorphism, there always could be subsets of the integers (each
subset of the integers can be seen as a real number) which are not
definable by the language we are using in our theory.
[/quote]
Ouch..I of course mean the subsets of the reals: for every FOl
language there are subsets of the reals not definable in our current
theory. So because we can`t quantify over all subsets of the reals,
our least upper bound property in FOl is always incomplete.

[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]

Gc · Guest

On 26 heinä, 17:46, Frederick Williams <frederick.willia...@tesco.net>
wrote:
[quote]Gc wrote:

On 26 heinä, 15:33, Gc <Gcut...@hotmail.com> wrote:
On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:

On Fri, 25 Jul 2008 05:13:44 -0700 (PDT), Gc <Gcut...@hotmail.com
wrote:

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.

Not in FOl. We can characterize the real closed fields by the Tarski
axioms.

David Ullrich I respect you a lot, but remember that we can`t express
the least upper bound property in FOl.

Which is why I asked "How?" in another branch, i.e. "How can we define
the set of real numbers in FOL?"
[/quote]
So are you saying that we can`t define them at all? We do define them
in set theory by dedekind cuts. FOl is the logic of foundations of
mathematics and there is no point using full second order logic for
foundations.

http://www.math.ucla.edu/~asl/bsl/0704/0704-003.ps

David C. Ullrich · Guest

On Fri, 25 Jul 2008 05:13:44 -0700 (PDT), Gc <Gcut667@hotmail.com>
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, even if
we can not characterize them.
[/quote]
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]Otherwise is there anything interesting
in FOl what we can actually define?
[/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 26 heinä, 17:50, Frederick Williams <frederick.willia...@tesco.net>
wrote:
[quote]Gc wrote:

Ouch..I of course mean the subsets of the reals: for every FOl
language there are subsets of the reals not definable in our current
theory. So because we can`t quantify over all subsets of the reals,
our least upper bound property in FOl is always incomplete.

In my country we have a saying: "when you>re in a hole, stop digging".
[/quote]
What do you mean? Don`t be ridiculous! There is no kappa-kategorical
axiomation of the reals, like I said. Let A be a uncountable model of
cardinality of the continuum of the theory of reals. We can extend A
to two uncountable models A` and A`` of cardinality k by follows. Let
subsets not definable in the present theory have the least upper bound
property in A` and not to have the least upper bound property in A`
´.Both are extensions of a countable model, where only every definable
set of the reals exist.

Gc · Guest

On 26 heinä, 19:46, Gc <Gcut...@hotmail.com> wrote:
[quote]On 26 heinä, 17:50, Frederick Williams <frederick.willia...@tesco.net
wrote:

Gc wrote:

Ouch..I of course mean the subsets of the reals: for every FOl
language there are subsets of the reals not definable in our current
theory. So because we can`t quantify over all subsets of the reals,
our least upper bound property in FOl is always incomplete.

In my country we have a saying: "when you>re in a hole, stop digging".

What do you mean? Don`t be ridiculous! There is no kappa-kategorical
axiomation of the reals, like I said. Let A be a uncountable model of
cardinality of the continuum of the theory of reals. We can extend A
to two uncountable models A` and A`` of cardinality k by follows. Let
subsets not definable in the present theory have the least upper bound
property in A` and not to have the least upper bound property in A`
´.Both are extensions of a countable model, where only every definable
set of the reals exist.
[/quote]
Correction: Let A be a uncountable model of cardinality of the
continuum of the theory of reals----> let A be a model of countable
cardinality of the theory of the reals.

David C. Ullrich · Guest

On Fri, 25 Jul 2008 11:11:09 -0700 (PDT), Gc <Gcut667@hotmail.com>
wrote:

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

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.
[/quote]
Interesting point of view. Either wrong or true but irrelevant,
depending on exactly what you mean.

The definition of 2 _does_ need to characterize 2, or it>s
not a definition. Similarly for the definition of set membership,
_if_ we>re taking it literally to be _the_ definition of set
membership.

Now, the definition of, say, "group" does not characterize
a group - many things satisfy the definition. That>s ok
because it>s not the definition of "the group", it>s the
definition of "a group" - the definition _does_ characterize
the meaning of the phrase "a group".

Similarly, (if we decide that a "definition" can be an infinite
set of sentence) the axioms of set theory _could_ be taken
to be the definition of "a membership relation". But
nobody here has been talking about the definition of
"_a_ membership relation", or if they have they haven>t
said so - people have been talking about the definition
of set membership, period.

[quote]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,
[/quote]
Saying that the axioms give "some sort of working definition"
is not the same as saying the conjunction of the axioms _is_
the definition.

You might also note an inconsistency in his "definition";
at first the definition was "the conjunction of the axioms
of set theory", then later he said something about how
any old model will do. The membership relation in
some old model of set theory is not characterized by
the axioms. So we have at least two "definitions"...

[quote]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.

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]
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.)

Frederick Williams · Guest

Gc wrote:
[quote]
On 26 heinä, 15:33, Gc <Gcut...@hotmail.com> wrote:
On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:

On Fri, 25 Jul 2008 05:13:44 -0700 (PDT), Gc <Gcut...@hotmail.com
wrote:

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.

Not in FOl. We can characterize the real closed fields by the Tarski
axioms.

David Ullrich I respect you a lot, but remember that we can`t express
the least upper bound property in FOl.
[/quote]
Which is why I asked "How?" in another branch, i.e. "How can we define
the set of real numbers in FOL?"

--
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.

Frederick Williams · Guest

Gc wrote:

[quote]
Ouch..I of course mean the subsets of the reals: for every FOl
language there are subsets of the reals not definable in our current
theory. So because we can`t quantify over all subsets of the reals,
our least upper bound property in FOl is always incomplete.
[/quote]
In my country we have a saying: "when you>re in a hole, stop digging".

--
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 27 heinä, 14:40, Frederick Williams <frederick.willia...@tesco.net>
wrote:
[quote]Gc wrote:

On 26 heinä, 17:46, Frederick Williams <frederick.willia...@tesco.net
wrote:
Gc wrote:

On 26 heinä, 15:33, Gc <Gcut...@hotmail.com> wrote:
On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:

On Fri, 25 Jul 2008 05:13:44 -0700 (PDT), Gc <Gcut...@hotmail.com
wrote:

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.

Not in FOl. We can characterize the real closed fields by the Tarski
axioms.

David Ullrich I respect you a lot, but remember that we can`t express
the least upper bound property in FOl.

Which is why I asked "How?" in another branch, i.e. "How can we define
the set of real numbers in FOL?"

So are you saying that we can`t define them at all?

Are you claiming that my claim that one can>t define the reals in FOL
implies that I think we can>t define them at all? Don>t be silly.
[/quote]
I mean that basically full second order logic is not that different
from mathematical english. If we wan`t to formalize things to the
bottom we must use FOL. That`s because the logical consequence of
second order logic is formalizable by provability or any other way.
There is nothing wrtong to define reals in FOl set theory. The
metamathemical fact that there are nonstandard models of reals on or
naturals in FOl is not going to spoil the thing that we can define
what ever there is definable in FOl using set theory. If we rely only
to full second order logic, we are not truly formalizing stuff IMHO.

[quote]We do define them
in set theory by dedekind cuts.

Set theory isn>t FOL.

FOl is the logic of foundations of
mathematics

A foundation is something one builds on, rather as set theory is built
on FOL. That does not mean that set theory is FOL.

and there is no point using full second order logic for
foundations.

http://www.math.ucla.edu/~asl/bsl/0704/0704-003.ps

--
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 26 heinä, 17:46, Frederick Williams <frederick.willia...@tesco.net
wrote:
Gc wrote:

On 26 heinä, 15:33, Gc <Gcut...@hotmail.com> wrote:
On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:

On Fri, 25 Jul 2008 05:13:44 -0700 (PDT), Gc <Gcut...@hotmail.com
wrote:

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.

Not in FOl. We can characterize the real closed fields by the Tarski
axioms.

David Ullrich I respect you a lot, but remember that we can`t express
the least upper bound property in FOl.

Which is why I asked "How?" in another branch, i.e. "How can we define
the set of real numbers in FOL?"

So are you saying that we can`t define them at all?
[/quote]
Are you claiming that my claim that one can>t define the reals in FOL
implies that I think we can>t define them at all? Don>t be silly.

[quote]We do define them
in set theory by dedekind cuts.
[/quote]
Set theory isn>t FOL.

[quote]FOl is the logic of foundations of
mathematics
[/quote]
A foundation is something one builds on, rather as set theory is built
on FOL. That does not mean that set theory is FOL.

[quote]and there is no point using full second order logic for
foundations.

http://www.math.ucla.edu/~asl/bsl/0704/0704-003.ps
[/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.

David C. Ullrich · Guest

On Sat, 26 Jul 2008 05:33:09 -0700 (PDT), Gc <Gcut667@hotmail.com>
wrote:

[quote]On 26 heinä, 14:43, David C. Ullrich <dullr...@sprynet.com> wrote:
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.

Not in FOl.
[/quote]
Of course not in the first-order theory of fields. I
assumed we were talking about set theory, since
that>s the topic, after all. We can certainly define
"complete ordered field" in set theory, and that
characterizes the reals (up to isomorphism).

Another reason for assuming that was what we were talking
about is your comment "I would say that in FOL we can define
the set of real numbers"...

[quote]We can characterize the real closed fields by the Tarski
axioms.

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]
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.)


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