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Brian
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 Posted: Mon Jul 28, 2008 12:25 am Post subject: structure of all structures: does it exist in any context |
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When I refer to the term structure, it is something along the lines as
this wiki article:
http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
I>m trying to investigate any way to formalize the concept of the
structure of all structures. This can>t be done in any well-founded
set theory, correct?
What else prohibits the structure of all structures from existing as a
structure?
Has this notion been investigated before and, if so, are there online
links to such investigations?
Perhaps some form of an ultraproduct can "glue together" the universes
of all structures and make it into a structure but I think one main
problem is that the universe of all universes is not a set in ZFC.
However, perhaps in some other set theory, such as the one linked to
below, the universe of all universes can be V, the universal set in
that set theory?
http://math.boisestate.edu/%7Eholmes/holmes/head.pdf |
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Guest
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| Posted: Mon Jul 28, 2008 1:35 am Post subject: Re: structure of all structures: does it exist in any contex |
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On Jul 27, 5:25 pm, Brian <tenn...@gmail.com> wrote:
[quote]When I refer to the term structure, it is something along the lines as
this wiki article:http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
I>m trying to investigate any way to formalize the concept of the
structure of all structures. This can>t be done in any well-founded
set theory, correct?
What else prohibits the structure of all structures from existing as a
structure?
Has this notion been investigated before and, if so, are there online
links to such investigations?
Perhaps some form of an ultraproduct can "glue together" the universes
of all structures and make it into a structure but I think one main
problem is that the universe of all universes is not a set in ZFC.
However, perhaps in some other set theory, such as the one linked to
below, the universe of all universes can be V, the universal set in
that set theory?http://math.boisestate.edu/%7Eholmes/holmes/head.pdf
[/quote]
See /\ class theory : in this usenet.
http://groups.google.com.jm/group/sci.logic/browse_thread/thread/afac14c26e7894a3?hl=en
It is only a preliminary trial of such a class theory.
But if it proves to be consistent, then it might be of help.
Zuhair |
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Brian
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| Posted: Mon Jul 28, 2008 2:03 am Post subject: Re: structure of all structures: does it exist in any contex |
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On Jul 27, 6:35 pm, Zaljo...@gmail.com wrote:
[quote]On Jul 27, 5:25 pm, Brian <tenn...@gmail.com> wrote:
When I refer to the term structure, it is something along the lines as
this wiki article:http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
I>m trying to investigate any way to formalize the concept of the
structure of all structures. This can>t be done in any well-founded
set theory, correct?
What else prohibits the structure of all structures from existing as a
structure?
Has this notion been investigated before and, if so, are there online
links to such investigations?
Perhaps some form of an ultraproduct can "glue together" the universes
of all structures and make it into a structure but I think one main
problem is that the universe of all universes is not a set in ZFC.
However, perhaps in some other set theory, such as the one linked to
below, the universe of all universes can be V, the universal set in
that set theory?http://math.boisestate.edu/%7Eholmes/holmes/head.pdf
See /\ class theory : in this usenet.http://groups.google.com.jm/group/sci.logic/browse_thread/thread/afac...
It is only a preliminary trial of such a class theory.
But if it proves to be consistent, then it might be of help.
Zuhair
[/quote]
Thanks. I>ll look into it. |
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William Elliot
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| Posted: Mon Jul 28, 2008 12:46 pm Post subject: Re: structure of all structures: does it exist in any contex |
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On Sun, 27 Jul 2008, Brian wrote:
[quote]When I refer to the term structure, it is something along the lines as
this wiki article:
http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
I>m trying to investigate any way to formalize the concept of the
structure of all structures. This can>t be done in any well-founded
set theory, correct?
What else prohibits the structure of all structures from existing as a
structure?
Has this notion been investigated before?
What prohibits the catagory of all catagories?[/quote]
---- |
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Jesse F. Hughes
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| Posted: Mon Jul 28, 2008 5:20 pm Post subject: Re: structure of all structures: does it exist in any contex |
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Brian <tennesb@gmail.com> writes:
[quote]On Jul 27, 6:35 pm, Zaljo...@gmail.com wrote:
On Jul 27, 5:25 pm, Brian <tenn...@gmail.com> wrote:
When I refer to the term structure, it is something along the lines as
this wiki article:http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
I>m trying to investigate any way to formalize the concept of the
structure of all structures. This can>t be done in any well-founded
set theory, correct?
What else prohibits the structure of all structures from existing as a
structure?
Has this notion been investigated before and, if so, are there online
links to such investigations?
Perhaps some form of an ultraproduct can "glue together" the universes
of all structures and make it into a structure but I think one main
problem is that the universe of all universes is not a set in ZFC.
However, perhaps in some other set theory, such as the one linked to
below, the universe of all universes can be V, the universal set in
that set theory?http://math.boisestate.edu/%7Eholmes/holmes/head.pdf
See /\ class theory : in this usenet.http://groups.google.com.jm/group/sci.logic/browse_thread/thread/afac...
It is only a preliminary trial of such a class theory.
But if it proves to be consistent, then it might be of help.
Zuhair
Thanks. I>ll look into it.
[/quote]
I wouldn>t. It>s a half-considered theory in infancy.
Try ZFA, non-well-founded set (and class) theory as presented in
Barwise and Moss>s "Vicious Circles". Warning: it>s a touch
technical.
--
Jesse F. Hughes
"The way that she did what she did to me
when she did what she did to me
made me think of you." --- Delbert McClinton |
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John Jones
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| Posted: Mon Jul 28, 2008 6:06 pm Post subject: Re: structure of all structures: does it exist in any contex |
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Brian wrote:
[quote]When I refer to the term structure, it is something along the lines as
this wiki article:
http://en.wikipedia.org/wiki/Structure_(mathematical_logic)
I>m trying to investigate any way to formalize the concept of the
structure of all structures. This can>t be done in any well-founded
set theory, correct?
What else prohibits the structure of all structures from existing as a
structure?
Has this notion been investigated before and, if so, are there online
links to such investigations?
Perhaps some form of an ultraproduct can "glue together" the universes
of all structures and make it into a structure but I think one main
problem is that the universe of all universes is not a set in ZFC.
However, perhaps in some other set theory, such as the one linked to
below, the universe of all universes can be V, the universal set in
that set theory?
http://math.boisestate.edu/%7Eholmes/holmes/head.pdf
[/quote]
I hadn>t intended to write any more posts. But the replies you got for
this post looked so academically driven and uninsightful that I just had
to reply.
Why is it that so often the first, the most obvious, answer is ignored
in favour of scurrying through structured academic texts?
Simply, there is no structure if a structure has not been defined. There
is no structure of structures if a meta-structure has not been defined.
What ZFC, etc, has to say about it is irrelevant. You can>t quote an
academic source as explanation.
The question you should be asking is why we should prefer mysteries and
intrigues in favour of a straightforward answer. |
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Brian
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| Posted: Mon Jul 28, 2008 9:35 pm Post subject: Re: structure of all structures: does it exist in any contex |
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Thanks everyone for the replies.
In reference to the Holmes book on NFU set theory, in chapter seven
there is a theorem which states that {x | P} where P is a stratified
well formed formula (definition provided in text) -is a set-.
Incidentally, when I write set in this thread, I mean an NFU set, as
mentioned in the Holmes text.
Here is that link again: http://math.boisestate.edu/~holmes/holmes/head.pdf
So, -if- I articulate a stratified wff P such that x is a logical
structure iff P(x), then the collections of all logical structures
would be an NFU-set. I have to see if P would be a first-order wff
and the main thing to use that theorem from chapter 7 is to see if P
is stratified. (An example of a non-stratified wff is x is an
element of x.)
Assuming that is possible, with or without the stratified
comprehension theorem, let L denote this (NFU) set of all logical
structures. My goal would be to talk about something which is
essentially the logical structure of all logical structures. Jesse>s
remarks come into play here because there would seem to be an
essential non-well-foundedness of this. Now to the end of thinking of
the logical structure of all logical structures, L would be the
universe of this structure. Here, L needs to be a set to qualify the
entity I refer to as a structure.
My motivation relates to Max Tegmark>s Mathematical Universe
Hypothesis paper. It would seem that assuming this hypothesis, the
concept of the structure of all structures, is the (level 4)
multiverse. |
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