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| Golden Boar |
 Posted: Fri Apr 25, 2008 7:56 pm |
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On 25 Apr, 21:53, magi...@math.berkeley.edu (Arturo Magidin) wrote:
Quote: In article <167458c2-3765-411d-80cc-6bc2c6b86...@i76g2000hsf.googlegroups.com>,
Golden Boar <goldenb...@hotmail.com> wrote:
[...]
A set is a collection of distinct objects considered as a whole, and
consists of a container and its contents.
The contents of a set are the elements or members belonging to the
set, which can be anything.
[...]
You have only introduced TWO sets: the minimal set, and the maximal
set. The minimal set is "the unique set that contains no content." The
maximal set is "the unique set that contains all the contents of all
the sets". You have not, however, said what "maximal set of the set X"
means, nor "minimal set of the set X", so the statements above can
only be taken to be definitions of "minimal set of the set of sets
with no members" (which is apparently just a synonym for "minimal
set") and "maximal set of the set of sets with no members". Though you
have no way to guarantee that {{}} is in fact a set, under your scheme
so far.
Yes, apart from the last sentence. Why not?
Because although you have said that a set consists of a container and
elements it contains, you have given no rules whereby one might
determine whether something is or is not a set, nor rules whereby one
can derive one set from another. You have specified by fiat that there
is a unique minimal and a unique maximal set, and said the minimal set has no
content (and said the minimal set consists of the minimal set consists
of the minimal set consists of the minimal set consists of the minimal
set....) and that the maximal set is the set that contains all members
of all sets. But ->THAT'S IT<-. You seem to think you will be able to
"grow" the maximal set as you go along, but that is incorrect; the
maximal set by definition is "the set that contains all contents of
all sets". It is both unique and static in the theory. You cannot
change what it is, though as you go along you may be able to justify
(or not) that something is or is not in that set.
I am not trying to introduce a rigorously defined mathematical theory
here, I thought that was plainly obvious.
The maximal set {...} will appear to be able to grow as new elements
are identified and constructed. If all elements were already known,
this would not be the case.
Quote:
Right now, you cannot even guarantee that the maximal set contains the
minimal set as a member, because the only set whose members you know
is the minimal set, which has no members. Because of that, you have
identified nothing which is an element of a set, and thus you have
identified nothing which is an element of the maximal set. Your
implicit assertion that the maximal set contains the minimal set (by
writing that it is {{}}) is unfounded.
Every set is a superset of the minimal set.
The maximal set is a superset of the minimal set.
Therefore, the minimal set is a subset of the maximal set.
Quote:
You have no warrant for asserting that there is a set whose only
element is the minimal set. Your only existing warrants are for the
minimal and the maximal set; and your warrant for the maximal set does
not allow you to assert that it is "{{}}" (presumably meaning a set
whose only member is the minimal set, given what you wrote later).
Yes I do.
Nothing exists.
The only possible sets that can be created from this initial
condition, is a set that contains nothing and a set that contains
everything. If nothing exists, then nothing is everything, and
therefore the only set that can be created is the minimal set which
contains nothing.
The initial condition has changed, now a set exists which contains
nothing.
From this condition, the set that contains everything will contain a
set that contains nothing.
Continuing in this fashion we arrive at the identity set.
Quote: So... sorry, but there is no reason to believe that what you write as
{{}} is in fact a set in your theory. Your assertions so far are all
satified in a model in which the only thing we have is the minimal
set, which also happens to be the maximal set, and which has no
members.
See above.
Quote: You are floundering; you need to become much more familiar with proper
axiomatic methods if you wish to create one yourself.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org |
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| Tim Little |
| Posted: Fri Apr 25, 2008 11:53 pm |
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On 2008-04-26, Golden Boar <goldenboar@hotmail.com> wrote:
Quote: I am not trying to introduce a rigorously defined mathematical theory
here, I thought that was plainly obvious.
You chose the subject line "Set theory from scratch". A set theory is
a rigorously defined mathematical theory. Maybe you should have
chosen "Maunderings on sets" instead.
- Tim |
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| Guest |
| Posted: Sat Apr 26, 2008 8:56 am |
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On Apr 25, 9:53 pm, Tim Little <t...@soprano.little-possums.net>
wrote:
Quote: On 2008-04-26, Golden Boar <goldenb...@hotmail.com> wrote:
I am not trying to introduce a rigorously defined mathematical theory
here, I thought that was plainly obvious.
You chose the subject line "Set theory from scratch". A set theory is
a rigorously defined mathematical theory.
Well, as usual, let me try to find a rigorous axiomatization
of the OP's theory.
Let's assume that we have the primitive "e" for "is an
element of," as usual. Based on the OP's comment:
Quote: If every element of set A is also an element of set B and every
element of set B is also an element of set A, then set A is equal to
set B.
we may assume that the OP accepts the Axiom of Extensionality
(which is more than we can say about WM, at least)!
So let us begin with the simple axiomatization:
Extensionality: Ax (Ay ((Az (zex <-> zey) -> x=y)))
Minimality: Ex (Ay (~yex))
Maximality: Ex (Ay ( yex))
By Extensionality, the sets whose existence is guaranteed
by Minimality and Maximality are unique and distinct. So
we may call the sets m (for minimum) and M (for maximum),
respectively, if we want.
Notice that these are the only two sets whose existence
is guaranteed by the axioms. |
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| Arturo Magidin |
| Posted: Sat Apr 26, 2008 9:22 am |
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In article <659e6847-650d-45e4-a21d-162a20191026@a70g2000hsh.googlegroups.com>,
Golden Boar <goldenboar@hotmail.com> wrote:
[...]
Quote: I am not trying to introduce a rigorously defined mathematical theory
here, I thought that was plainly obvious.
It is plainly obvious that you did not in fact introduce a correctly
defined (rigorous or otherwise) mathematical theory. However, the fact
that you attempted to call it "Set Theory" and presented it in
sci.math suggested that this is what you are ->attempting<- to do.
Since you now claim this is not in fact the case, then I suggest you
take it elsewhere. It does not belong in sci.math.
Quote: Right now, you cannot even guarantee that the maximal set contains the
minimal set as a member, because the only set whose members you know
is the minimal set, which has no members. Because of that, you have
identified nothing which is an element of a set, and thus you have
identified nothing which is an element of the maximal set. Your
implicit assertion that the maximal set contains the minimal set (by
writing that it is {{}}) is unfounded.
Every set is a superset of the minimal set.
False. Under your definition, the minimal set is not a superset of the
minimal set. I already pointed that out.
Quote: The maximal set is a superset of the minimal set.
You have no warrant for that assertion. Your deduction is faulty.
Quote: Therefore, the minimal set is a subset of the maximal set.
Therefore, you know not of what you speak, or else you are spinning
your wheels. Either way, you are wasting my (and your) time.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org |
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| Arturo Magidin |
| Posted: Sat Apr 26, 2008 9:25 am |
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In article <bf35ef1f-ebc8-4976-a996-07701578df9e@p25g2000hsf.googlegroups.com>,
Golden Boar <goldenboar@hotmail.com> wrote:
Quote: On 26 Apr, 05:53, Tim Little <t...@soprano.little-possums.net> wrote:
On 2008-04-26, Golden Boar <goldenb...@hotmail.com> wrote:
I am not trying to introduce a rigorously defined mathematical theory
here, I thought that was plainly obvious.
You chose the subject line "Set theory from scratch". A set theory is
a rigorously defined mathematical theory. Maybe you should have
chosen "Maunderings on sets" instead.
- Tim
Well, wouldn't the first step be to engage in thought and debate.
You are apparently not interested in debate, since the precise
mathematical criticisms I presented were dismissed as "being a
smartarse." So do kindly get off that ersatz moral high ground you
think you are on.
Quote: I posted some ideas about the first step or two for creating a model
(which should have been clearly obvious to anyone with any sense)
No, what is obvious is that you know little or not mathematical logic
or the necessary tools for engaging in this sort of endeavor. You
don't even know what a model is. So kindly, go learn it and then come back.
Quote: Usenet is quite funny in that way.
Yeah, full of clowns. I really like that big red nose you have on, by
the way.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org |
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| Guest |
| Posted: Sat Apr 26, 2008 6:46 pm |
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On Apr 25, 1:53 pm, magi...@math.berkeley.edu (Arturo Magidin) wrote:
Quote: In article <167458c2-3765-411d-80cc-6bc2c6b86...@i76g2000hsf.googlegroups.com>,
Golden Boar <goldenb...@hotmail.com> wrote:
Yes, apart from the last sentence. Why not?
So... sorry, but there is no reason to believe that what you write as
{{}} is in fact a set in your theory. Your assertions so far are all
satified in a model in which the only thing we have is the minimal
set, which also happens to be the maximal set, and which has no
members.
Actually, I think that Arturo Magidin and the OP are both partly
right.
If we use the axiomatization:
Extensionality: Ax (Ay ((Az (zex <-> zey) -> x=y)))
Minimality: Ex (Ay (~yex))
Maximality: Ex (Ay ( yex))
then we can definitely prove that the set whose existence is
guaranteed by Minimality (m, the empty set), and the set whose
existence is guaranteed by Maximality (M, the universal set) must
be distinct sets. But M = {{}} is clearly false, since as M is a
universal set, we must have meM and MeM, so that M cannot
possibly be a singleton.
If we were to replace Maximality by the following axiom:
Wellfounded Maximality: Ex (Ay (y=x or yex))
then M = {{}} is possible in a model, but then again, so is Magidin's
model in which m = M = {}.
(I call the axiom Wellfounded Maximality since it is more likely to be
consistent in a theory with the Axiom of Foundation/Regularity.) |
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