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.

There is a unique set that contains no content, which is called the
minimal set, {}.

There is a unique set that contains all the contents of all the sets,
which is called the maximal set, {...}.
The container of a set is the minimal set {}.

Given that we have only identified the existence of the minimal set {}
which contains nothing, and the maximal set {...} which contains
everything, then the maximal set must contain the minimal set, {...} =
{{}}.

The cardinality of a set represents how many elements the set
contains.

We shall assign the symbol 0 to the cardinality of {}, and the symbol
1 to the cardinality of {{}}.

{} is the minimal set of the set of sets with no members.
{{}} is the maximal set of the set of sets with no members.

We will call the set of sets with no members, the identity set.

The minimal set of the identity set is {},

which we shall call a virtual point.

The maximal set of the identity set is {{}}, which we shall call a
point.

In article <91a37007-4a1c-4ac6-92d4-2a7125c32...@34g2000hsh.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.

There is a unique set that contains no content, which is called the
minimal set, {}.

Under what conditions will you call two sets "the same"? Does the
the container matter? Do the contents matter? Does the name we give
matter?

There is a unique set that contains all the contents of all the sets,
which is called the maximal set, {...}.
The container of a set is the minimal set {}.

"The" container? It's unique? Okay...

So... since a set "consists of a container and its contents", and the
container is "the minimal set", that means that a set consists of
(i) the minimal set; and
b (ii) its contents.

In particular, the minimal set consists of the minimal set...

Given that we have only identified the existence of the minimal set {}
which contains nothing, and the maximal set {...} which contains
everything, then the maximal set must contain the minimal set, {...} =
{{}}.

This seems to be saying that the only content of the maximal set is
the minimal set.

It is hard to tell, however, since you have introduced no notation to
describe sets other than the minimal and maximal sets. So, right now,
what you have written is unintelligible since "{{}}" has no meaning in
your theory (given that you have 'started from scratch'). You cannot
even guarantee that {{}} denotes a set, under your theory so far.

Also, you know the minimal and maximal set exists by fiat, but you do
not have any way of determining that they are the only sets that
exist. So while you can say that the maximal set has, among its
constituents, the minimal set (you know that because "the container of
a set is the minimal set", and you know that any set, including the
maximal set, consists of a container and its elements, so you know
that one of the constituents of the maximal set is the minimal set),
you cannot say that this is the only constituent of the maximal set at
this stage.

The cardinality of a set represents how many elements the set
contains.

This is meaningless right now

We shall assign the symbol 0 to the cardinality of {}, and the symbol
1 to the cardinality of {{}}.

And what about other potential sets?

{} is the minimal set of the set of sets with no members.
{{}} is the maximal set of the set of sets with no members.

Do you mean "content"? You have no definition of "members", nor was it
introduced as a primitive concept.

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.

We will call the set of sets with no members, the identity set.

So now that set has THREE names...

The minimal set of the identity set is {},

You have not said what "minimal set of X" means.

which we shall call a virtual point.

So now it has FOUR names!

The maximal set of the identity set is {{}}, which we shall call a
point.

--
======================================================================

"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

On Apr 25, 12:45 pm, Golden Boar <goldenb...@hotmail.com> wrote:

Let's call it "less than super-naive set theory". Oops, it doesn't
even qualify to be a theory.

Anyway, what was that saying: people who don't learn the history are
doomed to repeat it? That's what you're doing right now.

Among other things, either your "sets" can't contain other "sets" and
also can only be finite (which makes it sort of 'baby set theory', or
you are running into the same problems others ran about a 100 years
ago.

But I find it cool that you define set as a 'collection'. The only
thing left is to define the collection. So what is the collection? A
set?

In article
91a37007-4a1c-4ac6-92d4-2a7125c32...@34g2000hsh.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.

You're going to run into some trouble with the idea that the container
is part of the set. For example if I have the set consisting of the
numbers 1 and 2, and you have the set consisting of the numbers 1 and 2,
you are not in a position to say that these are the same set; because
even if the contents are the same, the containers, whatever those are,
may not be the same.

On 25 Apr, 18:07, magi...@math.berkeley.edu (Arturo Magidin) wrote:
In article <91a37007-4a1c-4ac6-92d4-2a7125c32...@34g2000hsh.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.

There is a unique set that contains no content, which is called the
minimal set, {}.

Under what conditions will you call two sets "the same"? Does the
the container matter? Do the contents matter? Does the name we give
matter?

The foundation for creating anything must be that nothing exists. An
empty canvas, so to speak.

We represent this as a set which contains no elements.

If we use the symbol {}, to represent the container, then {} also
represents the set which contains no elements.

Any symbol within the container is an element of the set, eg, {?} or
{red}.

For any set which contains multiple elements, the elements will be
separated by a comma, eg, {red,green,blue}.

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.

If every element of set A is also an element of set B and not every
element of set B is an element of set A, then set A is a subset of set
B.

If not every element of set A is an element of set B and every element
of set B is also an element of set A, then set A is a superset of set
B.

A subset is a set within a set, and has a lower cardinality than its
parent set.

A superset is a set that contains a set, and has a higher cardinality
than its child.

The cardinality of a set represents how many elements the set
contains.

There is a unique set that contains all the contents of all the sets,
which is called the maximal set, {...}.
The container of a set is the minimal set {}.

"The" container? It's unique? Okay...

Yes.

So... since a set "consists of a container and its contents", and the
container is "the minimal set", that means that a set consists of
(i) the minimal set; and
b (ii) its contents.

In particular, the minimal set consists of the minimal set...

Yes.

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?

In article <167458c2-3765-411d-80cc-6bc2c6b86...@i76g2000hsf.googlegroups.com>,
Golden Boar <goldenb...@hotmail.com> wrote:



On 25 Apr, 18:07, magi...@math.berkeley.edu (Arturo Magidin) wrote:
In article <91a37007-4a1c-4ac6-92d4-2a7125c32...@34g2000hsh.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.

There is a unique set that contains no content, which is called the
minimal set, {}.

Under what conditions will you call two sets "the same"? Does the
the container matter? Do the contents matter? Does the name we give
matter?

The foundation for creating anything must be that nothing exists. An
empty canvas, so to speak.

From nothing comes nothing. Every theory begins with some undefined
terms and undefined relations between them. You began with "set",
"container", "contains", and "contents".

We represent this as a set which contains no elements.

You are now jumping well ahead. No. You cannot represent something as
"a set which contains no elements" until you either DEFINE what a set,
what element, and what contain mean, or else you declare them to be
undefined/primitive terms and relations. In either case, you are not
"starting from nothing".

If we use the symbol {}, to represent the container, then {} also
represents the set which contains no elements.
Any symbol within the container is an element of the set, eg, {?} or
{red}.

Is "within" a primitive relation, or one that is defined? Don't know,
because you don't say.

If any symbol within the container is an element of the set, does that
mean that the set you have attempted to exemplify this by contains
both '?' and "red", given that both of them are instances of symbols
within the "container". I doubt, however, that this was your intent.

For any set which contains multiple elements, the elements will be
separated by a comma, eg, {red,green,blue}.

By your definition, the comma is also an element of the set.

So, presumably, you are not saying what you mean. That, certainly, is
a problem (since you asked about problems).

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.

Ah! Finally. This should have been well closer to the top.

If every element of set A is also an element of set B and not every
element of set B is an element of set A, then set A is a subset of set
B.

So, no set is a subset of itself, then. Okay.

If not every element of set A is an element of set B and every element
of set B is also an element of set A, then set A is a superset of set
B.

Again, no set is a superset of itself. Okay...

A subset is a set within a set, and has a lower cardinality than its
parent set.

You have not defined "within", "cardinality" (or how 'lower' may apply
to it), or "parent", so this statement is either meaningless, or a new
source of undefined terms.

A superset is a set that contains a set, and has a higher cardinality
than its child.

You have not defined "cardinality" (or how 'higher' may apply to it),
or "child", so this statment is either meaningless or a new source of
undefined terms.

The cardinality of a set represents how many elements the set
contains.

But what ->is it<-? You are telling me what you want it to
"represent", but you don't say what it is. And you should NEVER define
a term AFTER you have used it repeatedly.

I would say you should start over from scratch again. Perhaps take a
look at a real book in set theory so you can see how to set things up
(even if you do not wish to use their axioms, the way to set up a
discourse would be instructive).

There is a unique set that contains all the contents of all the sets,
which is called the maximal set, {...}.
The container of a set is the minimal set {}.

"The" container? It's unique? Okay...

Yes.

And yet, you did not say so when you started. How silly of you to just
assert it by innuendo.

So... since a set "consists of a container and its contents", and the
container is "the minimal set", that means that a set consists of
(i) the minimal set; and
b (ii) its contents.

In particular, the minimal set consists of the minimal set...

Yes.

So... the minimal set consists of SOMETHING, then. Which of course
contradicts your assertion that the minimal set is (the unique set
that) has no contents. Therefore, your theory is inconsistent.

And that's the game, right there.

--
======================================================================
"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

A set is a collection of distinct objects considered as a whole, and
consists of a container and its contents.

On 25 Apr, 21:35, magi...@math.berkeley.edu (Arturo Magidin) wrote:
In article <167458c2-3765-411d-80cc-6bc2c6b86...@i76g2000hsf.googlegroups.com>,
Golden Boar <goldenb...@hotmail.com> wrote:



On 25 Apr, 18:07, magi...@math.berkeley.edu (Arturo Magidin) wrote:
In article <91a37007-4a1c-4ac6-92d4-2a7125c32...@34g2000hsh.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.

There is a unique set that contains no content, which is called the
minimal set, {}.

Under what conditions will you call two sets "the same"? Does the
the container matter? Do the contents matter? Does the name we give
matter?

The foundation for creating anything must be that nothing exists. An
empty canvas, so to speak.

From nothing comes nothing. Every theory begins with some undefined
terms and undefined relations between them. You began with "set",
"container", "contains", and "contents".

First sentence.

We represent this as a set which contains no elements.

You are now jumping well ahead. No. You cannot represent something as
"a set which contains no elements" until you either DEFINE what a set,
what element, and what contain mean, or else you declare them to be
undefined/primitive terms and relations. In either case, you are not
"starting from nothing".

First sentence.

If we use the symbol {}, to represent the container, then {} also
represents the set which contains no elements.
Any symbol within the container is an element of the set, eg, {?} or
{red}.

Is "within" a primitive relation, or one that is defined? Don't know,
because you don't say.

It is just the English word.

If any symbol within the container is an element of the set, does that
mean that the set you have attempted to exemplify this by contains
both '?' and "red", given that both of them are instances of symbols
within the "container". I doubt, however, that this was your intent.

For any set which contains multiple elements, the elements will be
separated by a comma, eg, {red,green,blue}.

By your definition, the comma is also an element of the set.


Any symbol(except the comma), within the container is an element of
the set, eg, {?} or {red}.

So, presumably, you are not saying what you mean. That, certainly, is
a problem (since you asked about problems).

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.

Ah! Finally. This should have been well closer to the top.

If every element of set A is also an element of set B and not every
element of set B is an element of set A, then set A is a subset of set
B.

So, no set is a subset of itself, then. Okay.

Not enough sets exist to determine so far.

The minimal set has no subsets.

If not every element of set A is an element of set B and every element
of set B is also an element of set A, then set A is a superset of set
B.

Again, no set is a superset of itself. Okay...

Same again.

A subset is a set within a set, and has a lower cardinality than its
parent set.

You have not defined "within", "cardinality" (or how 'lower' may apply
to it), or "parent", so this statement is either meaningless, or a new
source of undefined terms.

That is because no special meaning is attributed to these words, they
are just words used within the context of the English language.

I was thinking about replacing cardinality with amount of elements.

In article <f06f2575-3ac4-40f1-802e-f4410c251...@a23g2000hsc.googlegroups.com>,
Golden Boar <goldenb...@hotmail.com> wrote:



On 25 Apr, 21:35, magi...@math.berkeley.edu (Arturo Magidin) wrote:
In article <167458c2-3765-411d-80cc-6bc2c6b86...@i76g2000hsf.googlegroups.com>,
Golden Boar <goldenb...@hotmail.com> wrote:

On 25 Apr, 18:07, magi...@math.berkeley.edu (Arturo Magidin) wrote:
In article <91a37007-4a1c-4ac6-92d4-2a7125c32...@34g2000hsh.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.

There is a unique set that contains no content, which is called the
minimal set, {}.

Under what conditions will you call two sets "the same"? Does the
the container matter? Do the contents matter? Does the name we give
matter?

The foundation for creating anything must be that nothing exists. An
empty canvas, so to speak.

From nothing comes nothing. Every theory begins with some undefined
terms and undefined relations between them. You began with "set",
"container", "contains", and "contents".

First sentence.

I'm pointing out that you are incorrect in your assertion thbat you
are starting "from nothing." The statement "The foundation for
creating anything must be that nothing exists" is little but empty
rhetoric. May sound nice to you, but it is an utter waste of breath in
a mathematical discussion, It is also leading you astray in thinking
that your theory is being "built up". That's not how a mathematical
theory works.

I realize that you introduced primitive terms. The person who seems to
be unaware of this, however, is you, first in asserting that you are
starting "from nothing", and later when you insist on conflating your
expectations of those terms with what you actually manage to define.

We represent this as a set which contains no elements.

You are now jumping well ahead. No. You cannot represent something as
"a set which contains no elements" until you either DEFINE what a set,
what element, and what contain mean, or else you declare them to be
undefined/primitive terms and relations. In either case, you are not
"starting from nothing".

First sentence.

Are you really this thick?

If we use the symbol {}, to represent the container, then {} also
represents the set which contains no elements.
Any symbol within the container is an element of the set, eg, {?} or
{red}.

Is "within" a primitive relation, or one that is defined? Don't know,
because you don't say.

It is just the English word.

Which means you are now conflating your expectations with what you
actually define. Your primitive terms do not allow you to simply use
English words as synonyms whenever you feel like it just because you
have an intuitive feeling that your "membership" relation means
"within" or "inside". There is no warrant for such an assertion.

If any symbol within the container is an element of the set, does that
mean that the set you have attempted to exemplify this by contains
both '?' and "red", given that both of them are instances of symbols
within the "container". I doubt, however, that this was your intent.

For any set which contains multiple elements, the elements will be
separated by a comma, eg, {red,green,blue}.

By your definition, the comma is also an element of the set.

Any symbol(except the comma), within the container is an element of
the set, eg, {?} or {red}.

Are you attempting a new definition? Perhaps you might be better off
actually stating them formally. That way you might see the serious
drawbacks in your attempt.

For starters: one cannot define something by listing examples. Your
attempt at definition here is lousy and useless.

So, presumably, you are not saying what you mean. That, certainly, is
a problem (since you asked about problems).

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.

Ah! Finally. This should have been well closer to the top.

If every element of set A is also an element of set B and not every
element of set B is an element of set A, then set A is a subset of set
B.

So, no set is a subset of itself, then. Okay.

Not enough sets exist to determine so far.

No, you are quiet simply wrong, probably because you are utterly
confused about what you have said as opposed to what you think you
were saying. Comes from your inability to state things correctly or
precisely.

We know there are sets, because you declared by fiat that there are
sets. Under the definitions you have given, "No set is a subseet of
itself" Is a theorem of your theory.

Proof. We have two possibilities:

(i) There exists an element x and a set A such that x is an element
of A and x is not an element of A. Then the theory is inconsistent,
and any well-formed sentence is a theorem of the theory. In
particular, "No set is a subset of itself" is a theorem.

(ii) There does not exist any element x and set A such that x is an
element of A and x is not an element of A. Then, if A is any set,
then there does not exist x such that x is an element of A and x is
not an element of A. By the given definition of "subset", it follows
that A is not a subset of itself. By universal quantification, "No
set is a subset of itself."

QED

The minimal set has no subsets.

So what? No set is a subset of itself according to your definition.

If not every element of set A is an element of set B and every element
of set B is also an element of set A, then set A is a superset of set
B.

Again, no set is a superset of itself. Okay...

Same again.

Indeed, same again: you don't know what you are talking about. It is
trivial that A is a superset of B if and only if B is a subset of
A. Since we have established that "No set is a subset of itself" is a
theorem of your theory, it follows that "No set is a superset of
itself" is a theorem of your theory. QED

You really need to learn some basic logic if you are going to try to
continue along this road.

A subset is a set within a set, and has a lower cardinality than its
parent set.

You have not defined "within", "cardinality" (or how 'lower' may apply
to it), or "parent", so this statement is either meaningless, or a new
source of undefined terms.

That is because no special meaning is attributed to these words, they
are just words used within the context of the English language.

I was thinking about replacing cardinality with amount of elements.

I think you should start thinking about learning some basic logic
instead of continuing to waste your time. I have no desire to continue
to waste mine.

--
======================================================================
"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

"Golden Boar" <goldenb...@hotmail.com> wrote in message

news:91a37007-4a1c-4ac6-92d4-2a7125c32707@34g2000hsh.googlegroups.com...

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.

There is a unique set that contains no content, which is called the
minimal set, {}.
There is a unique set that contains all the contents of all the sets,
which is called the maximal set, {...}.

Unfortunately you have just eliminated the possibility of infinite sets. In
normal set theory (with the axiom of infinity) there cannot be a maximal
"set" according to the normal definition.

If you are trying to build standard set theory using your definitions, then
this will doom you to failure. If you are trying to build a different type
of set theory (not equivalent to "normal" set theory), you will have to be a
lot tighter with your axioms.

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

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.

There is a unique set that contains no content, which is called the
minimal set, {}.
There is a unique set that contains all the contents of all the sets,
which is called the maximal set, {...}.