A hidden assumption

[doronshadmi]

Great! So statements you made in that PDF you referenced a few times in this thread are flatly wrong.

I just want to be clear on that.

If you know the statements are wrong, why not correct them?

If you mean to this statment:

By the standard notion a set is determined by its members, which is the same as saying that the union of all members of X is set X.

It does not mean that X is a member of itself. It means that the standard notion of the concept of Membership looks only on the content and use the container as a hidden assumption.

 

[jsfisher]

If you mean to this statment:

By the standard notion a set is determined by its members, which is the same as saying that the union of all members of X is set X.

It does not mean that X is a member of itself. It means that the standard notion of the concept of Membership looks only on the content and use the container as a hidden assumption.

Two points:

(1) Distilled down, your statement, above, is "By the standard notion [of set] ... the union of all members of X is set X."

That is flatly wrong. Do you really believe otherwise?


(2) You keep repeating this notion of set container as a hidden assumption.

This is also flatly wrong. You seem to be hung up attributing physical properties to abstract constructs. Were what you say true, then in the "standard notion of set", { B } and B would be indistinguishable. That would be nonsense.

 

[doronshadmi]

I've read this material before. Are you plagiarizing?

PDF

It seems to me that you are discussing philosophy and not math. If I understand you correctly, you are basically saying two-value logic is not enough to describe itself.

Hi Richard Masters,

Thank you for your reply and for the links (I am going to read it).

I am basically saying that the multiple is the outcome of total symmetry\total asymmtery synthesis,
where 2-valued logic is some particular case of this synthesis.

Math and Philosophy are disconected by the current formal paradigm.

 

[doronshadmi]

Two points:

(1) Distilled down, your statement, above, is "By the standard notion [of set] ... the union of all members of X is set X."

That is flatly wrong. Do you really believe otherwise?

Please povide some link that clearly show that by the standard notion a set is not the union of its members (in the case of the empty set the union is simply empty).

Here is some link:

"What the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the members of the members of A. By the axiom of extensionality this set B is unique and it is called the union of A, and denoted ∪A. Thus the essence of the axiom is:

The union of a set is a set." ( http://en.wikipedia.org/wiki/Axiom_of_union )

In other words, since a set is determined by its elements(members) ( http://plato.stanford.edu/entries/set-theory/ ) we can conclude that the union of all members of X is set X.

(2) You keep repeating this notion of set container as a hidden assumption.

This is also flatly wrong.

Please explain why |{}|=0 and not 1 , |{{}}|=1 and not 2 , |{{},{{}}}|=2 and not 3 , etc. ... by the standard notion? (I claim that the standard notion is right in this case. The difference between the standard framework and my framework, in this case, is this:

My framework explicitly provides the logical difference between being content (a content is a XOR outcome) and being a container (the container is a NXOR outcome).

The standard framework uses the container but it is not explicitly defined by it, or in other words, it is used as a hidden assumption.

You seem to be hung up attributing physical properties to abstract constructs.

In my framework there is no dichotomy between the abstract and the non-abstract. Both of them are generalized to Symmetry\Asymmetry relation which is the universal principle of my framework.

Were what you say true, then in the "standard notion of set", { B } and B would be indistinguishable. That would be nonsense.

Not at all, you look at the notations instead of the notion that they represent.

If you wish to accurately understend the concept of set, then please use the most accurate representation of it.

The best way is to use some notation to represent the domain (the container) and some notation(s) (or their absence in the case of emptiness) to represent the content.

If you do that then (for example):

B={}, and {B}={{}}≠{}

 

[jsfisher]

Please povide some link that clearly show that by the standard notion a set is not the union of its members (in the case of the empty set the union is simply empty).

Whoa! Doronshadmi, do you not realize that "union" is a set operator? "A union B" is only meaningful when both A and B are sets. The union of the members of { {A,B}. {B,C}, {A,C} } is {A, B, C}. To ask the union of the members of {A,B,C} is meaningless.

You do not understand a very basic set theory concept. How embarrassing for you.

Here is some link:

"What the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the members of the members of A. By the axiom of extensionality this set B is unique and it is called the union of A, and denoted ∪A. Thus the essence of the axiom is:

The union of a set is a set." ( http://en.wikipedia.org/wiki/Axiom_of_union )

Did you completely miss the part I bolded? Did you completely miss the part where the Axiom of Union deals with sets of sets?

In other words, since a set is determined by its elements(members) ( http://plato.stanford.edu/entries/set-theory/ ) we can conclude that the union of all members of X is set X.

You were fine until the "we can conclude" part, then you completely fell apart.

Please explain why |{}|=0 and not 1 , |{{}}|=1 and not 2 , |{{},{{}}}|=2 and not 3 , etc. ... by the standard notion?

Why would I explain something when there is no controversy?

(I claim that the standard notion is right in this case. The difference between the standard framework and my framework, in this case, is this:

My framework explicitly provides the logical difference between being content (a content is a XOR outcome) and being a container (the container is a NXOR outcome).

The standard framework uses the container but it is not explicitly defined by it, or in other words, it is used as a hidden assumption.

Nope, the standard notion of sets fully recognizes the difference between "set" and "element of set".

If you do that then (for example):

B={}, and {B}={{}}≠{}

Huh? You equate any element to the null set? That is bizarre beyond belief.

 

[doronshadmi]

Whoa! Doronshadmi, do you not realize that "union" is a set operator? "A union B" is only meaningful when both A and B are sets. The union of the members of { {A,B}. {B,C}, {A,C} } is {A, B, C}. To ask the union of the members of {A,B,C} is meaningless.

You are right in this case, because by the standard notion, a set is a collection of distinct objects, so the union of the members of {{A,B,C},{A,B,C}} is {A,B,C} is indeed meaningless because {{A,B,C},{A,B,C}} is a multiset.

Thank you for your correction.

By the way is simplifies my claim, which is now based only on the fact that a set is determined by its (distinct) members.

Nope, the standard notion of sets fully recognizes the difference between "set" and "element of set".

A set is determined by its members (by its content). Please show me a definition of the container.

Huh? You equate any element to the null set? That is bizarre beyond belief.

B={{},{{}}}, and { {{},{{}}} } ≠ {{},{{}}}

 

[jsfisher]

You are right in this case, because by the standard notion, a set is a collection of distinct objects, so the union of the members of {{A,B,C},{A,B,C}} is {A,B,C} is indeed meaningless because {{A,B,C},{A,B,C}} is a multiset.

Thank you for your correction.

I offered two examples, neither of which is the one you cited, above.

(1) Union of the members of { {A,B}, {B,C}, {A,C} } is {A, B, C} as follows:
Members of the original set are {A,B}, {B,C}, and {A,C}.
( {A,B} union {B,C} ) union {A,C} = {A,B,C} union {A,C} = {A,B,C}

(2) Union of the members of {A,B,C} is meaningless because the members of the original set, A, B, and C are not themselves sets.

(The not-set nature of A, B, and C was not explicitly stated, only implied in the orginal. Please, let's not waste thread space quibbling over that point.)

By the way is simplifies my claim, which is now based only on the fact that a set is determined by its (distinct) members.

That's fundamental to set theory, that the elements of a set are distinct. Your bogus claim is unchanged.

A set is determined by its members (by its content). Please show me a definition of the container.

Again with this "container" nonsense? A set is a collection of things according to some well-defined rule for membership. Membership requires no containment, it simply is. Why do you feel a container is necessary?

B={{},{{}}}, and { {{},{{}}} } ≠ {{},{{}}}

Was there a point to this last bit?

 

[doronshadmi]

One of the beautiful things is to learn from a mistake, at least because of two reasons.

1) To understand a mistake is to understand better the rules of a given system.

2) If you understand better the rules of a given system, you are in a better position in order to examine it and maybe define a new point of view that can improve it.

By improvement I mean fewer rules that at least achieve what the current rules achieve (and if we are lucky, than the fewer rules even achieve more interesting\accurate results and\or help us to solve problems which are unsolved from the previuos point of view).

As for the union case, please look at this:

If A and B are sets, then the union of A and B is the set that contains all elements of A and all elements of B, but no other elements. The union of A and B is usually written "A ∪ B". Formally:
x is an element of A ∪ B if and only if
x is an element of A or
x is an element of B.
(This is an inclusive "or".)

For example, the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. The number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.

http://en.wikipedia.org/wiki/Union_(set_theory)

In that case the union is not limited to the members of the members of a set as defined in the case of ZF axiom of union (http://en.wikipedia.org/wiki/Axiom_of_union )

In other words if A=B={x,y,z} then A ∪ B={x,y,z}∪{x,y,z}={x,y,z}

In that case the union of all members of X (where X must be a set, otherwise it does not have members) is set X ( http://www.geocities.com/complementa...y/NXOR-XOR.pdf ).

If I missed something, then please show it.

 

[doronshadmi]

Part 2:

By examine the difference between the union between members of sets http://en.wikipedia.org/wiki/Union_(set_theory) and the union between the members of the members of a set (http://en.wikipedia.org/wiki/Axiom_of_union ) I have found this:

 

[doronshadmi]

part 3:

A set is a collection of things according to some well-defined rule for membership. Membership requires no containment, it simply is. Why do you feel a container is necessary?

Membership is a comparison between elements according to some rule.

The rule determines what set we get by the comparison, for example:

If any given element is not a member of the domain, than the domain is called "the empty set", otherwise "a non-empty set".

Other rules determine the exact property of each domain by determine the properties of its non-empty content.

No rule can be used if nothing is comparable, so the most fundamental principle is the principle of comparison.

Now we ask:

When comparison is impossible?

We have found that comparison is impossible if we are in total Symmetry, because all we have is sameness.

Also we have found that comparison is impossible if we are in total Asymmetry, because all we have is isolation.

So no mathematical rule is expressible without Symmetry\Asymmetry synthesis, which enables the comparable.

Symmetry is used as the invariant side of the comparison (what is called the domain or the container).

Asymmetry is used as the variant side of the comparison (what is called a member or a content).

 

[jsfisher]

<snip>
In that case the union of all members of X (where X must be a set, otherwise it does not have members) is set X

If I missed something, then please show it.

You were doing quite well up until this last bit. The union of all members of a set X is not (in general) the set X.

For example, let X = { {A,B}, {A,C,D} }

The set X has two members. They are {A,B} and {A,C,D}.

The union of all members of set X = {A,B} union {A,C,D}.

{A,B} union {A,C,D} = {A,B,C,D}.

{ {A,B}, {A,C,D} } = set X and therefore set X ≠ {A,B,C,D}.

A set is not the union of all its members.

 

[doronshadmi]

You were doing quite well up until this last bit. The union of all members of a set X is not (in general) the set X.

For example, let X = { {A,B}, {A,C,D} }

The set X has two members. They are {A,B} and {A,C,D}.

The union of all members of set X = {A,B} union {A,C,D}.

{A,B} union {A,C,D} = {A,B,C,D}.

{ {A,B}, {A,C,D} } = set X and therefore set X ≠ {A,B,C,D}.

A set is not the union of all its members.

You are right, the "Union case" is closed and I'll fix my paper.

Please reply to http://www.internationalskeptics.com/forums/showpost.php?p=3541102&postcount=90 .

 

[jsfisher]

Membership is a comparison between elements according to some rule.

No. Membership is a simple abstract property not involving comparison. Either A is a member of set B, or it isn't. There is no comparison of A, for example, to the members of B looking for a match.

In fact, there is no meaningful way to formulate the question "Is A a member of set B?" as a comparison with and among members. That would require assembling the membership of B of comparison, and that would presuppose already knowing whether A were a member.

(By the way, this is not to say that the membership property isn't governed by some comparison-based rule, but that is a different matter.)

 

[doronshadmi]

You wrote:

Either A is a member of set B, or it isn't.

Let C be a rule and let A be an element.

If there is a comparison between C and A, we are enable to conclude if either A is a member of B (under rule C), or it isn't (not under rule C).

Furthermore, both states (under rule C\not under rule C) must simultaneously share the same framework in order to be distinguished.

Try to use only XOR (either C OR A) and you cannot define A's membership because if you get C you cannot get A, and if you get A you cannot get C (the XOR connective does not enables it).

So the framework that enables Membership is not less than simultaneity(Symmetry)\non-simultaneity(Asymmetry) synthesis.

I call simultaneity the non-local atom of that framework, and I call non-simultaneity the local atom of that framework.

So this realm is not less than Non-locality\Locality synthesis, where the synthesis understood in terms of comparison.

I logically addressed Non-locality by NXOR connective and Locality as XOR connective.

From the synthesis point of view, any other logical connective is the outcome of NXOR\XOR synthesis. In this case NXOR\XOR synthesis is the universal principle of this Logic.

What do you think?

 

[ArmillarySphere]

I think C is completely unnecessary.

A set can be validly described as "the set of every positive, odd integer except 37 and 29, the Heisenberg uncertainty principle, the colour blue, and my cat". That these four members share no comparable criteria whatsoever is completely irrelevant. A rule may be an easy shorthand to describe what the members of a set are, but it's just as valid just to list them.

 

[doronshadmi]

I think C is completely unnecessary.

A set can be validly described as "the set of every positive, odd integer except 37 and 29, the Heisenberg uncertainty principle, the colour blue, and my cat". That these four members share no comparable criteria whatsoever is completely irrelevant. A rule may be an easy shorthand to describe what the members of a set are, but it's just as valid just to list them.

Rule C is rule C no matter how many conditions it has.

Each element has to be compared with rule C in order to determine if it is a member of set B.

No membership is definable without the comparison between rule C and element A.

In that case, Comparison is the fundamental principle of Membership.


Rule C and element A comparison is used as a fundamental principle also in the case of Fuzzy Logic's Membership_function ( http://en.wikipedia.org/wiki/Membership_function_(mathematics) )

 

[ben m]

You wrote:

Let C be a rule and let A be an element.

Try to use only XOR (either C OR A) and you cannot define A's membership because if you get C you cannot get A, and if you get A you cannot get C (the XOR connective does not enables it).

You're comparing a rule with an element. If B is the set of all integers greater than 10, then C is "(> 10)". A is something like 3, or 11, or 50001. "either (> 10) or 50001" doesn't mean anything; neither does "(>10) or 50001", etc.

 

[jsfisher]

Let C be a rule and let A be an element.

You need to differentiate between the value of something and how that value might be determined.

Whether something is a member of a given set is a property of that something. There may be a rule that you can use to determine the property's value, but the the property has a value whether you know it or not.

Just as I am of a certain height. You may need to use some sort of measuring device to assess my height (a comparison I suppose), but that doesn't change what my height is, only whether you know what it is.

 

[jsfisher]

Let C be a rule and let A be an element.

If there is a comparison between C and A, we are enable to conclude if either A is a member of B (under rule C), or it isn't (not under rule C).

You know, the bolded part, above, has no meaning. "Comparison" is definitely the wrong word.

A membership rule is essentially a function, as in C(A) with value of either "A is a member" or "A isn't a member" of some set. There is no binary operator (comparison or otherwise) that interacts with C and A.


ETA: Ben_M beat me to this point. Doh!!

 

[jsfisher]

Rule C and element A comparison is used as a fundamental principle also in the case of Fuzzy Logic's Membership_function ( http://en.wikipedia.org/wiki/Membership_function_(mathematics) )

By the way, let's not jump to other areas of Mathematics. For the most part, we've been discussion set theory with in the naive set theory framework with occasional excursions into ZFC. Fuzzy logic, on the other hand, is a different animal all together, so let's leave it aside.