A hidden assumption

[doronshadmi]

What do you mean "dichotomy"? Set and member are two different concepts.

No. Set is the result of (the concept of Function) AND (the concept of Member).

The concept of Function is notated as the outer "{" and "}" and the concept of Member is what's between "{" and "}" .

Any union between sets (including the union of a set with itself) is between elements that each one of them is the result of (the concept of Function) AND (the concept of Member) (if each one of them is also a set).

If you omit one of the concepts, the result (which is the concept of Set) does not hold water.

If you show evan a one set that is not the result of (the concept of Function) AND (the concept of Member) then and only then my argument does not hold.

By the way the member of a set is not necessarily the result of (the concept of Function) AND (the concept of Member), but a set that some or all of its members are ur-elements is the result of (the concept of Function) AND (the concept of Member) no matter if each member is itself a set or not.

You may say:

"so by your own words you have found a member which is not necessarily a set, so you can distinguish between the concept of Set and the concept of Member."

My answer is:

"x is a member X of only if (the concept of Function) AND (the concept of Member) are used, without any connection to the property of the member (it can be a ur-element or not, it does not matter).

Let X={set1,ur-element1}

Union(set1,ur-element1) is not the union between X members, but it is between set1 and ur-element1.

Union({set1,ur-element1}) is the union between X members because X is not less than the result of (the concept of Function, notated by the outer "{" and "}") AND (the concept of Member, notated be anything between "{" and "}") where "{""}" and anything between is an one organic object (the result of whole\parts complementation).

Explicitness is valuable, but coherence is essential.

There is no coherence if not all possible terms that define X are explicitly expressed.

 

[ArmillarySphere]

Your definitions are not the same as used in standard set theory.
If you don't use the same definitions, there is nothing to discuss, as you are not talking about the same thing as anyone else.

FWIW, it's perfectly fine working with sets even if they're undefined, as long as you don't need to look at the members.

 

[doronshadmi]

Your definitions are not the same as used in standard set theory.
If you don't use the same definitions, there is nothing to discuss, as you are not talking about the same thing as anyone else.

FWIW, it's perfectly fine working with sets even if they're undefined, as long as you don't need to look at the members.

You cannot conclude that without (the concept of Function) AND (the concept of Member) complementation, for example:

Function(Standard theory,My theory) = {Standard theory,My theory}

Let us return to the example of 2:

2 has many properties that can be a set.

In that case 2 is the function (notated by the outer "{" and "}") and its properties are its members (what's between "{" and "}").

So also in this case (the concept of Function) AND (the concept of Member) are used as complement concepts.

We can ask:

"And what if a function is it own member?"

My answer:

Also in this case it is wrong to claim that (the concept of Function) AND (the concept of Member) is meaningless (think of Klein's bottle, for example).

---------------------------------------------------

FWIW, it's perfectly fine working with sets even if they're undefined, ...

What a beautiful day (oopss... sorry, a contradiction). I like it.

I challenge you to define a researchable thing, which is not the result of (the concept of Function) AND (the concept of Member) complementation (you can use "Variable" insted of "Member", if you wish).

 

[PixyMisa]

No. Set is the result of (the concept of Function) AND (the concept of Member).

Functions don't come into it. Sets and members are related concepts, but what you are actually saying is nonsense.

The concept of Function is notated as the outer "{" and "}" and the concept of Member is what's between "{" and "}" .

Nope.

If you show evan a one set that is not the result of (the concept of Function) AND (the concept of Member) then and only then my argument does not hold.

Your argument is drivel.

You may say:

"so by your own words you have found a member which is not necessarily a set, so you can distinguish between the concept of Set and the concept of Member."

No thanks.

My answer is:

"x is a member X of only if (the concept of Function) AND (the concept of Member) are used, without any connection to the property of the member (it can be a ur-element or not, it does not matter).

At best this is a tautology. x is a member of X if x is a member of X.

Let X={set1,ur-element1}

Union(set1,ur-element1) is not the union between X members, but it is between set1 and ur-element1.

It's the exact same thing.

Union({set1,ur-element1}) is the union between X members because X is not less than the result of (the concept of Function, notated by the outer "{" and "}") AND (the concept of Member, notated be anything between "{" and "}") where "{""}" and anything between is an one organic object (the result of whole\parts complementation).

Total drivel. You can't have a union of one item. You can have a union of an item with itself - which is an identity function - but that's different.

There is no coherence if not all possible terms that define X are explicitly expressed.

There is no coherence.

 

[PixyMisa]

We can ask:

"And what if a function is it own member?"

Functions don't have members.

My answer:

Also in this case it is wrong to claim that (the concept of Function) AND (the concept of Member) is meaningless (think of Klein's bottle, for example).

Okay, thinking of a Klein bottle...

Huh. Your argument is still total nonsense.

 

[doronshadmi]

Sets and members are related concepts,...

And how they are related?

Ansewr: By Function(the concept of Set,the concept of Member).

It is inevitable for any researchable thing.

You can't have a union of one item. You can have a union of an item with itself - which is an identity function - but that's different.

You are right. it has to be written as Union({set1,ur-element1},{set1,ur-element1})={set1,ur-element1}

It does not change my argument.

Functions don't have members.

http://en.wikipedia.org/wiki/Membership_function_(mathematics)

A particular case of the Membership_function is Membership(0)=not a member of ...
or Membership(1)=a member of ....

 

[PixyMisa]

You are right. it has to be written as Union({set1,ur-element1},{set1,ur-element1})={set1,ur-element1}

It does not change my argument.

Agreed. Correcting this one invalid statement does nothing to correct all your other invalid statements.

http://en.wikipedia.org/wiki/Membership_function_(mathematics)

A particular case of the Membership_function is Membership(0)=not a member of ...
or Membership(1)=a member of ....

Did you even look at that page? Once again: Functions don't have members.

 

[doronshadmi]

Agreed. Correcting this one invalid statement does nothing to correct all your other invalid statements.

Did you even look at that page? Once again: Functions don't have members.

You did not read my argument.

Function() can also be notated by "{" and "}".

A set is the result of (the concept of Function) AND (the concept of Member).

 

[PixyMisa]

You did not read my argument.

Function() can also be notated by "{" and "}".

And sets can be notated by f(x). But they aren't.

A set is the result of (the concept of Function) AND (the concept of Member).

No. This makes no sense at all.

A set is an unordered collection. A function is a transformation. A member is an object viewed in the context of a collection.

Actual sets have nothing to do with the concepts of functions or members. And the concept of sets is not a set.

You have created an appalling mishmash of misunderstood mathematical concepts, non-standard mathematical notation, and irrelevant examples. To say that your argument lacks rigour is akin to stating that custard is an unsuitable building material for reactor containment vessels. I award you no points, and may God have mercy on your soul.

 

[doronshadmi]

And sets can be notated by f(x). But they aren't.

You are talking about notations and names and I am talking about notions and generalization.

From the general point of view any unordered collection (of distinct objects, if it is a set) cannot be defined if there is no invariant togetherness ( http://en.wikipedia.org/wiki/Element_(mathematics) ) which is not changed by any transformation, manipulation, or any other ...ation.

"In mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the set (or class)."
( http://en.wikipedia.org/wiki/Element_(mathematics) )

Without this invariant basis (notated as "{" and "}" in the case of a set, or as "(" and ")" in the case of a function etc. ... ) nothing is related even to itself.

Also no collection can be defined if the concept of Member is not defined.

A set is the result of Invariance\Variance complementation.

I award you no points, and may God have mercy on your soul.

Even God cannot help you to understand it, without your help.

A set is an unordered collection.

No, a set is an unordered collection of distinct members. The question is: how do you know that (what enables you to know that)?

Without the understanding of the Researchable\Non-researchable complementation, your "rigorus" definitions do not hold water.

Actual sets have nothing to do with the concepts of functions or members. And the concept of sets is not a set.

Ho, we have here a Platonist.

Please tell us what divine proprty do you have that enables you to get things without any concept of them?

 

[drkitten]

You are talking about notations and names and I am talking about notions and generalization.

From the general point of view ....

Your idiosyncratic mishmash of definitions is hardly "general," and your collection of drivel is by no possible stretch of imagination a "point of view."

And you're not "talking." You're dribbling.

So, zero for three.

 

[jsfisher]

Function() can also be notated by "{" and "}".

Telling us the notation for something doesn't define it. So far, you have simply drawn a circle that includes "set", "function", and "{...}".

You get no points for a complete thought.

A set is the result of (the concept of Function) AND (the concept of Member).

Still hung up on verbs, aren't you?. A set is not the result of any action, be it some function or otherwise; it simply is. A set is a collection of objects. A set is not the result of putting anything in a container, and it is totally independent of whatever notation we might use to express a set.

 

[doronshadmi]

A set is not the result of any action

Complelentation is a state, not an action, and it is timeless.

 

[doronshadmi]

Your idiosyncratic mishmash of definitions is hardly "general," and your collection of drivel is by no possible stretch of imagination a "point of view."

And you're not "talking." You're dribbling.

So, zero for three.

FWIW, it's perfectly fine working with sets even if they're undefined, as long as you don't need to look at the members.

Are you kidding?

A set is an undefined object in your traditional universe (as we learn from ArmillarySphere), so your conclusions do not hold water.

You may claim:

" A set, a Number, a Point, a Line are all context dependent things deduced
from carefully (rigorously) defined self-evident true statements known as axioms.

This context dependent approach enables these undefined concepts to be researched from many points of view, which enrich the mathematical science that in general is the result of rigorous definitions and relations based on axioms."

This is really fine, but a simple question is asked?

What enables the researchable?

If this simple question is not answered, then all these beautiful phrases above have no rigorous basis.

My theory a possible answer of this question inspired by Hilbert's organic paradigm of the mathematical science, and it is based on Researchable\Non-researchable complementation (where complementation is a timeless concept of my theory).

When observed from this complementation, a set, a number, a point, a line are concepts of a one organic universe that has infinitely many points of view that consistently enrich each one of its organs, and enables them to be developed by paradigm-shifts (known as mutations of old paradigms).

 

[jsfisher]

Complelentation is a state, not an action, and it is timeless.

You really like that word, "complementation". Care to share with the rest of us what you mean by it?


ETA: This is another of those reading comprehension issues you have, isn't it? Your response has absolutely nothing to do with the portion of my post you quoted.

 

[ArmillarySphere]

Should we make a list? These are the undefined terms that came to my mind:

* complementation
* researchable
* symmetry
* assymmetry
(fill in the blanks)


And what's so wrong about working with undefined sets? They're sets, so it's meaningful to speak of their members (could be zero or more). Any object is a potential member, but the question can't be answered as long as the set is undefined, nor does it probably matter.

Of course, you may define the sets, in which case you can start talking about cardinality, answering the membership question, and getting explicit results from playing with set algebra. But you don't have to.

 

[PixyMisa]

Are you kidding?

A set is an undefined object in your traditional universe (as we learn from ArmillarySphere), so your conclusions do not hold water.

No. Once again, you are confusing the concept with the instance.

The concept of sets is defined. The individual set in question isn't. That's fine; we still know it's a set.

You may claim:

" A set, a Number, a Point, a Line are all context dependent things deduced
from carefully (rigorously) defined self-evident true statements known as axioms.

An axiom doesn't need to be self-evident.

This context dependent approach enables these undefined concepts to be researched from many points of view, which enrich the mathematical science that in general is the result of rigorous definitions and relations based on axioms."

Yep.

This is really fine, but a simple question is asked?

What enables the researchable?

The Invisible Pink Unicorn.

If this simple question is not answered, then all these beautiful phrases above have no rigorous basis.

You mean, apart from propositional logic?

My theory a possible answer of this question inspired by Hilbert's organic paradigm of the mathematical science, and it is based on Researchable\Non-researchable complementation (where complementation is a timeless concept of my theory).

And again we retreat into meaninglessness.

When observed from this complementation, a set, a number, a point, a line are concepts of a one organic universe that has infinitely many points of view that consistently enrich each one of its organs, and enables them to be developed by paradigm-shifts (known as mutations of old paradigms).

And pass beyond into self-parody...

 

[PixyMisa]

You are talking about notations and names and I am talking about notions and generalization.

You are making up notations and generalization. That's the problem. If you want to communicate with someone, you have to use words with a mutually agreed definition.

"In mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the set (or class)."
( http://en.wikipedia.org/wiki/Element_(mathematics) )

Without this invariant basis (notated as "{" and "}" in the case of a set, or as "(" and ")" in the case of a function etc. ... ) nothing is related even to itself.

And?

Also no collection can be defined if the concept of Member is not defined.

That's backwards. A member is defined in terms of a collection. Without the context of the collection, a member is just any old object.

A set is the result of Invariance\Variance complementation.

More nonsense.

No, a set is an unordered collection of distinct members.

Yep.

The question is: how do you know that (what enables you to know that)?

That's how sets are defined.

Please tell us what divine proprty do you have that enables you to get things without any concept of them?

The concept is not the thing. You continually fail to grasp this.

 

[jsfisher]

Should we make a list? These are the undefined terms that came to my mind:

* complementation
* researchable
* symmetry
* assymmetry
(fill in the blanks)

non-local
dichotomy
organic
paradigm
sameness
variant
invariant
real-line
ur-element
superposition
function
togetherness

(Those are just ones I recall off the top of my head.) Still, I'd prefer doronshadmi simply use words and phrases with their standard meaning. It is unnecessary to invent things the way he does, especially since doronshadmi has demonstrated his inability to explain what he means nor does he always use his invented terms consistently.

Then again, expecting him to use standard terminology may be a bit much to ask. Heck, he can't even work with the basic concept of set union without making a mess of it.

 

[tsig]

Are you kidding?

A set is an undefined object in your traditional universe (as we learn from ArmillarySphere), so your conclusions do not hold water.

You may claim:

" A set, a Number, a Point, a Line are all context dependent things deduced
from carefully (rigorously) defined self-evident true statements known as axioms.

This context dependent approach enables these undefined concepts to be researched from many points of view, which enrich the mathematical science that in general is the result of rigorous definitions and relations based on axioms."

This is really fine, but a simple question is asked?

What enables the researchable?

If this simple question is not answered, then all these beautiful phrases above have no rigorous basis.

My theory a possible answer of this question inspired by Hilbert's organic paradigm of the mathematical science, and it is based on Researchable\Non-researchable complementation (where complementation is a timeless concept of my theory).

When observed from this complementation, a set, a number, a point, a line are concepts of a one organic universe that has infinitely many points of view that consistently enrich each one of its organs, and enables them to be developed by paradigm-shifts (known as mutations of old paradigms).

You make me feel old-fashioned since I only have the traditional universe to live in. I think your organically enriched universe is full of bull enriched food.