Deeper than primes

[Little 10 Toes]

Again,

By the standard notion a proper class is a collection that logically cannot be or it is too big in order to be considered as a set.

" Logically can't be " or "Too big" means that it is impossible to define such a collection in terms of set.

Funny, that's not what Proper_class^WP says. Once again, you are abusing standard terms to be something they're not.

Since by OM, the cardinality of a collection is the magnitude of existence of its objects, then the full set is definable and proper classes are avoided, simply because only the full set is an actual non-finite.

See above.

It is done by using an ontological point of view of collections (it cannot be done by using the standard notion of cardinality) and as a result we get simpler and richer mathematical framework, which is much more interesting than the cantorean transfinite framework.

Collections can be done using the standard notion of cardinality. It's you that makes the water muddy.

<snip>

The fact is this: we have an existing {} that its cardinality is 0.

From an ontological point of view (where we first of all care about the existence (or non-existence) of things) an existing thing cannot have cardinality 0.

But you just said that there is an existing set that has a cardinality of 0. Most people call it the empty set.

In that case the cardinality of {} is detemined by the magnitude of existence of its members, where {} is en existing measurment tool, that is always excluded from the measurment.

Please define "the magnitude of existence of its members". You have been asked several times yet you avoid it.

Ok, let us close this dialog for now.

No. Let's not.

The fact is that jsfisher and The Man try to get OM by using the standard notions of set.

As a result they can't get OM's new notions about set.

It's because you can't define it.

By using an ontological viewpoint of the foundations of the mathematical science we distinguish between 3 levels of existence:
1) Emptiness
2) Intermediate
3) Fullness

Emptiness or Fullness can be researched only indirectly by using the intermediate level of existence.

Ok then, back to basics. Please define "Emptiness", "Intermediate", and "Fullness".

A set is a level 2 (Intermediate) thing.

Let's remember that.

By OM, the cardinality of a set is determined by the magnitude of existence of its members.

Please define "the magnitude of existence of its members".

A set exists even if it is empty, but this existence is excluded from its cardinality value.

So if a set has no members, the cardinality of that set is zero.

Since a set is a level 2 (Intermediate) thing, then:
1) It is above the level of Emptiness ( for example: {} )
2) It is at the level of sets ( for example: {a,b,c,...} )
3) It is below the level of Fullness ( for example: {_}_ )

You have not established why a set is a level 2 thing. The rest of this post depends on providing proof that a set is a level 2 thing. Just because you say it is, doesn't mean it's so.

Some claims "there is nothing below set".

He is right because "there is nothing" is Emptiness.

By following the ontological notion, we get the opposite of Emptiness, called Fullness.

Some claims "there is nothing above set".

Well this is ontologically wrong because "nothing" is below set.

Some claims "there is everything above set".

Site?

In summation, please define

• the magnitude of existence of its members
• Emptiness
• Intermediate
• Fullness

 

[dlorde]

Some claims "there is nothing below set".

He is right because "there is nothing" is Emptiness.

By following the ontological notion, we get the opposite of Emptiness, called Fullness.

Some claims "there is nothing above set".

Well this is ontologically wrong because "nothing" is below set.

Surely this is just gibberish, or to be generous, a misunderstanding - nothing is, by definition, not 'a thing' that can only be in one place or another...

 

[doronshadmi]

Simple concepts seem to confuse you very much, Doron. It's not that we "don't get it". Dispute your logic circles, contradictions, inconsistencies, misuse of terminology, and all the rest, we do "get it". Not only do we "get it", we find it contorted, nonsensical, unnecessary, and trivial.

See? We do "get it". We also reject it. Those are not mutually exclusive concepts.

You did not support the claim that you get it.

As a result your post is meaningless.

 

[doronshadmi]

The Ontological basis of Logic​

No logical consequence is determined unless A is comparable with not-A. Let not-A be represented as B. There can be 16 different logical connectives that are the result of A B comparison.

Flag 0 = the compared holds

Flag 1 = the compared does not hold

F = the binary result of the comparison does not hold

T = the binary result of the comparison holds

R = the binary result

Each logical connective is determined by 4 A B comparisons, as follows:

A B R   A B R   A B R   A B R   A B R   A B R   A B R   A B R  
0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T   
0 1 F   0 1 F   0 1 T   0 1 T   0 1 F   0 1 F   0 1 T   0 1 T   
1 0 F   1 0 F   1 0 F   1 0 F   1 0 T   1 0 T   1 0 T   1 0 T 
1 1 F   1 1 F   1 1 F   1 1 F   1 1 F   1 1 F   1 1 F   1 1 F


A B R   A B R   A B R   A B R   A B R   A B R   A B R   A B R  
0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T   
0 1 F   0 1 F   0 1 T   0 1 T   0 1 F   0 1 F   0 1 T   0 1 T   
1 0 F   1 0 F   1 0 F   1 0 F   1 0 T   1 0 T   1 0 T   1 0 T 
1 1 T   1 1 T   1 1 T   1 1 T   1 1 T   1 1 T   1 1 T   1 1 T

Each binary T result of the 16 logical connectives is ontologically based on identical flags (0 0 , 1 1) or different flags (0 1 , 1 0). "0A,0B" and "1A,1B" are ontologically reduced to "flagA is as flagB", where " 0A,1B" and " 1A,0B" are ontologically reduced to "flagA is not as flagB".

By ontological reduction the two building-blocks of Logic are:

flagA is as flagA

flagA is not as flagB

"flagA is as flagB" is the non-local aspect of any binary comparison, where "flagA is not as flagB" is the local aspect of any binary comparison.

 

[doronshadmi]

Funny, that's not what Proper_class^WP says. Once again, you are abusing standard terms to be something they're not.

Read This:

http://en.wikipedia.org/wiki/Cantor's_paradox

Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we also conclude that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".

 

[doronshadmi]

Such a declaration, that the ‘nature’ of your ‘research’ is ‘pre-axiomatic’, is just, well, an axiom of your notions and one that specifically lacks self consistency. In that, being self inconsistent, it certainly does represent the nature of your ‘research’.

You do not understand the difference between defining\using an axiom and ontologically research it.

 

[jsfisher]

You did not support the claim that you get it.

As a result your post is meaningless.

My claim has been well-supported throughout this and other threads. You, on the other hand, have failed repeatedly to support yours. You cannot even define your terms, so absolutely nothing from you beyond that has meaning.

 

[jsfisher]

The Ontological basis of Logic​

No logical consequence is determined unless A is comparable with not-A. Let not-A be represented as B.

No. Let's represent not-A by not-A. No point trying to obscure it's origins.

There can be 16 different logical connectives that are the result of A B comparison.

No. The only free variable is A. Not-A is dependent on A.

Flag 0 = the compared holds

Flag 1 = the compared does not hold

F = the binary result of the comparison does not hold

T = the binary result of the comparison holds

Such over-abundance of notation. How about you pick one system - either 0/1 or F/T - and stick with it? I favor F/T for this application.

R = the binary result

Each logical connective is determined by 4 A B 2 A/not-A comparisons, as follows:

[strike]A B R   A B R   A B R   A B R   A B R   A B R   A B R   A B R 
0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T
0 1 F   0 1 F   0 1 T   0 1 T   0 1 F   0 1 F   0 1 T   0 1 T
1 0 F   1 0 F   1 0 F   1 0 F   1 0 T   1 0 T   1 0 T   1 0 T
1 1 F   1 1 F   1 1 F   1 1 F   1 1 F   1 1 F   1 1 F   1 1 F


A B R   A B R   A B R   A B R   A B R   A B R   A B R   A B R
0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T   0 0 F   0 0 T
0 1 F   0 1 F   0 1 T   0 1 T   0 1 F   0 1 F   0 1 T   0 1 T
1 0 F   1 0 F   1 0 F   1 0 F   1 0 T   1 0 T   1 0 T   1 0 T
1 1 T   1 1 T   1 1 T   1 1 T   1 1 T   1 1 T   1 1 T   1 1 T[/STRIKE]

 A  not-A  R     A  not-A  R     A  not-A  R    A  not-A  R
=== ===== ===   === ===== ===   === ===== ===  === ===== ===
 F    T    F     F    T    T     F    T    F    F    T    T 
 T    F    F     T    F    F     T    F    T    T    F    T

Ok, I fixed that part for you. You should try fixing the rest if you can. Here's some help: Note that the formulae for the above truth tables are F, not-A, A, and T, respectively.

It's all kind of trivial, though. I don't know why you are bothering.

 

[doronshadmi]

Ok, I fixed that part for you.

No.

You fixed nothing, you simply do not understand the ontological reduction of Logic.

Jsfisher you simply cannot get things beyond your box.

 

[doronshadmi]

My claim has been well-supported throughout this and other threads.

Not even in your dearms.

 

[jsfisher]

No.

You fixed nothing, you simply do not understand the ontological reduction of Logic.

Jsfisher you simply cannot get things beyond your box.

At least I understand the meaning of things like "not-A".

 

[jsfisher]

Let's say, just for a moment, Doron, we look beyond the contradictions (not-A is different from not-A), inconsistencies (notation shifts from 0/1 to F/T), and misuse of terminology (ontological). What are we left with? Nothing more than a recitation of truth tables from any introductory text on boolean logic.

Why do you dwell on the trivial, and why do you first contort it so?

 

[The Man]

You do not understand the difference between defining\using an axiom and ontologically research it.

You do not understand the difference between simply making inconsistent declarations and actually doing ‘research’.

No.

You fixed nothing, you simply do not understand the ontological reduction of Logic.

Jsfisher you simply cannot get things beyond your box.

Well your ‘ontological thinking’ certainly appears to have reduced your ‘logic’.

 

[doronshadmi]

You do not understand the difference between simply making inconsistent declarations and actually doing ‘research’.



Well your ‘ontological thinking’ certainly appears to have reduced your ‘logic’.

Let's say, just for a moment, Doron, we look beyond the contradictions (not-A is different from not-A), inconsistencies (notation shifts from 0/1 to F/T), and misuse of terminology (ontological). What are we left with? Nothing more than a recitation of truth tables from any introductory text on boolean logic.

Why do you dwell on the trivial, and why do you first contort it so?

At least I understand the meaning of things like "not-A".

Your reasoning is "nothing more, nothing less" than X, where X is reseachable.

My reasoning is "more, less" than X, where X is reseachable.

By "more,less" reasoning we think out of the box of X, which enables us to get X better:

[qimg]http://www.geocities.com/complementarytheory/OMF.jpg[/qimg]​

My reasoning replaces your reasoning.

 

[Christian Klippel]

Ok, I fixed that part for you. You should try fixing the rest if you can. Here's some help: Note that the formulae for the above truth tables are F, not-A, A, and T, respectively.

I'm wondering what these truth-tables are supposed to represent? Unless you give an operator, no one knows how things are compared. That applies to doron's as well as your TT's.

In the current form they simply mean that a given set of two values can have any result. Don't see any logic there. So, what's the comparison? Equal, not equal, bigger than, less than, bigger or equal than, less or equal than, and, or, not, exclusive or?

Maybe i'm missing something, but i have learned that a truth table applies to a certain function/comparison/logic. So, one could state that 1 AND 0 = 0, or that 1 XOR 0 = 1, or that 1 AND NOT 0 = 1 ..... But things like 0 0 = 0 followed by 0 0 = 1 is a bit of confusing.

Greetings,

Chris

 

[doronshadmi]

I'm wondering what these truth-tables are supposed to represent? Unless you give an operator, no one knows how things are compared. That applies to doron's as well as your TT's.

In the current form they simply mean that a given set of two values can have any result. Don't see any logic there. So, what's the comparison? Equal, not equal, bigger than, less than, bigger or equal than, less or equal than, and, or, not, exclusive or?

Maybe i'm missing something, but i have learned that a truth table applies to a certain function/comparison/logic. So, one could state that 1 AND 0 = 0, or that 1 XOR 0 = 1, or that 1 AND NOT 0 = 1 ..... But things like 0 0 = 0 followed by 0 0 = 1 is a bit of confusing.

Greetings,

Chris

The standard view of the 16 logical connectives:

A	 0 0 1 1	
B	 0 1 0 1	

----------------
0	 F F F F	Contradiction
A AND B	 F F F T	Conjunction
A not→ B F F T F	A not implies B
A	 F F T T	Proposition A
A←not B	 F T F F	B not implies A
B	 F T F T	Proposition B
A XOR B	 F T T F	Exclusive disjunction
A OR B	 F T T T	Disjunction
A NOR B	 T F F F	Joint denial
A NXOR B T F F T	Biconditional
NOT B	 T F T F	not B
A ← B	 T F T T	 B implies A
NOT A	 T T F F	not A
A → B	 T T F T	A implies B
A NAND B T T T F	Alternative denial
1	 T T T T	Tautology

The ontological view of the 16 logical connectives, where each connective
is some combination of the non-local with the local:

A	 0 0 1 1	
B	 0 1 0 1	

----------------
0	 F F F F	No measurement
A AND B	 F F F T	Full non-local
A not→ B F F T F	Local A
A	 F F T T	Full non-local , Local A
A←not B	 F T F F	Local B
B	 F T F T	Full non-local , Local B
A XOR B	 F T T F	Local A, Local B
A OR B	 F T T T	Full non-local , Local A , Local B
A NOR B	 T F F F	Empty non-local
A NXOR B T F F T	Full non-local , Empty non-local
NOT B	 T F T F	Local A , Empty non-local
A ← B	 T F T T	Full non-local , Local A , Empty non-local
NOT A	 T T F F	Local B , Empty non-local
A → B	 T T F T	Full non-local , Local B , Empty non-local
A NAND B T T T F	Local A , Local B , Empty non-local
1	 T T T T	F. non-local , Local A , E. non-local , Local B

If Non-locality (0 0 , 1 1) or Locality (0 1 , 1 0) are related to the same connective (by T value) more than once, then their unique properties are ignored (Fullness and Emptiness or A and B are ignored):

A	 0 0 1 1	
B	 0 1 0 1	

----------------
0	 F F F F	No measurement
A AND B	 F F F T	Full non-local
A not→ B F F T F	Local A
A	 F F T T	Full non-local , Local A
A←not B	 F T F F	Local B
B	 F T F T	Full non-local , Local B
A XOR B	 F T T F	Local  
A OR B	 F T T T	Full non-local , Local  
A NOR B	 T F F F	Empty non-local
A NXOR B T F F T	Non-local   
NOT B	 T F T F	Local A , Empty non-local
A ← B	 T F T T	Non-local , Local A  
NOT A	 T T F F	Local B , Empty non-local
A → B	 T T F T	Non-local , Local B  
A NAND B T T T F	Local  , Empty non-local
1	 T T T T	Non-local , Local

 

[jsfisher]

Your reasoning is "nothing more, nothing less" than X, where X is reseachable....

You again completely ignore any discussion of deficiencies in your ideas and instead just sling insult.

Some things just never change.

 

[doronshadmi]

You again completely ignore any discussion of deficiencies in your ideas and instead just sling insult.

Some things just never change.

I do not ignore your "nothing more, nothing less then X" reasoning.

On the contrary, I replace it by the better "more, less then X" reasoning.


"nothing more, nothing less then X" reasoning cannot get the "more, less then X" reasoning.

On the contrary "more, less then X" reasoning gets the "nothing more, nothing less then X" reasoning, exposes its fundamental fallacies and its inability to understand fundamentals like Actual non-finite, Collection, Fusion, The ontological foundations of Logic, The limits of the researchable, and more …, and replaces it.

Your "nothing more, nothing less then X" dies from entropy like any closed system.

This is exactly the reason why you are unable to understand the first post of this thread and in particular the reasoning of Entropy of mathematical frameworks as shown, for example, by this part:

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

The non-local ur-element is the maximum entropy of itself (no differences can be found within it). Also a local ur-element is the maximum entropy of itself (no differences can be found within it).

Maximum entropy exists in both non-locality and locality, but they are opposite by their self nature, so if non-locality and locality are associated, then a non-entropic domain is created.

The history of such a domain is written by symmetry, where at the first stage symmetry is so strong that no outcome of this domain has a unique identity, and all we have is a superposition of identities.

Symmetry is collapsed because the opposite properties of non-locality and locality are expressed more and more until each local ur-element has a unique identity of its own.

This uniqueness, which is anti-entropic by nature, cannot exist without the association between the non-local and the local.

Opposite properties do not contradict each other, if they are based on NXOR connective.

A NXOR connective enables the existence of NXOR\XOR logic (non-locality and locality are associated, and associated realms have more than one entropy level).

A XOR connective does not enable the existence of NXOR\XOR logic (non-locality and locality are isolated, and isolated realms have maximum entropy).

Please read pages 13-14 of my work called Eventors ( http://www.geocities.com/complementa...y/Eventors.pdf ).

I think that the organic approach (the associations between the non-local and the local) is the accurate way to understand the realm that we are an inseparable part of it.

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

Let us re-examine these cases:

Case 1: associated realms have more than one entropy level.

Case 2: isolated realms have maximum entropy.

In case 1 NXOR is associated with XOR and we get an open realm because both NXOR and XOR go beyond their self state of maximum (and opposite state of) entropy.

In case 2 there is no association between NXOR and XOR, and each opposite is closed upon its own maximum entropy, and nothing exists beyond these closed and isolated opposite maximum entropies.

In a complementary realm, each opposite is opened to an "off spring" outcome, which is beyond its own isolated state (an isolated realm has maximum entropy).

EDIT:

The "off spring" outcome of NXOR-XOR reasoning ( pages 26-28 of http://www.geocities.com/complementarytheory/OMPT.pdf ) is exactly the intermediate and researchable realm of collections, that are able to measure the magnitude of their ontological limits, known as Emptiness and Fullness.

 

[jsfisher]

I do not ignore your "nothing more, nothing less then X" reasoning.

On the contrary, I replace it by the better "more, less then X" reasoning.

More dodging and weaving, I see. Can't address, for example, the contradiction head-on?

You based your whole recent recitation on having A and not-A independent of each other. Mutually independent, as you have been known to say. That's a contradiction you've yet to reconcile.

And how about that explanation of what you mean by "distinction is a first-order property". Still working on that, are you?

 

[doronshadmi]

You based your whole recent recitation on having A and not-A independent of each other.

You are wrong again.

Mutually independent is exactly connected and isolated, and your closed "nothing more, nothing less" closed entropic reasoning cannot get it.

For example: By your closed reasoning you deal only with A because you determine that not-A is the complement of A.

Well you are wrong, not-A is not necessarily the complement of A (and I am not talking about Intuitionistic logic).

Still working on that, are you?

No.