Deeper than primes

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jsfisher said:
I suppose if I state, "Two is a prime number", you'll counter with, "No, you are wrong because two is an even number".

I claimed you have a certain property, a property you continue to demonstrate you have. That, however, in no way excludes you from having other properties.



You show nothing. You merely assert. And your assertions are not founded in very much evidence.
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In other words, you understand nothing (in the negative sense).
 
jsfisher said:
Not true. I understand what you mean by it just fine. It is another of your fantastical inventions, founded in contradiction and nonsense.
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Since your understanding is "nothing more, nothing less" than a collection, you do get the full set.

Furthermore, you do not get the empty set.
 
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zooterkin said:
You do realise that in this context, "axiomatic" means pretty much the same as "ontological", don't you?

Axiom: Logic, maths, a self-consistent, self-evident statement that is a universally accepted truth.

Ontology: Logic, the set of entities presupposed by a theory
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Ontology (from the Greek ὦν, genitive ὄντος: of being <part. of εἶναι: to be> and -λογία: science, study, theory) is the philosophical study of the nature of being, existence or reality in general,... ( http://en.wikipedia.org/wiki/Ontology )

I research, for example, why and how an axiom is considered as "self-consistent, self-evident statement that is a universally accepted truth".
 
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jsfisher said:
This relates to a point I was intending to (re-)make. Doron is spellbound by anthropomorphism. Sets can't simply have cardinality as a property; instead, it must be measured (by whom or what isn't yet clear) and for there to be a measurement, there must be a measurement tool.
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Jsfisher is a realist-only.

But a realist only cannot explain how independent things can be known in the first place. For example the property of a set , where a set is independent of any research (nothing has to be known by a research) according to the realist-only.

Pay attention that I do not claim that the property is necessarily created by the researcher.
What I claim is that the property must be measured in order to be known.

In this case a set is used as a measurement tool "nothing more, nothing less".
 
doronshadmi said:
Possible = exists.

Impossible = does not exist.

{} cardinality is determined by the magnitude of existence of the impossible.

It is impossible to do it without the existing {}.
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doronshadmi said:
You did not get it.

Possible = exists as a possibility ( for example possibility X, or {X} )

Impossible = does not exist ( for example, the member of {} )
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Well here we have the crux of your misunderstanding doron; cardinality is not a ‘magnitude of existence’ but simply a representation of the size of a set. The empty set is not a collection of impossibilities or any singular impossibility, like your ‘empty fusion’. It has no members and that means no impossible members as well. A set of the impossible would in fact be infinite as there are at least as many, if not more, impossibilities then there are possibilities. Possibilities have strict requirements of limited conditions leaving the unlimited conditions as resulting in impossibly (for those desiring a more strict probabilistic interpretation you can simply replace possible with probable and impossible with improbable). In fact the set of the impossible (or improbable) would be infinite and uncountable (if I am not mistaken) and thus the cardinality of that set would be ∞.
 
The Man said:
Well here we have the crux of your misunderstanding doron; cardinality is not a ‘magnitude of existence’ but simply a representation of the size of a set. The empty set is not a collection of impossibilities or any singular impossibility, like your ‘empty fusion’. It has no members and that means no impossible members as well. A set of the impossible would in fact be infinite as there are at least as many, if not more, impossibilities then there are possibilities. Possibilities have strict requirements of limited conditions leaving the unlimited conditions as resulting in impossibly (for those desiring a more strict probabilistic interpretation you can simply replace possible with probable and impossible with improbable). In fact the set of the impossible (or improbable) would be infinite and uncountable (if I am not mistaken) and thus the cardinality of that set would be ∞.
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You are talking about what is called proper class, which is too big or logically cannot be considered as a set

Well, by OM a set is a measurement tool that its cardinality is the magnitude of existence of its members.

From this ontological point of view the full set is definable and proper subsets are avoided, because only the full set is an actual non-finite.
 
doronshadmi said:
You are talking about what is called proper class,
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No, I am not.

doronshadmi said:
which is too big or logically cannot be considered as a set
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So you just do not understand the meaning of ‘class’ let alone ‘proper class’ (a class that is not a set) in set theory, how surprising.

doronshadmi said:
Well, by OM a set is a measurement tool that its cardinality is the magnitude of existence of its members.
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Again cardinality does not ‘measure existence’ so you’ve got the wrong ‘tool’.

doronshadmi said:
From this ontological point of view the full set is definable and proper subsets are avoided, because only the full set is an actual non-finite.
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Your “ontological point of view” only seems to avoid one thing, understanding.
 
The Man said:
No, I am not.



So you just do not understand the meaning of ‘class’ let alone ‘proper class’ (a class that is not a set) in set theory, how surprising.



Again cardinality does not ‘measure existence’ so you’ve got the wrong ‘tool’.



Your “ontological point of view” only seems to avoid one thing, understanding.
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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.

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.

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.

Please read http://www.internationalskeptics.com/forums/showpost.php?p=4661711&postcount=2613 in order to understan better the natue of my research.

EDIT:

I tell you the same things that I said to jsfisher.


If this is your entire abstract ability, then this is not of my concern, I do not care about your limitations.


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.

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.
 
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doronshadmi said:
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.
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Well, I've spent 5 minutes looking at this, and I can see you're just making things up. Too big to be a set? How big would that be, exactly?
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.
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Could I have ranch dressing with my word salad, please?
 
doronshadmi said:
Please read http://en.wikipedia.org/wiki/Cantor's_paradox .
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Curious Doron would cite that particular paradox. The so-called Cantor's Paradox is not a paradox at all in any formal set theory, but a simple theorem.

And you know what the theorem proves? It proves the non-existence of Doron's "full set".

I suppose that just means Doron will make it a member of the null set, since that's where other impossible things have ended up.
 
doronshadmi said:
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.

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.

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.

Please read http://www.internationalskeptics.com/forums/showpost.php?p=4661711&postcount=2613 in order to understan better the natue of my research.

EDIT:

I tell you the same things that I said to jsfisher.


If this is your entire abstract ability, then this is not of my concern, I do not care about your limitations.


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.

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.
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Doron this is just you usual contradiction again, first your say that…


doronshadmi said:
...proper classes are avoided…
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Which would mean that any class must be a set.

Then you immediately follow that with..

doronshadmi said:
...simply because only the full set is an actual non-finite.
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Meaning that other infinite classes are not sets (or as you put it are incomplete) in your consideration, thus making them proper classes in your consideration.

So which is it Doron ‘proper classes are avoided’ or just any proper classes other then yours are avoided? Certainly you seem to have managed to avoid some ‘proper classes’ on set theory.
 
Ok, let us close this dialog for now.

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.

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.

A set is a level 2 (Intermediate) thing.

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

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

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: {_}_ )

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".

Well this is ontologically wrong because "everything" is at the level of set (and this is the state that some paradoxes have to be avoided if we are using the standard notion of set, because by the standard notion the universal quantifier "for all" is used also on non-finite sets. These problems are avoided by using the ontological notion of the full set because only the full set is an actual non-finite thing and any set of many members cannot be but a potential non-finite).

Some claims "there is Fullness above collection".

This time he is ontologically right.

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As for potential non-finite sets:

a = Emptiness

b = Intermadiate

c = Fullness

We know that the Cantorean transfinite universe is considered as an actual infinity, where the limiting "process" is a is considered as potential infinity. However, by the new notion of the non-finite since card(b) < card(c), then only (c) is considered as an actual infinity (card(c) has no successor). Here any non-finite set is no more than a potential infinity because card(b) < card(c) is invariant. The model of Tachyon (a hypothetic particle that its minimal speed is not less than the speed of light) is used here in order to show that operations like Subtraction, Division, Square root etc. do not reduce a non-finite collection into a finite collection. On the contrary, arithmetical operations like Addition, Multiplication, Exponent etc. increase the magnitude of a non-finite collection, which is incomplete because no collection has the magnitude of (c). Let @ be a cardinal of a non-finite set such that:

Sqrt(@) = @
@ - x = @
@ / x = @
If |A|=@ and |B|=@ + or * or ^ x , then |B| > |A| by + or * or ^ x

Some comparisons:

By Cantor aleph0 = aleph 0+1 , by the new notion @+1 > @.
By Cantor aleph0 < 2^aleph0 , by the new notion @ < 2^@.
By Cantor aleph0-2^aleph0 is undefined, by the new notion @-2^@ = @.
By Cantor 3^aleph0 = 2^aleph0 > aleph0 and aleph0-1 is problematic.
By the new notion 3^@ > 2^@ > @ = @-1 etc.

This new approach to the non-finite is not counter intuitive as the Cantorean transfinite universe, because it clearly distinguishes between the continuum (which is not less than (c) ) and the discrete (which is no more than (b) ).

From a Cantorean point of view, cardinals are commutative (1+aleph0 = aleph0+1) and ordinals are not (1+ω ≠ ω +1). By using the new notion of the non-finite, both cardinals and ordinals are commutative because of the inherent incompleteness of any non-finite set. In other words, @ is used for both ordered and unordered non-finite sets; moreover, the equality x+@ = @+x holds in both cases.
 
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doronshadmi said:
The fact is that jsfisher and The Man try to get OM by using the standard notions of set.
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I guess it is so much easier to think deep thoughts when you are unconstrained by things like logic and consistency nor with the formality of deciding what anything means, isn't it Doron. It's just so convenient to make it up as you go along.

By the way, how are those definitions coming along? You remember, the meaning of "distinction is a first-order property". Can we expect anything of substance any time soon, or just more of your rambling contradictory gibberish?
 
jsfisher said:
I guess it is so much easier to think deep thoughts when you are unconstrained by things like logic and consistency nor with the formality of deciding what anything means, isn't it Doron. It's just so convenient to make it up as you go along.

By the way, how are those definitions coming along? You remember, the meaning of "distinction is a first-order property". Can we expect anything of substance any time soon, or just more of your rambling contradictory gibberish?
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It is not easy to be grasped at all, for example: you don't get it. ( http://www.internationalskeptics.com/forums/showpost.php?p=4663885&postcount=2636 ).

In other words: wrong guess.
 
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doronshadmi said:
It is not easy to be grasped at all, for example: you don't get it. ( http://www.internationalskeptics.com/forums/showpost.php?p=4663885&postcount=2636 ).
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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.
 
Doron's 'pre-axiomatic' axiom

doronshadmi said:
Please read http://www.internationalskeptics.com/forums/showpost.php?p=4661711&postcount=2613 in order to understan better the natue of my research.
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Well since you feel that post represents the ‘nature’ of your research I will comment.


doronshadmi said:
In order to do that a pre-axiomatic research has to be done, and this is exactly the nature of my work.
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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’.
 
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