Cont: Deeper than primes - Continuation 1/3*9

doronshadmi said:
Ok here is my corrected version.

D is a placeholder for any given set.

Definition 1: Cardinality is the 'size' of D iff |D| is at von Neumann domain.
Click to expand...


You continue to abuse that poor if-and-only-if relation.

What is "von Neumann domain"? What does it mean to be "at" it?

(No forward references, remember.)
 
Little 10 Toes said:
There are three other definitions. Please arrange them in the order that you wish. No forward references please.
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They are ordered, such that definition 2 is based on definition 1, definition 3 is based on definition 2 and definition 4 is based on definitions 1 and 2.
 
jsfisher said:
(1) That would remedy none of the three defects I cited.
Click to expand...

"von Neumann domain" is the 'horizontal' and 'vertical' directions of the following inductive set:

Code: 
{
   -------------->
    | 	   ∅, 
    | 	   {∅}, 
    | 	   { ∅, {∅} }, 
    | 	   { ∅, {∅} , {∅, {∅}} }, 
    | 	   { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, 
... V

}

In both cases one is at von Neumann domain.

D is a placeholder for any given set.

Definition 1: Cardinality is the 'size' of D iff |D| is at von Neumann domain.

Code: 
p="Cardinality is the 'size' of D" 

q="|D| is at von Neumann domain"
  
p iff q
-------
F     F  T
F     T  F
T     F  F
T     T  T

Definition 2: N is the set of finite cardinalities iff each given n ∈ N is determined by its corresponding von Neumann member ( as follows: )

Code: 
{
   -------------->
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
... 

}

Definition 3: D is called finite iff it is bijective with some particular von Neumann member, where |D| is the corresponding N member.

Definition 4: D is called non-finite iff |D| is not any particular N member (the fact that von Neumann set does not have the greatest successor, is taken, yet one is at N domain, which is equivalent to von Neumann domain).
 
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doronshadmi said:
"von Neumann domain" is the 'horizontal' and 'vertical' directions of the following inductive set...
Click to expand...

Sets don't have directions. What do you mean by "von Neumann domain"? Vague pictures won't help you define it.

In both cases one is at von Neumann domain.
Click to expand...

What does "at" mean in "one is at von Neumann domain"?

D is a placeholder for any given set.

Definition 1: Cardinality is the 'size' of D iff |D| is at von Neumann domain.
Click to expand...

What does "'size' of D" mean? How does one determine whether |D| is "at von Neumann domain" without knowing what the cardinality of D is?
 
jsfisher said:
Sets don't have directions.
Click to expand...

I do not mean direction in terms of space, but in terms of guidance of notions.

von Neumann inductive set is not just a collection of finite sets (what I call "the 'horizontal' direction" in my diagram) but also the fact that the existence of the biggest set of this collection is not satisfied (what I call "the 'vertical downward' direction" in my diagram).

jsfisher said:
What do you mean by "von Neumann domain"? Vague pictures won't help you define it.

What does "at" mean in "one is at von Neumann domain"?
Click to expand...

Both directions are called "von Neumann domain", where being "at" means that one takes both directions of von Neumann set and not only the fact that each member is a finite set.

jsfisher said:
What does "'size' of D" mean? How does one determine whether |D| is "at von Neumann domain" without knowing what the cardinality of D is?
Click to expand...
Please look at the rest of my definitions in http://www.internationalskeptics.com/forums/showpost.php?p=13180373&postcount=166 .
 
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doronshadmi said:
I do not mean direction in terms of space, but in terms of guidance of notions.
Click to expand...

Doesn't matter what type of direction you meant, sets do not have direction.

von Neumann inductive set
Click to expand...

What set is that?

...is not just a collection of finite sets (what I call "the 'horizontal' direction" in my diagram)
Click to expand...

If you mean "collection of finite sets", then just say that. This continual interjection of meaningless terms as substitutes for less meaningless terms doesn't advance you point at all.

...but also the fact that the existence of the biggest set of this collection is not satisfied (what I call "the 'vertical downward' direction" in my diagram).
Click to expand...

What does "biggest set" mean? In what what does this "is not satisfied" characteristic have any bearing on anything?

Both directions
Click to expand...

Sets still don't have directions.

...
Click to expand...

And so on.
 
jsfisher said:
If you mean "collection of finite sets", then just say that.
Click to expand...
Ok.

D is a placeholder for any given set.

V is the von Neumann set of ordinals.

Definition 1: Cardinality is the 'size' of D iff |D| is determined by V.

Definition 2: N is the set of finite cardinalities iff each given n ∈ N is determined by its corresponding V member.

For example:
Code: 
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
...

Definition 3: D is called finite iff it is bijective with some particular V member, where |D| is the corresponding N member.

Definition 4: D is called non-finite iff |D| is not any particular N member since V does not have the greatest successor.
 
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doronshadmi said:
Ok.

D is a placeholder for any given set.

V is the von Neumann set of ordinals.
Click to expand...

Which set would that be? Seems unlikely you would be referring to the ordinals recognized by von Neumann since there are considerably more of those than you'd like to admit.

Definition 1: Cardinality is the 'size' of D iff |D| is determined by V.
Click to expand...

You still have that bogus "iff" in there. What does "is determined by" mean?
 
jsfisher said:
... since there are considerably more of those than you'd like to admit.
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Well, you can't admit (as seen in http://www.internationalskeptics.com/forums/showpost.php?p=13172730&postcount=139) that ZF did not establish ω or |N| exactly because of the bogus "all" that was arbitrarily added to it.

"all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied.

jsfisher said:
You still have that bogus "iff" in there. What does "is determined by" mean?
Click to expand...

Moreover, |N| does not exist exactly because each member of N is determined by (easily understood term) its corresponding V member (the existence of the greatest successor is not satisfied).

The highlighted part is bogus since ZF did not define N as a non-finite set (in the actual sense).
jsfisher said:
So, a set X is a non-finite set if and only if |N| <= |X|, where N is the minimal set established by the Axiom of Infinity.
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The minimal set established by the Axiom of Infinity is V, such that any given N member is the cardinality of its corresponding V member.

This is exactly the reason why you can't complete the following definition:

|N| > any given n iff ...
 
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doronshadmi said:
Well, you can't admit (as seen in http://www.internationalskeptics.com/forums/showpost.php?p=13172730&postcount=139) that ZF did not establish ω or |N| exactly because of the bogus "all" that was arbitrarily added to it.
Click to expand...

Your continual attempts to disprove definitions are pointless. Axiomatic set theories allow for infinite sets, and those sets contain "all" of there members.

"all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied.
Click to expand...

You still haven't defined this set, V. You haven't yet defined what "greatest successor" means. You haven't shown any relevance to set theory.
 
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jsfisher said:
Your continual attempts to disprove definitions are pointless.
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Look jsfisher, by this categorical determination, you put the notion of definition as some kind of religious dogma.

It is not hard to understand that it is pointless to discuss with a given person about his\her dogmas.

jsfisher said:
Axiomatic set theories allow for infinite sets, and those sets contain "all" of there members.
Click to expand...
You did not show all along our discussion that, for example, ZF consistently and rigorously establishes an infinite set that contains all of its members.

In order to be more concrete, please show exactly how ZF establishes ω as a limit ordinal of all smaller ordinals (https://en.wikipedia.org/wiki/Ordinal_number#Successor_and_limit_ordinals):
Successor and limit ordinals

Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is α ∪ { α } since its elements are those of α and α itself.[3]

A nonzero ordinal that is not a successor is called a limit ordinal. One justification for this term is that a limit ordinal is the limit in a topological sense of all smaller ordinals (under the order topology).
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Time after time the term "all" is used "out of the blue" without any reasoning behind it.

So, please demonstrate ZF reasoning, which actually consistently and rigorously establishes an infinite set that contains all of its members (for example: by ZF reasoning {0,1,2,3,4,5,...} is an infinite set that contains all of its members in spite of the fact that there is no largest natural number.

Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:

There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.

So in the following sequence:

0, 1, 2, …, ω, ω+1

ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.
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No matter what symbolic maneuvers are used, the fact that the, so called, infinite {0,1,2,3,4,5,...} does not have the largest member (the terms "all" does not hold) actually prevents the existence of ω as a limit ordinal after 0, 1, 2, …

Since ω is not established, the very notion of the difference between cardinal numbers and ordinal numbers, in case of infinite sets, has no basis whatsoever.
 
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doronshadmi said:
Look jsfisher, by this categorical determination, you put the notion of definition as some kind of religious dogma.

It is not hard to understand that it is pointless to discuss with a given person about his\her dogmas.
Click to expand...

Interesting bit of projection, there.

You did not show all along our discussion that, for example, ZF consistently and rigorously establishes an infinite set that contains all of its members.
Click to expand...

Well, that would get back to defining what is meant by "an infinite set", which you have failed to express. My only claim related to this is that the Axiom of Infinity establishes the existence of a certain non-empty set. Of necessity, set's existence guarantees it has all of its members. That goes along with the definition of "all", but since you feel free to disregard meaning....

In order to be more concrete, please show exactly how ZF establishes ω as a limit ordinal of all smaller ordinals (https://en.wikipedia.org/wiki/Ordinal_number#Successor_and_limit_ordinals):
Click to expand...

If there is a point you would like to make, you make it. Again, your water, you carry it.

Time after time the term "all" is used "out of the blue" without any reasoning behind it.
Click to expand...

Out of the dictionary would be a more accurate statement.

So, please demonstrate ZF reasoning, which actually consistently and rigorously establishes an infinite set that contains all of its members (for example: by ZF reasoning {0,1,2,3,4,5,...} is an infinite set that contains all of its members in spite of the fact that there is no largest natural number.
Click to expand...

You keep bringing up that last bit as if it were somehow important. Mathematics is not bound to your lack of imagination about the characteristics of infinite sets.


By the way, all of this has been a nice dodge by you. You still have a task of defining your version of cardinality, a task that has so far proven impossible for you.
 
jsfisher said:
Mathematics is not bound to your lack of imagination about the characteristics of infinite sets.
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Then please demonstrate exactly how ZF consistently and rigorously establishes an infinite set that includes all of its members, in spite of my lack of imagination. (b.t.w since when imagination is a factor in your arguments about actually doing mathematics?)

jsfisher said:
Interesting bit of projection, there.
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Well, unlike you, I do not claim that definitions can't be disproved in case that new notions about X are fundamentally changed.

Actually Cantor's approach about infinite sets was an attempt to fundamentally change our understanding about them.

ZF is an example of how to do math according to this attempt.

But when asked to consistently and rigorously use ZF in order to establish an infinite set that includes all of its members, you simply can't do that.

Instead, my lack of imagination is involved etc. ("all" is impossible in case of V, exactly because the existence of the greatest successor is not satisfied, exactly as the largest natural number does not exit, and no special kind of imagination is needed in order to understand it, just simple common sense (where being simple is far from being trivial)).

You claim that ZF consistently and rigorously establishes an infinite set that includes all of its members.

Your claim, your water; you carry it.
 
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D is a placeholder for any given set.

∃ V ( ∅ ∈ V ∧ for any given x ∈ V ( ( x ∪ { x } ) ∈ V ) )

Code: 
V={
    
       ∅, 
       {∅}, 
       { ∅, {∅} }, 
       { ∅, {∅} , {∅, {∅}} }, 
       { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }, 
...  

}

and it does not have the largest member.

Definition 1: Cardinality is the 'size' of D iff |D| is defined by V.

Definition 2: N is the set of finite cardinalities iff any given n ∈ N is defined by its corresponding V member.

For example:
Code: 
0    	= |∅|
1    	= |{∅}|
2    	= |{ ∅, {∅} }|
3    	= |{ ∅, {∅} , {∅, {∅}} }|
4    	= |{ ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }|
...

Definition 3: D is called finite iff it is bijective with some particular V member, where |D| is the corresponding N member.

Definition 4: D is called non-finite iff |D| is not any particular N member since V does not have the largest member.


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


jsfisher said:
Which set would that be? Seems unlikely you would be referring to the ordinals recognized by von Neumann since there are considerably more of those than you'd like to admit.
Click to expand...
Well, please show consistently and rigorously, how ω is established as a limit ordinal in spite of the equivalent fact that V and N do not have their largest members (it means that the term "all" in both sets is not satisfied, where this simple fact actually prevents the existence of ω as a size (called limit ordinal) that claimed to exist by traditional mathematics as follows:
ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.
Click to expand...

Such claim artificially establishes ω as a size larger than the sizes of "all" natural numbers, in spite of the fact that the term "all" is not satisfied, exactly because N does not have its largest member (ω as a size > "all" natural numbers, is not satisfied).

By using only imagination one can establish whatever he/she likes, by ignoring fundamental facts (as shown above) of the considered subject.

So the problem here, jsfisher, is not my "lack of imagination about the characteristics of infinite sets", but it is your inability to consistently and rigorously establish ω or |N|, exactly because your axiomatic framework (ZF in this case) artificially eliminates fundamental characteristics of infinite sets, where the lack of the term "all" is one of them.

"ω is called limit ordinal iff ω > all n ∈ N" is no more than a string of symbols based only on imagination, without any understanding of fundamental characteristics of N as a non-finite collection.
 
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