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

[jsfisher]

So N is a general set of infinitely many finite sets...

Is it now? If only you would tell us what you mean by finite set (or its complement) they maybe we'd know what you mean.

You will also need to define "general set" since you've added that to the lexicon.

 

[doronshadmi]

Is it now? If only you would tell us what you mean by finite set (or its complement) they maybe we'd know what you mean.

You will also need to define "general set" since you've added that to the lexicon.

If only you would tell us what you mean by finite set

I follow your definition of set of finite sets (as the complementary set of set X) notated by you as N.

N is a general set of finite sets

N = {
 ∅,                    ---> {{∅}}, ---> ...    
 ↓  [B]No-bijection[/B]
{∅},                  ---> {{∅}, {{∅}} }, ---> ...   
 ↓  [B]No-bijection[/B]
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...   
 ↓  [B]No-bijection[/B]
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...   
 ↓  [B]No-bijection[/B]
...                         ...
}

V is the requested set

V = {
       	    ∅, 
            ↓  [B]No-bijection[/B]
     	   {∅},
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} },
            ↓  [B]No-bijection[/B] 
...  

}

Please define the membership function of V.

V is the set of all the finite sets of von Neumann set of ordinals iff

 

[doronshadmi]

B.T.W I know that N members can also be infinite sets, but since we are talking about the minimal set satisfying the requirements of the Axiom of Infinity, only finite sets (which are the complements of set X) are the members of N.

N as a set of finite sets:

N = {
 ∅,                    ---> {{∅}}, ---> ...    
 ↓  [B]No-bijection[/B]
{∅},                  ---> {{∅}, {{∅}} }, ---> ...   
 ↓  [B]No-bijection[/B]
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...   
 ↓  [B]No-bijection[/B]
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...   
 ↓  [B]No-bijection[/B]
...                         ...
}

V is the requested set:

V = {
       	    ∅, 
            ↓  [B]No-bijection[/B]
     	   {∅},
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} },
            ↓  [B]No-bijection[/B] 
...  

}

Please define the membership function of V.

V is the set of all the finite sets of von Neumann set of ordinals iff

 

[jsfisher]

I follow your definition of set of finite sets (as the complementary set of set X) notated by you as N.

And just what would your definition be, then?

By the way, my definition for non-finite sets was based on a definition for relative cardinality. Are you now adopting my definition of cardinality, too?

And just what did you mean by "(as the complementary set of set X) notated by you as N"? In particular, what is a "complementary set"?

We are still all wondering, too, what you mean by "general set".

 

[doronshadmi]

V is the set of all the finite sets of von Neumann set of ordinals iff (∅ ∈ V) AND (∀v ∈ V ( ( v ∪ { v } ) ∈ V )) AND (∀v ∈ V (|v| < |V|))

 

[doronshadmi]

And just what would your definition be, then?

By the way, my definition for non-finite sets was based on a definition for relative cardinality. Are you now adopting my definition of cardinality, too?

And just what did you mean by "(as the complementary set of set X) notated by you as N"? In particular, what is a "complementary set"?

We are still all wondering, too, what you mean by "general set".

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=13224656&postcount=483 and http://www.internationalskeptics.com/forums/showpost.php?p=13224813&postcount=485

http://www.internationalskeptics.com/forums/showpost.php?p=13224613&postcount=482 is irrelevant.

 

[Little 10 Toes]

And just what would your definition be, then?

By the way, my definition for non-finite sets was based on a definition for relative cardinality. Are you now adopting my definition of cardinality, too?

And just what did you mean by "(as the complementary set of set X) notated by you as N"? In particular, what is a "complementary set"?

We are still all wondering, too, what you mean by "general set".

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=13224656&postcount=483 and http://www.internationalskeptics.com/forums/showpost.php?p=13224813&postcount=485

http://www.internationalskeptics.com/forums/showpost.php?p=13224613&postcount=482 is irrelevant.

I notice that you failed to answer jsfisher's questions. Is there a reason why?

 

[doronshadmi]

By the way, my definition for non-finite sets was based on a definition for relative cardinality. Are you now adopting my definition of cardinality, too?

No, I take a ride on it with you in order to find if it holds water.

And just what did you mean by "(as the complementary set of set X) notated by you as N"? In particular, what is a "complementary set"?

Since finite and non-finite are complementary terms, defining one automatically defines the other.

In that case, finite sets' cardinality and X cardinality are complements, isn't it?

For example, |v| < |V| are complementary terms, since both are needed in order to define that one is smaller than the other or that one is bigger than the other.

We are still all wondering, too, what you mean by "general set".

Who is "We"? Do you need some group in order to discus with me?

As for "general set" it is not relevant anymore since I wrote a new post, but you replied to the old one.

Please look at the relevant one http://www.internationalskeptics.com/forums/showpost.php?p=13224656&postcount=483 and also at http://www.internationalskeptics.com/forums/showpost.php?p=13224813&postcount=485 .

Thank you.

 

[jsfisher]

V is the set of all the finite sets of von Neumann set of ordinals iff (∅ ∈ V) AND (∀v ∈ V ( ( v ∪ { v } ) ∈ V )) AND (∀v ∈ V (|v| < |V|))

If only that were a way to define a set. Sadly, it is not.

 

[jsfisher]

By the way, my definition for non-finite sets was based on a definition for relative cardinality. Are you now adopting my definition of cardinality, too?

No, I take a ride on it with you in order to find if it holds water.

So, then, what are the definitions you a taking a ride upon? In your own words, please. Well, symbols, actually, are preferable.

And just what did you mean by "(as the complementary set of set X) notated by you as N"? In particular, what is a "complementary set"?

Since finite and non-finite are complementary terms, defining one automatically defines the other.

In that case, finite sets' cardinality and X cardinality are complements, isn't it?

No, "finite set" and "non-finite set" are complementary terms. A set that is not finite is non-finite, and a set that is not non-finite is finite. Complements.

I have no idea what you meant by "complementary set", and I suspect you don't either.

We are still all wondering, too, what you mean by "general set".

Who is "We"? Do you need some group in order to discus with me?

As for "general set" it is not relevant anymore since I wrote a new post, but you replied to the old one.

This, then, is just another term you inserted in to a sentence because it sounded important, but in reality it has no meaning whatsoever. You used an invented term; you got caught; so now you run away from it.

How about not making up terms in the first place?

Please look at the relevant one http://www.internationalskeptics.com/forums/showpost.php?p=13224656&postcount=483 and also at http://www.internationalskeptics.com/forums/showpost.php?p=13224813&postcount=485 .

Thank you.

As has been pointed out to you before, your posts may seem really good when you compose them, but they are in fact a confused collection of misinformation, made-up and meaningless terms, and non sequiturs.

Nevertheless, I do read your posts in their entirety. If I choose to respond, I usually will limit myself to up to the first significant error, which is my right and habit. If the beginning of a post is garbage, there is seldom a reason to dignifying the trash after that point with any sort of response.

You are welcome.

 

[jsfisher]

Who knows? This may become relevant at some point, too:

∃I (Ø∈I ∧ ∀x (x∈I ⇒ x ⋃ {x} ∈ I))
N = {x∈I : ∀y ((Ø∈y ∧ ∀z ((z∈y ⇒ z ⋃ {z} ∈ y))) ⇒ x∈y)}​

 

[doronshadmi]

No, "finite set" and "non-finite set" are complementary terms. A set that is not finite is non-finite, and a set that is not non-finite is finite. Complements.

In other words, "finite set" is a complement of "non-finite set", and vice versa.

I have no idea what you meant by "complementary set", and I suspect you don't either.

Non-finite set is a complementary set of finite set, and vice versa.

 

[doronshadmi]

If only that were a way to define a set. Sadly, it is not.

Ok, lets replace iff by if ... then.

p="(∅ ∈ V) AND (∀v ∈ V ( ( v ∪ { v } ) ∈ V )) AND (∀v ∈ V (|v| < |V|))"

q="V is the set of all the finite sets of von Neumann set of ordinals"

[B][I]p[/I][/B] → [B][I]q[/I][/B] 	 
T   T	  T
T   F	  F
F   T	  T
F   F	  T

If (∅ ∈ V) AND (∀v ∈ V ( ( v ∪ { v } ) ∈ V )) AND (∀v ∈ V (|v| < |V|)), then V is the set of all the finite sets of von Neumann set of ordinals.

V is the requested set:

V = {
       	    ∅, 
            ↓  [B]No-bijection[/B]
     	   {∅},
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} },
            ↓  [B]No-bijection[/B] 
...  

}

Please define the (yes/no) iff membership function of V.

V is the set of all the finite sets of von Neumann set of ordinals iff ...

 

[jsfisher]

In other words, "finite set" is a complement of "non-finite set", and vice versa.

No. The term "finite set" is the complement of the term "non-fnite set".

 

[doronshadmi]

V = {v : (∅ ∈ V) ∧ (∀v ∈ V ( ( v ∪ { v } ) ∈ V )) ∧ (∀v ∈ V (|v| < |V|))}

 

[doronshadmi]

No. The term "finite set" is the complement of the term "non-fnite set".

A = "infinite set"

~A is (the complement of A) = "finite set"


A = "finite set"

~A is (the complement of A) = "infinite set"

The above holds only in case of excluded middle framework.

 

[jsfisher]

Ok, lets replace iff by if ... then.

Still not a set definition. You could not do this before; why to you suddenly believe you can do it now?

 

[jsfisher]

V = {v : (∅ ∈ V) ∧ (∀v ∈ V ( ( v ∪ { v } ) ∈ V )) ∧ (∀v ∈ V (|v| < |V|))}

Did you not notice you are defining the set V in terms of itself?

ETA: ...not to mention the symbolic gibberish you've included nor the reference to cardinality.

 

[jsfisher]

A = "infinite set"

~A is (the complement of A) = "finite set"

No, being a finite set is a property.

A is a finite set​

The complement of that property would then be:

A is not a finite set​

and that is equivalent to:

A is a non-finite set​

 

[doronshadmi]

That would be back when I defined a set X to be non-finite when |N| <= |X| (where N is the minimal set satisfying the requirements of the Axiom of Infinity). Since finite and non-finite are complementary terms, defining one automatically defines the other.

Since N is the minimal set satisfying the requirements of the Axiom of Infinity, does it mean that it is at least the set of all sets that are not non-finite sets, which are the members of the von Neumann set of ordinals?