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

[doronshadmi]

{∅, {∅}, {∅, {{∅}}}}

p={ ∅, {∅}, {∅, {{∅}}} }

In that case, since S=p∪{p} it is

{ ∅, {∅}, {∅, {{∅}}}, { ∅, {∅}, {∅, {{∅}}} } }

which is not the same as

{ ∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}} }

 

[jsfisher]

p={ ∅, {∅}, {∅, {{∅}}} }

In that case, since S=p∪{p} it is

{ ∅, {∅}, {∅, {{∅}}}, { ∅, {∅}, {∅, {{∅}}} } }

which is not the same as

{ ∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}} }

What difference do you see? They look like identical sets to me.

How is that latest Definition #3 coming along?

 

[doronshadmi]

What difference do you see? They look like identical sets to me.

They are identical.

What does your Definition #3 look like these days?

It looks like a strawberry pie, want some? because tomorrow it will look like a roller-blade.

Ok let's see:

Definition 3: A successor set K is a largest successor set in X iff K∈X AND K successor set ∉ X

 

[Little 10 Toes]

Lining up sets to show there is a difference

How exactly {∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} is a successor set by the following definition? :

Definition: S is a successor set ⇔ ∃p∈S (S = p∪{p})

In order to define S={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} you first have to define p.

So what is p if S={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} ?

{∅, {∅}, {∅, {{∅}}}}

p={ ∅, {∅}, {∅, {{∅}}} }

In that case, since S=p∪{p} it is

{ ∅, {∅}, {∅, {{∅}}}, { ∅,  {∅},      {∅, {{∅}}} }  }  which is not the same as 
a    b b  b   cd dcb   b     c  c      c   de edc b  a
{ ∅, {∅}, {∅   {∅},   { ∅, {{∅}}}},   {∅, {{∅}}} }
a    b b  b    c c     c    de edcb    b   cd dcb a

What difference do you see? They look like identical sets to me.

They are identical.

Edit reasons: Purposely edited messages to remove extra topics, added

 tag in a post to help line up the elements, and added letters below the brackets show the "level" of the brackets.

 

[jsfisher]

Definition 3: A successor set K is a largest successor set in X iff K∈X AND K successor set ∉ X

Ah! Back to progressing towards that dead-end that's just a few steps away.

You are using "successor set" in two different ways -- one as a set property; the other as a set function. The former usage you have defined; you'll need something else for the later.

Note for later, too, that to apply this definition you'll need to specify both K and X.

 

[doronshadmi]

Lining up sets to show there is a difference

...

Edit reasons: Purposely edited messages to remove extra topics, added

 tag in a post to help line up the elements, and added letters below the brackets show the "level" of the brackets.[/QUOTE]
Maybe a simpler way is to use, for example, colors.

S is a set which its members are (the members that are induced in p) AND (the members that are induced in {p} (which is actually p)).

p = { [COLOR="Red"]∅, {∅}, {∅, {{∅}}}[/COLOR] } 

{p} = { [COLOR="Blue"]{ ∅, {∅}, {∅, {{∅}}} }[/COLOR] }

In that case

S={ [COLOR="Red"]∅, {∅}, {∅, {{∅}}}[/COLOR] , [COLOR="Blue"]{ ∅, {∅}, {∅, {{∅}}} }[/COLOR] }

or

S={[COLOR="Red"]∅, {∅}[/COLOR], [COLOR="Blue"]{∅, {∅}, {∅, {{∅}}}}[/COLOR], [COLOR="red"]{∅, {{∅}}[/COLOR]}

since order is irrelevant.

 

[doronshadmi]

Let's do some diet:

Definition 3: A successor set K is a largest successor set iff ∃K ∈ X AND K∪{K} ∉ X

For example:

K={∅}

K successor = {∅,{∅}} = {∅}∪{{∅}} = K∪{K}

Now, ∃K ∈ X = {{∅}} = {K} AND {∅,{∅}} = {∅}∪{{∅}} = K∪{K} ∉ X

 

[jsfisher]

Let's do some diet to:

Definition 3: A successor set K is a largest successor set iff ∃K ∈ X AND {{K}}∪{{{K}}} ∉ X

Good! That didn't take very long at all. However, the stuff after the AND is wrong, but easily fixed. The existential quantifier is completely wrong, too, and must be removed. K, after all, is a given. You also need to indicate that being a largest successor set is with respect to some set. Perhaps you meant:

Definition 3: A successor set K is a largest successor set in X iff K ∈ X AND K∪{K} ∉ X​

You might consider, too, this simplification. It is only so slightly different in what can be a largest successor set (i.e., base sets), but I think it is closer to what you really mean:

Definition 3: Given X and K, K is a largest successor set in X iff K ∈ X AND K ∪ {K} ∉ X​

Now, you just need a membership function, M(x), so you can declare N = { x : M(x) }. Remember, too, any reliance on Definition #3 needs to identify two sets.

ETA: Actually, it is probably better to leave the set membership condition on the right. I've edited my second definition offering accordingly.

 

[doronshadmi]

Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})

Definition 2: B is a base set ⇔ ~(∃p∈B (B = p∪{p}))

Definition 3: A successor set L is a largest successor set iff ∃L ∈ X AND L∪{L} ∉ X

N = {x : (x=base set) ∨ (x=successor set ∧ ~largest successor set)}

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

No set is its successor set since for any given set S, S ≠ S∪{S}, or in other words, we avoid Russell's Paradox without any need for special axioms.

 

[jsfisher]

Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})

Definition 2: B is a base set ⇔ ~(∃p∈S (S = p∪{p}))

Definition 3: A successor set K is a largest successor set iff ∃K ∈ X AND K∪{K} ∉ X

Stuff needed to be added in bold; stuff to remove struck through:

Definition 3: A successor set K is a largest successor set in X iff ∃ K ∈ X AND K∪{K} ∉ X​

The "successor set" requirement for K is unnecessary.

 

[doronshadmi]

Stuff needed to be added in bold; stuff to remove struck through:

Definition 3: A successor set K is a largest successor set in X iff ∃ K ∈ X AND K∪{K} ∉ X​

The "successor set" requirement for K is unnecessary.

Thank you a lot jsfisher.

Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})

Definition 2: B is a base set ⇔ ~(∃p∈B (B = p∪{p}))

Definition 3: A successor set L is a largest successor set iff ∃L ∈ X AND L∪{L} ∉ X

N = {x : (x=base set) ∨ (x=successor set ∧ ~largest successor set)}

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

No set is its successor set since for any given set S, S ≠ S∪{S}, or in other words, we avoid Russell's Paradox without any need for special axioms.

 

[jsfisher]

Thank you a lot jsfisher.

And yet you didn't correct any of the problems in Definition #3.

N = {x : (x=base set) ∨ (x=successor set ∧ ~largest successor set)}

x is a set; base set is a property. "x = base set" is gibberish. Did you perhaps means "x is a base set"?

No set is its successor set since for any given set S, S ≠ S∪{S}, or in other words, we avoid Russell's Paradox without any need for special axioms.

This might be more convincing if Russell's Paradox actually involved successor sets. It doesn't.

 

[doronshadmi]

The "successor set" requirement for K is unnecessary.

It is necessary, otherwise K (I have changed it to L) can be a base set that does not have its successor set in X, and my argument against ∀ in case of infinite sets, is not satisfied.

 

[doronshadmi]

And yet you didn't correct any of the problems in Definition #3.



x is a set; base set is a property. "x = base set" is gibberish. Did you perhaps means "x is a base set"?



This might be more convincing if Russell's Paradox actually involved successor sets. It doesn't.

Well, Ok let's correct it.

Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})

Definition 2: B is a base set ⇔ ~(∃p∈B (B = p∪{p}))

Definition 3: A successor set L is a largest successor set in X iff L ∈ X AND L∪{L} ∉ X

N = {x : (x is a base set) ∨ (x is successor set ∧ ~largest successor set)}

N can't be its own member by definition 1, so N is not the set of all sets.

Here is some example of set N:

N = {
∅,                    ---> {{∅}}, ---> ...   <-- base sets
{∅},                  ---> {{∅}, {{∅}} }, ---> ...  <-- successor sets
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...  <-- successor sets
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...  <-- successor sets
...                         ...
}

The cardinality of set N is not any particular value exactly because for any given successor set in N, it is successor set ∧ ~largest successor set

 

[doronshadmi]

Now about Russell's Paradox.

∀ is the cause of Russell's Paradox, whether a given collection of distinct objects is finite, or not.

For example:

U is a set of two distinct members, such that one of the members, called u, shaves ∀ the members of set U that do not shave themselves and only these members of set U (this is supposed to be his property in order to be a member of set U).

Who shaves u?

If u shaves himself, then he must not shave himself (shaves AND ~shaves himself, which is a contradiction) exactly because of the term ∀.

If u does not shave himself, then he must shave himself (~shaves AND shaves himself, which is a contradiction) exactly because of the term ∀.

So, because the term ∀ is used as a part of the terms that define u as a member U, u must be referred to himself, and we get the contradictions that actually prevents to well-define ∀ the members of set U (the term ∀ itself is actually not well-defined in case of U).

The same problem holds also among infinite sets that the term ∀ is one of their properties, therefore the Axiom of Restricted Comprehension was add to ZF in order to avoid Russell's Paradox, but it is done without being aware of the fact that the term ∀ is the cause of any given contradictory self reference, whether it is used among finite or infinite sets.

 

[doronshadmi]

The set N in question there was your set N, a set you were claiming was defined directly by the Axiom of Infinity. Your rules, notice.

Please show where exactly I used N before you used it in http://www.internationalskeptics.com/forums/showpost.php?p=13163695&postcount=83 ?

 

[jsfisher]

It is necessary, otherwise K (I have changed it to L) can be a base set that does not have its successor set in X, and my argument against ∀ in case of infinite sets, is not satisfied.

If ∅ is in the set, don't you want {∅}, too? Seems like an important thing not to omit.

 

[jsfisher]

Well, Ok let's correct it.

Yes, please.

Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})

Definition 2: B is a base set ⇔ ~(∃p∈B (B = p∪{p}))

Definition 3: A successor set L is a largest successor set in X iff L ∈ X AND L∪{L} ∉ X

N = {x : (x is a base set) ∨ (x is successor set ∧ ~largest successor set)}

Better, but you need to fix that last part. Did you mean "~ (x is a largest successor set in N)"?

N can't be its own member by definition 1, so N is not the set of all sets.

Definition 1 does no such thing. It simply distinguishes sets as "successor sets" or "not successor sets".

 

[doronshadmi]

If ∅ is in the set, don't you want {∅}, too? Seems like an important thing not to omit.

{∅} is a successor set in some set by definition 1.

Definition 3 is only about successor sets.

 

[doronshadmi]

Better, but you need to fix that last part. Did you mean "~ (x is a largest successor set in N)"?

Yes, thank you

N = {x : (x is a base set) ∨ ~(x is a largest successor set)}

Definition 1 does no such thing. It simply distinguishes sets as "successor sets" or "not successor sets".

Let's check it.

p=N

By definition 1 S=N∪{N}≠N

N can't be its own member by definition 1, so N is not the set of all sets.