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

[jsfisher]

Here we come again to ∀.

Seems like an inconvenient choice, but, hey, knock yourself out.

 

[doronshadmi]

Seems like an inconvenient choice, but, hey, knock yourself out.

Can you tell me please for what set the expression "x" stands for, by adding "∀" to "x"?

If you can't do it, it means that x is a placeholder for any given set (without any information about its members).

 

[jsfisher]

Can you tell me please for what set the expression "x" stands for, by adding "∀" to "x"?

You might have a better time of it if you focused on the second definition.

 

[Little 10 Toes]

My mistake of writing a misleading post , I mean that generally w meaning can also be "no members", where w in jsfishr's example and x in my post are used for the same purpose, to define a set according its structure and not according to its members.

And this is wrong.

Edit: and VERY Dishonest. You changed the post at 10:56am after 3 other posts.

 

[doronshadmi]

You might have a better time of it if you focused on the second definition.

{x : x∈A OR x∈B} is the general logical structure of A∪B, where x is a placeholder for any given member (including no members at all), and so is the case about A and B, they are placeholders for sets, without any information about their members, exactly because x is such placeholder.

Here it is:

A∪B = {x : x∈A OR x∈B}

Now let's go to the general logical structure of x∪{x}

x is a placeholder for any given set (without any information about its members).

x∪{x} = {x : x∈x AND {x∈x}}

Definition 1: A set K is a base set iff K ≠ {x : x∈x AND {x∈x}}.

Definition 2: A set K is a successor set iff K = {x : x∈x AND {x∈x}}.

 

[doronshadmi]

And this is wrong.

You are right, my use of x is different than jsfisher's use of w.

jsfisher uses w in order to define the members of a given set.

I use x as a part of the general logical structure of x∪{x}, which enables me to define N members in terms base sets or successor sets that are not largest successor sets, according to their structures, no matter what members N members have.

 

[Little 10 Toes]

I notice that once again, you do not mention your dishonesty.

 

[doronshadmi]

EDIT:


Let's put what we have until now, in one post.


{x : x∈A OR x∈B} is the general logical structure of A∪B, where x is a placeholder for any given member (including no members at all), and so is the case about A and B, they are placeholders for sets, without any information about their members, exactly because x is such placeholder.

Here it is:

A∪B = {x : x∈A OR x∈B}

Now let's go to the general logical structure of x∪{x}

x is a placeholder for any given set (without any information about its members).

x∪{x} = {x : x∈x AND {x∈x}}

Definition 1: A set K is a base set iff K ≠ {x : x∈x AND {x∈x}}.

Definition 2: A set K is a successor set iff K = {x : x∈x AND {x∈x}}.

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

Let M be the membership function of set N.

If y is largest successor set, then M : false.

If y is not largest successor set, then M : true.

Not being largest successor set in N is:

(successor set that has its successor set) OR (base set)

An example:

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

As can be seen, set N members are (base sets) OR (successor sets that have their successor sets) (M is true), but no largest successor set is a member of set N (M is false).

Moreover, if x=N, then N is not the same as (N∪{N}), and we can't define N as its own member, exactly as we can't define, for example, {} as its own member, since {}∪{{}}={{}}≠{}.

In other words, ∀ and N have nothing to do with each other, which enables to define N as a set of infinitely many members that its cardinality is not any particular size.



Without definition 2, the following is actually unknown:

Any set that is the member of itself is actually a successor set of that set, for example:

x={a,b,c,...}

If {a,b,c,...} is a member of itself, then we get the set {a,b,c,...{a,b,c,...}} which is actually ({a,b,c,...}∪{{a,b,c,...}}) = (x∪{x}) ≠ x = {a,b,c,...}

More general, no set is its successor set.


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

jsfisher, another unknown thing by mathematicians that do not define successor set, is ∀ as the cause of Russell's Paradox, and this time please do not skip on it.


∀ 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.

 

[jsfisher]

A∪B = {x : x∈A OR x∈B}

The left-hand side provides the meaning for A and B used on the right.

Definition 1: A set K is a base set iff K ≠ {x : x∈x AND {x∈x}}.

Definition 2: A set K is a successor set iff K = {x : x∈x AND {x∈x}}.

So, what's x? We are set on what K means; not at all set on x.

 

[doronshadmi]

The left-hand side provides the meaning for A and B used on the right.

Yet, x does not stand for any particular members, including no members at all.

So, what's x? We are set on what K means; not at all set on x.

K is defined by its structure, not by its members.

Please carefully read http://www.internationalskeptics.com/forums/showpost.php?p=13212306&postcount=345 from start to end.

 

[jsfisher]

Yet, x does not stand for any particular members.

What has that to do with anything? You clearly do not understand the notation, so please stop pretending you do.

The set builder notation gives fully quantified meaning to your bold/italic x.

K is defined by its structure, not by its members.

No, K is defined by neither its structure nor its members. All we know about K, all we care about K, is that K is a set.

Be that as it may, though, the question you are ducking is what is x. It is sitting as a free variable in your definitions; it needs to be bound or quantified in some way.

What is x?

 

[doronshadmi]

No, K is defined by neither its structure nor its members. All we know about K, all we care about K, is that K is a set.

K is indeed a set that is defined by its structure.

Be that as it may, though, the question you are ducking is what is x. It is sitting as a free variable in your definitions; it needs to be bound or quantified in some way.

x is a placeholder for any given set, all we care about is that x is a set.

What is x?

x is a set.

x is a placeholder for any given member, including no member at all.

x∪{x} = {x : x∈x AND {x∈x}}

The left-hand side provides the meaning for x used on the right.

 

[jsfisher]

K is indeed a set that is defined by its structure.

No. K is just any set. Your definitions impose no restrictions on K other than it being a set.

The left-hand side provides the meaning for x used on the right.

Not in any of your definitions.

Definition 1: A set K is a base set iff ...
Definition 2: A set K is a successor set iff ...​

There's K on the left in both cases. No x, though.

So what is x? You need to quantify it somehow.

 

[doronshadmi]

What has that to do with anything? You clearly do not understand the notation, so please stop pretending you do.

The set builder notation gives fully quantified meaning to your bold/italic x.

What do you mean by "fully quantified meaning"?

No, K is defined by neither its structure nor its members. All we know about K, all we care about K, is that K is a set.

Really? In that case what is w if not a condition that defines the members of a given set, as seen in your example in http://www.internationalskeptics.com/forums/showpost.php?p=13212145&postcount=334 .

 

[doronshadmi]

No. K is just any set. Your definitions impose no restrictions on K other than it being a set.

Yes, it imposes on set K a structure.

Not in any of your definitions.

Definition 1: A set K is a base set iff ...
Definition 2: A set K is a successor set iff ...​

There's K on the left in both cases. No x, though.

So what is x? You need to quantify it somehow.

x is a placeholder for any given set, all we care about is that x is a set.

 

[Little 10 Toes]

Yes, it imposes on set K a structure.



x is a placeholder for any given set, all we care about is that x is a set.

No! It does not matter if K has "a structure".

 

[jsfisher]

What do you mean by "fully quantified meaning"?

Everyone is better in Comic Sans:

S = { x : P(x) } ⇔ ∀x ( x∈ S ⇔ P(x) )
​

Really? In that case what is w if not a condition that defines the members of a given set

The set K was the subject. No w was involved. (And w, by the way, is a variable, not a condition.)

 

[doronshadmi]

jsfisher, as seen in http://www.internationalskeptics.com/forums/showpost.php?p=13212361&postcount=348 there can be more than one notion about strings of symbols, but in your case, you do not understand the potential damage of ∀ on sets, finite or not.

Also you do not understand the difference between N and N∪{N} that actually prevents N as a member of itself, without the need of any ad hoc axioms like the ZF axiom of specification https://en.wikipedia.org/wiki/Axiom_schema_of_specification .

 

[jsfisher]

Yes, it imposes on set K a structure.

Where, exactly do either of your first two definitions impose any structure on K?

x is a placeholder for any given set, all we care about is that x is a set.

So, I get free choice as to what set I want it to be? Excellent!!! I chose the empty set.

Definition 1: A set K is a base set iff K ≠ {x : x∈∅ AND x∈{∅}}.
Definition 2: A set K is a successor set iff K = {x : x∈∅ AND x∈(∅}}.​

By the way, I corrected, let's be generous and call it a typo, a typo in your definitions. Not quite up there with negated sets, but still....

 

[doronshadmi]

The set K was the subject. No w was involved. (And w, by the way, is a variable, not a condition.)

Call it a variable, yet it is used to define the members of a given set.