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

[doronshadmi]

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



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....

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

 

[jsfisher]

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.

Let's address one bit of totally-wrongness at a time, please. Right now you are still working on these definitions you feel are required to define your set N that you need for you definition of cardinality.

 

[jsfisher]

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

Is there a point you are trying to make about x? You introduced the set builder notation. I was perfectly happy with the straightforward union of two sets expression.

Do you accept my choice for non-bold, non-italic x?

 

[jsfisher]

Yes, it imposes on set K a structure.

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

Well?

 

[doronshadmi]

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



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....

Nothing has to be corrected in the following definitions:

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}}.

As long as you "correct" them you are simply missing http://www.internationalskeptics.com/forums/showpost.php?p=13212361&postcount=348 .

 

[jsfisher]

Nothing has to be corrected in the following definitions:

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}}.

As long as you "correct" them you are simply missing http://www.internationalskeptics.com/forums/showpost.php?p=13212361&postcount=348 .

This simple sequence of symbols, {x∈x}, is gibberish. It is on par with your ~(x∪{x}) nonsense.

 

[jsfisher]

Looking over the past many pages of this thread, it is clear, Doronshadmi, you are struggling. You are trying desperately to piece together snippets from a language you don't understand to express ideas you don't understand in opposition to concepts you don't understand.

At some point in the past a raised an analogy to Pakleds. I'll let others decide if the analogy has any present merit.

Still, in the hopes of some minimal progress past the barrier that now stymies you, Doronshadmi, I offer this: Is this what you are trying to say for Definition 2?

Definition 2: S is a successor set ⇔ ∃p∈S (S = p∪{p})​

(It has to be Comic Sans. It just has to be.)

Possible responses:

• "No. My definitions are correct in every way" followed by continued gibberish like what we saw for negated sets.
• "Yes, thank you; I understand" following by a sequence of new and different gibberish further showing no understanding of any of this.

Surprise me, Doronshadmi. Go for "none of the above" and prove me wrong.

Otherwise, please stop pretending.

 

[doronshadmi]

This simple sequence of symbols, {x∈x}, is gibberish. It is on par with your ~(x∪{x}) nonsense.

Not if x is a placeholder for members, or for not members at all (for example, in the case of {}).

Since the notion of emptiness or nothingness is not directly formalized by ZF (ZF deals only with, so called, 'existing' objects, where the 'minimal existing' object is {}) {x∈x} or x∈x are indeed taken as gibberish, by the standard formalization.

More details about this issue are already given in http://www.internationalskeptics.com/forums/showpost.php?p=13212414&postcount=352.

 

[doronshadmi]

Looking over the past many pages of this thread, it is clear, Doronshadmi, you are struggling. You are trying desperately to piece together snippets from a language you don't understand to express ideas you don't understand in opposition to concepts you don't understand.

At some point in the past a raised an analogy to Pakleds. I'll let others decide if the analogy has any present merit.

Still, in the hopes of some minimal progress past the barrier that now stymies you, Doronshadmi, I offer this: Is this what you are trying to say for Definition 2?

Definition 2: S is a successor set ⇔ ∃p∈S (S = p∪{p})​

(It has to be Comic Sans. It just has to be.)

Possible responses:

• "No. My definitions are correct in every way" followed by continued gibberish like what we saw for negated sets.
• "Yes, thank you; I understand" following by a sequence of new and different gibberish further showing no understanding of any of this.

Surprise me, Doronshadmi. Go for "none of the above" and prove me wrong.

Otherwise, please stop pretending.

jsfisher, first "Yes, thank you; I understand" you have formalized successor set by the standard way without the need for ∀.

Second, please hold your horses of criticism about me, maybe you forgot that couple of days ago, you said that there is no formal definition for successor set.

Moreover, now N members are formally defined by their structures as (base sets) OR (successor sets that have their successor sets) and we can move on in our discussion.

 

[jsfisher]

jsfisher, first "Yes, thank you; I understand" you have formalized successor set by the standard way without the need for ∀.

You're welcome. It was clear you were not going to ever get there. By the way, although I've expressed what I think you were trying to express, it still isn't what you meant. There are hidden assumptions you have yet to expose.

Second, please hold your horses of criticism about me, maybe you forgot that couple of days ago, you said that there is no formal definition for successor set.

Really? Where did I do that?

Moreover, now N members are formally defined by their structures as (base sets) OR (successor sets that have their successor sets) and we can move on in our discussion.

Slow down, Tex. You now need to express your three definitions in a relatively complete and cohesive way. Then you need to express your definition for N in a relatively complete and cohesive.

 

[jsfisher]

Not if x is a placeholder for members, or for not members at all (for example, in the case of {}).

Nope. Gibberish. Boolean values do not make for very good set elements.

 

[doronshadmi]

Boolean values do not make for very good set elements.

Please give some example.

 

[doronshadmi]

You're welcome. It was clear you were not going to ever get there. By the way, although I've expressed what I think you were trying to express, it still isn't what you meant. There are hidden assumptions you have yet to expose.



Really? Where did I do that?



Slow down, Tex. You now need to express your three definitions in a relatively complete and cohesive way. Then you need to express your definition for N in a relatively complete and cohesive.

Let's start from your first use of the terms base elements and their successors.

I apologize. It seems I have introduced some confusion I did not intend by using the term "von Neumann ordinal" in a non-standard way.

The Axiom of Infinity establishes the existence of a set with certain properties. The set contains the empty set and what I will call other base elements. The set contains the successors of all of its elements (which would be its base elements, and their successors, and their successors, and ...).

A set which can be considered the minimal set satisfying the axiom's conditions is the set I will call N. It includes only the empty set as a base element. The empty set is the only element of N that is not the successor another element.

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.

As can be seen you are using these terms in order to define set N, without first formally define a base element or its successors (which means that by this its there is an inseparable linkage between the two definitions, where X is a set by your own words (you are using it in your definition, after all)).

By reducing N base elements into a single element (the empty set) you established N as the set of von Neumann ordinals, and then you defined injection from N to X, in order to formally define X as a non-finite set (where N is a proper subset of X and the minimal non-finite set established by the axiom of infinity).

You did not formally define base element or its successors, and so is the case of set X, you did not define its membership function, yet you used it in your definition of X as a non-finite set.

When I struggled to define N as the minimal set, by trying to formally define base element (or base set), you "have changed the rules of the game" and imposed base set {{{}}} on it, as if N is actually X and not the minimal set (the set of von Neumann ordinals), as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=13191454&postcount=202.

By doing that you have created a confusion in our discussion, and I had to change N into X.

When it was done I became aware of the beautiful verity of base sets and their successors as bijective proper subsets of set X (was called N by you from now on, as seen in the link above, where V became what was N (V became the set of von Neumann ordinals)).

You ask me where you said that there is no formal definition for what I called successor set.

Well, it is my mistake, you did not directly said that but indirectly, because of the linkage between the definitions of base set and a successor set.

Here is the relevant quote, taken from http://www.internationalskeptics.com/forums/showpost.php?p=13192465&postcount=206 :

How do you propose to express "as its one and only base member" in the predicate calculus of set theory? (Keep in mind, too, 'base member' has no formal definition, either.)

Here is your definition of successor set:

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

B is a base set ⇔ not a successor set.

So as can be seen, now there is a formal definition of base set, which is something that you said that it has no formal definition, either .

In other words, without my struggle to define formal definitions to successor set (which stands at the basis of the formal definition of base set) and base set as members of set N (which was first named by you as set X), we can't be at the current point in our discussion, which enables to formally define largest successor, and use it in order to define the yes/no membership function for set N (which was actually used by you as set X, in order to define X as a non-finite set, without first define the membership function of X).

Moreover, my non-standard use of x as a placeholder for members, or no members at all (as seen in case of {}), may open a door for direct formalization of nothingness.

Furthermore, by formally define successor set we naturally avoid Russell's paradox, since no set is its successor set, for example: N ≠ N∪{N}.

No ad hoc axioms ( as done in ZF by establish the axiom of specification https://en.wikipedia.org/wiki/Axiom_schema_of_specification ) are needed.

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

jsfisher, without being open to each other, even if we are doing mistakes, there is less chance that new notions are emerged in order to be formalized later.

In my opinion, this is how the evolution of ideas works, by mutations.

During the past 12 years I gradually tried to learn how to take it as a journey and not as a war between egos.

I invite you to establish a paradise of base sets and their successor sets:

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

 

[jsfisher]

Second, please hold your horses of criticism about me, maybe you forgot that couple of days ago, you said that there is no formal definition for successor set.

Again, I ask, exactly where did I do that?

 

[jsfisher]

As can be seen you are using these terms in order to define set N, without first formally define a base element or its successors

Are you completely oblivious to the fact that was in informal description of a particular set I had in mind? Informal, key word there. I invented the term, base element, for the sole purpose of conveying a concept. I had no intention of going any deeper than creating a mental image.

When I struggled to define N as the minimal set, by trying to formally define base element (or base set), you "have changed the rules of the game" and imposed base set {{{}}} on it, as if N is actually X and not the minimal set (the set of von Neumann ordinals), as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=13191454&postcount=202.

I changed no rules. I simply pointed out that the "rules of the game" you were imposing allowed for things you did not intend. The fact you still have not figured that out is no surprise.

Here is your definition of successor set:

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

Nope, not my definition. I offered it as a possible version of what you meant for your definition.

B is a base set ⇔ not a successor set.

So as can be seen, now there is a formal definition of base set, which is something that you said that it has no formal definition, either .

"Base element," actually, and of course it had no formal definition when I stated that: It was a term I made up. (ETA: And my meaning for the term isn't quite captured by your base set definition, not that that really matters.)

Be that as it may, Now that you have acceptable-to-you definitions for base set and successor set, will you be proceeding to Definition #3, now?

Furthermore, by formally define successor set we naturally avoid Russell's paradox, since no set is its successor set, for example: N ≠ N∪{N}.

All you have done is given something a name. That does not in any way empower it with wondrous properties. Besides, Russell's Paradox has nothing to do with successor sets. It has everything to do with self-reference, as to all paradoxes of that ilk.

 

[jsfisher]

While waiting for Doronshadmi's new and improved Definition #3, it might be useful considering some implications of his first two definitions.

Definition: S is a successor set ⇔ ∃p∈S (S = p∪{p})
Definition: B is a base set ⇔ not a successor set.

The two definitions are complementary, so every set within the set theory is either a successor set or a base set. No set is both, nor is any set neither.

Here are some examples of base sets:
∅, and {{∅}}, and {∅, {{∅}}}, and {∅, {∅}, {∅, {{∅}}}}

Here are some examples of successor sets:
{∅}, and {{∅}, {{∅}}}, and {∅, {∅}}. and {∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}}

 

[doronshadmi]

I changed no rules. I simply pointed out that the "rules of the game" you were imposing allowed for things you did not intend. The fact you still have not figured that out is no surprise.

Here are your first terms about set N :

A set which can be considered the minimal set satisfying the axiom's conditions is the set I will call N. It includes only the empty set as a base element. The empty set is the only element of N that is not the successor another element.

In http://www.internationalskeptics.com/forums/showpost.php?p=13191454&postcount=202 you included in N another base set ( {{∅}} ), which does follow after your first terms about set N.

I call such move "changing the rules of the game in the middle of the game".

You actually meant to set X, but used set N instead, which caused a confusion in our discussion.

Moreover, where is exactly the membership function of X that is used in your definition, which defines X as a non-finite set?

As long as you do not do that, X is no more than a letter in your definition of non-finite set (and so is N, it is no more than a letter in your definition of non-finite set).

Here is your definition of non-finite set:

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.

Are you completely oblivious to the fact that was in informal description of a particular set I had in mind? Informal, key word there.

Shell we also take it as an informal definition, in spite of the iff there ?

 

[doronshadmi]

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

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={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} ?

 

[jsfisher]

In http://www.internationalskeptics.com/forums/showpost.php?p=13191454&postcount=202 you included in N another base set ( {{∅}} ), which does follow after your first terms about set N.

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.

 

[jsfisher]

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={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} ?

{∅, {∅}, {∅, {{∅}}}}


What does your Definition #3 look like these days?