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Old 4th September 2020, 02:09 PM   #361
doronshadmi
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Originally Posted by jsfisher View Post
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.
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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 4th September 2020, 02:10 PM   #362
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Originally Posted by doronshadmi View Post
jsfisher, as seen in http://www.internationalskeptics.com...&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.
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Old 4th September 2020, 02:13 PM   #363
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Originally Posted by doronshadmi View Post
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?
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Old 4th September 2020, 02:17 PM   #364
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Originally Posted by jsfisher View Post
Originally Posted by doronshadmi View Post
Yes, it imposes on set K a structure.
Where, exactly do either of your first two definitions impose any structure on K?
Well?
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Old 4th September 2020, 02:19 PM   #365
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Originally Posted by jsfisher View Post
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...&postcount=348 .
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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 4th September 2020 at 02:21 PM.
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Old 4th September 2020, 02:25 PM   #366
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Originally Posted by doronshadmi View Post
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...&postcount=348 .

This simple sequence of symbols, {x∈x}, is gibberish. It is on par with your ~(x∪{x}) nonsense.
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Old 4th September 2020, 07:56 PM   #367
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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.
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Old 5th September 2020, 12:36 PM   #368
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Originally Posted by jsfisher View Post
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...&postcount=352.
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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 5th September 2020 at 12:50 PM.
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Old 5th September 2020, 01:12 PM   #369
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Originally Posted by jsfisher View Post
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.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 5th September 2020 at 01:16 PM.
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Old 5th September 2020, 01:44 PM   #370
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Originally Posted by doronshadmi View Post
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.

Quote:
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?

Quote:
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.
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Old 5th September 2020, 01:45 PM   #371
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Originally Posted by doronshadmi View Post
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.
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Old 5th September 2020, 11:36 PM   #372
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Originally Posted by jsfisher View Post
Boolean values do not make for very good set elements.
Please give some example.
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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 6th September 2020, 06:08 AM   #373
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Originally Posted by jsfisher View Post
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.

Originally Posted by jsfisher View Post
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...&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...&postcount=206 :
Originally Posted by jsfisher View Post
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_..._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:

Code:
N = {
∅,                    ---> {{∅}}, ---> ...   <-- base sets
{∅},                  ---> {{∅}, {{∅}} }, ---> ...  <-- successor sets
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...  <-- successor sets
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...  <-- successor sets
...                         ...
}
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 6th September 2020 at 06:47 AM.
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Old 6th September 2020, 08:37 AM   #374
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Originally Posted by doronshadmi View Post
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?
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Old 6th September 2020, 09:01 AM   #375
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Originally Posted by doronshadmi View Post
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.

Quote:
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...&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.

Quote:
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.

Quote:
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?

Quote:
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.
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Last edited by jsfisher; 6th September 2020 at 09:36 AM.
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Old 6th September 2020, 09:15 AM   #376
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While waiting for Doronshadmi's new and improved Definition #3, it might be useful considering some implications of his first two definitions.

Quote:
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 {∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}}
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Last edited by jsfisher; 6th September 2020 at 09:58 AM.
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Old 7th September 2020, 03:48 AM   #377
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Originally Posted by jsfisher View Post
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 :
Originally Posted by jsfisher
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...&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:

Originally Posted by jsfisher
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.
Originally Posted by jsfisher
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 ?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 05:23 AM.
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Old 7th September 2020, 03:55 AM   #378
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Originally Posted by jsfisher View Post
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={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} ?
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 04:01 AM.
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Old 7th September 2020, 05:50 AM   #379
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Originally Posted by doronshadmi View Post
In http://www.internationalskeptics.com...&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.
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Old 7th September 2020, 05:54 AM   #380
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Originally Posted by doronshadmi View Post
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?
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Old 7th September 2020, 06:26 AM   #381
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Originally Posted by jsfisher View Post
{∅, {∅}, {∅, {{∅}}}}
p={ ∅, {∅}, {∅, {{∅}}} }

In that case, since S=p∪{p} it is

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

which is not the same as

{ ∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}} }
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Old 7th September 2020, 06:35 AM   #382
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Originally Posted by doronshadmi View Post
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?
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Old 7th September 2020, 07:27 AM   #383
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Originally Posted by jsfisher View Post
What difference do you see? They look like identical sets to me.
They are identical.

Originally Posted by jsfisher View Post
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
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 07:32 AM.
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Old 7th September 2020, 08:11 AM   #384
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Lining up sets to show there is a difference

Originally Posted by doronshadmi View Post
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={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}} ?
Originally Posted by jsfisher View Post
{∅, {∅}, {∅, {{∅}}}}
Originally Posted by doronshadmi View Post
Code:
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
Originally Posted by jsfisher View Post
What difference do you see? They look like identical sets to me.
Originally Posted by doronshadmi View Post
They are identical.
Edit reasons: Purposely edited messages to remove extra topics, added [code] tag in a post to help line up the elements, and added letters below the brackets show the "level" of the brackets.

Last edited by Little 10 Toes; 7th September 2020 at 08:15 AM.
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Old 7th September 2020, 08:43 AM   #385
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Originally Posted by doronshadmi View Post
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.
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Old 7th September 2020, 08:56 AM   #386
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Originally Posted by Little 10 Toes View Post
Lining up sets to show there is a difference

...

Edit reasons: Purposely edited messages to remove extra topics, added [code] tag in a post to help line up the elements, and added letters below the brackets show the "level" of the brackets.
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 = { ∅, {∅}, {∅, {{∅}}} }

{p} = { { ∅, {∅}, {∅, {{∅}}} } }

In that case

S={ ∅, {∅}, {∅, {{∅}}} , { ∅, {∅}, {∅, {{∅}}} } }

or

S={∅, {∅}, {∅, {∅}, {∅, {{∅}}}}, {∅, {{∅}}}

since order is irrelevant.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 08:57 AM.
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Old 7th September 2020, 09:20 AM   #387
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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
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 09:50 AM.
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Old 7th September 2020, 09:48 AM   #388
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Originally Posted by doronshadmi View Post
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.
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Last edited by jsfisher; 7th September 2020 at 09:54 AM.
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Old 7th September 2020, 09:58 AM   #389
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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.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 10:29 AM.
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Old 7th September 2020, 10:09 AM   #390
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Originally Posted by doronshadmi View Post
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.
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Old 7th September 2020, 10:55 AM   #391
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Originally Posted by jsfisher View Post
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.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 7th September 2020, 11:01 AM   #392
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Originally Posted by doronshadmi View Post
Thank you a lot jsfisher.
And yet you didn't correct any of the problems in Definition #3.

Quote:
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"?

Quote:
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.
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Old 7th September 2020, 11:08 AM   #393
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Originally Posted by jsfisher View Post
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.
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That is also over the matrix, is aware of the matrix.

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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 7th September 2020, 11:20 AM   #394
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Originally Posted by jsfisher View Post
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:

Code:
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
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 12:07 PM.
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Old 7th September 2020, 11:32 AM   #395
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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.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 7th September 2020, 11:53 AM   #396
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Originally Posted by jsfisher View Post
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...5&postcount=83 ?
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That is also over the matrix, is aware of the matrix.

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For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 7th September 2020, 12:03 PM   #397
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Originally Posted by doronshadmi View Post
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.
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Old 7th September 2020, 12:11 PM   #398
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Originally Posted by doronshadmi View Post
Well, Ok let's correct it.
Yes, please.

Quote:
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)"?


Quote:
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".
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Old 7th September 2020, 12:16 PM   #399
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Originally Posted by jsfisher View Post
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.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
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Old 7th September 2020, 12:39 PM   #400
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Originally Posted by jsfisher View Post
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)}

Originally Posted by jsfisher View Post
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.
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That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

Last edited by doronshadmi; 7th September 2020 at 01:06 PM.
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