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Old 3rd September 2020, 04:13 AM   #321
jsfisher
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Originally Posted by doronshadmi View Post
I take the property (x ∪ {x}) right from the axiom above, as follows:

Definition 1: A set K is a base set iff K property is ~(x∪{x}).

How can you determine whether K is of the form x ∪ {x} or not?

(By the way, (x ∪ {x}) is not a property and ~(x∪{x}) is not a valid formula. You don't negate sets.)
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Old 3rd September 2020, 04:16 AM   #322
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Originally Posted by doronshadmi View Post
Each one of the given definitions simply declares the existence of some set that satisfies a given property.
No, they don't. They give names to things. Neither "declares the existence" of anything.
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Old 3rd September 2020, 06:25 AM   #323
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Originally Posted by jsfisher View Post
How can you determine whether K is of the form x ∪ {x} or not?
Originally Posted by jsfisher;
No, the formula (x ∪ {x}) appears without "the need to first define it as a successor" because it is unnecessary to give it a name.
Is (x∪{x}) a wff?
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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; 3rd September 2020 at 06:31 AM.
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Old 3rd September 2020, 06:53 AM   #324
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Originally Posted by doronshadmi View Post
Is (x∪{x}) a wff?
Not in propositional calculus, no. "Cup" is a set-valued binary operator, and x and {x} are sets. That makes x ∪ {x} a set and not a true/false ;proposition.
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Old 3rd September 2020, 07:23 AM   #325
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Originally Posted by jsfisher View Post
Not in propositional calculus, no. "Cup" is a set-valued binary operator, and x and {x} are sets. That makes x ∪ {x} a set and not a true/false ;proposition.
A∪B = {x : x∈A OR x∈B}

A=x
B={x}

and we get A∪B = {x : x∈x OR x∈{x}}

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

Definition 2: A set K is a successor set iff K = {x : x∈x OR x∈{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; 3rd September 2020 at 07:33 AM.
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Old 3rd September 2020, 08:57 AM   #326
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Originally Posted by doronshadmi View Post
Definition 1: A set K is a base set iff K ≠ {x : x∈x OR x∈{x}}
So, what is "x" thing (the non-bold, non-italic version)?
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Old 4th September 2020, 08:33 AM   #327
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Originally Posted by jsfisher View Post
So, what is "x" thing (the non-bold, non-italic version)?
Whet are the iff yes/no logical foundations of "x" and "x ∪ {x}" things in the following expression? :

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])
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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 4th September 2020, 09:05 AM   #328
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Originally Posted by doronshadmi View Post
Whet are the iff yes/no logical foundations of "x" and "x ∪ {x}" things in the following expression? :

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])
Yes/no logical foundations? Where did that come from? I asked what was x in your definition. You cannot just introduce x ∪ {x} (in whatever form you care to represent set union) without telling us what x is. If you leave x unknown, as you have, then your definition isn't.

The Axiom of Infinity does include x ∪ {x}, but if first tells us about x. That's the ∀x∈I part.

So, what's your x?
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Old 4th September 2020, 09:13 AM   #329
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Originally Posted by doronshadmi View Post
Whet are the iff yes/no logical foundations of "x" and "x ∪ {x}" things in the following expression? :

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])
That makes no sense.

"What are the 'if and only if' yes/no foundations of 'x' and 'x ∪ {x}" things in the following expression"?

There are none.

Axiom_of_infinityWP: "∃I (∅ ∈ I ∧ ∀ x ∈ I [(x ∪ {x}) ∈ I]). In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set." (I use square brackets in the for readability).

No iff or yes/no there.
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Old 4th September 2020, 09:24 AM   #330
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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}}.
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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; 4th September 2020 at 09:43 AM.
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Old 4th September 2020, 09:37 AM   #331
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Originally Posted by doronshadmi View Post
x is a placeholder for a set.

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}}.
Define x.
Define x.
What is the difference between x and x?
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Old 4th September 2020, 09:37 AM   #332
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Originally Posted by doronshadmi View Post
x is a placeholder of any given set.
You need to quantify that in some way. Otherwise, x is just an unknown. The "right hand side" of your definitions are not allowed to contain unknowns.

You can have unknowns on the left, after all those are the things you intend to define, but by the time you get to the right, it all needs to be fixed.
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Old 4th September 2020, 09:43 AM   #333
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Originally Posted by jsfisher View Post
You need to quantify that in some way. Otherwise, x is just an unknown. The "right hand side" of your definitions are not allowed to contain unknowns.

You can have unknowns on the left, after all those are the things you intend to define, but by the time you get to the right, it all needs to be fixed.
x is a placeholder for any given set (without any information about its members).

Originally Posted by jsfisher View Post
The Axiom of Infinity does include x ∪ {x}, but if first tells us about x. That's the x∈I part.
Once again you force "for all" (universal quantification).
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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; 4th September 2020 at 09:52 AM.
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Old 4th September 2020, 09:47 AM   #334
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Originally Posted by Little 10 Toes View Post
Define x.
Define x.
What is the difference between x and x?
In the set-builder notation Doronshadmi is using, x has a defined meaning. It is simply a variable representing each possible member of the set being defined; everything following the colon provides the conditions all members of the set must meet (and which all non-members fail to meet).

E.g.,
{ w : w is an even whole number } = { 0, 2, 4, 6, ... }
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Old 4th September 2020, 09:54 AM   #335
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Originally Posted by doronshadmi View Post
x is a placeholder for any given set (without any information about its members).
Then your definition fails to define anything. Whether K is equal to some set involving x cannot be decided since x is not known. You'd need to quantify x in some way, formally, right there in the definition.

(This is the second time I have provided that hint.)
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Old 4th September 2020, 09:58 AM   #336
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Originally Posted by jsfisher View Post
{ w : w is an even whole number } = { 0, 2, 4, 6, ... }
In this example w meaning can also be "no members".
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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 4th September 2020, 10:02 AM   #337
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Originally Posted by doronshadmi View Post
In this example w meaning can also be "no members".
How? Are you saying that even whole numbers don't exist?
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Old 4th September 2020, 10:04 AM   #338
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Originally Posted by jsfisher View Post
Then your definition fails to define anything. Whether K is equal to some set involving x cannot be decided since x is not known. You'd need to quantify x in some way, formally, right there in the definition.

(This is the second time I have provided that hint.)
Here we come again to ∀.
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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 4th September 2020, 10:05 AM   #339
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Originally Posted by doronshadmi View Post
In this example w meaning can also be "no members".
It is correct exactly as presented. You didn't understand the example at all, did you?

Stop pretending.
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Old 4th September 2020, 10:06 AM   #340
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Originally Posted by Little 10 Toes View Post
How? Are you saying that even whole numbers don't exist?
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.
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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; 4th September 2020 at 10:56 AM.
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Old 4th September 2020, 10:12 AM   #341
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Originally Posted by doronshadmi View Post
Here we come again to ∀.
Seems like an inconvenient choice, but, hey, knock yourself out.
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Old 4th September 2020, 10:42 AM   #342
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Originally Posted by jsfisher View Post
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).
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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 4th September 2020, 10:49 AM   #343
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Originally Posted by doronshadmi View Post
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.
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Old 4th September 2020, 11:24 AM   #344
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Originally Posted by doronshadmi View Post
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.

Last edited by Little 10 Toes; 4th September 2020 at 11:26 AM.
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Old 4th September 2020, 11:35 AM   #345
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Originally Posted by jsfisher View Post
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}}.
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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; 4th September 2020 at 01:24 PM.
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Old 4th September 2020, 11:50 AM   #346
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Originally Posted by Little 10 Toes View Post
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.
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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 4th September 2020, 12:01 PM   #347
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I notice that once again, you do not mention your dishonesty.
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Old 4th September 2020, 12:08 PM   #348
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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(y) be the membership function of set N.

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

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

Not being largest successor set in N is:

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

An example:

Code:
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(y) is true), but no largest successor set is a member of set N (M(y) 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.
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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; 4th September 2020 at 01:21 PM.
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Old 4th September 2020, 12:08 PM   #349
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Originally Posted by doronshadmi View Post
A∪B = {x : x∈A OR x∈B}
The left-hand side provides the meaning for A and B used on the right.

Quote:
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.
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Old 4th September 2020, 12:17 PM   #350
doronshadmi
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Originally Posted by jsfisher View Post
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.


Originally Posted by jsfisher View Post
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...&postcount=345 from start to end.
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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; 4th September 2020 at 12:33 PM.
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Old 4th September 2020, 12:46 PM   #351
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Originally Posted by doronshadmi View Post
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.

Quote:
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?
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Old 4th September 2020, 01:04 PM   #352
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Originally Posted by jsfisher View Post
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.

Originally Posted by jsfisher View Post
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.

Originally Posted by jsfisher View Post
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.
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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; 4th September 2020 at 01:27 PM.
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Old 4th September 2020, 01:19 PM   #353
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Originally Posted by doronshadmi View Post
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.

Quote:
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.
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Old 4th September 2020, 01:39 PM   #354
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Originally Posted by jsfisher View Post
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"?


Originally Posted by jsfisher View Post
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...&postcount=334 .
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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 4th September 2020, 01:46 PM   #355
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Originally Posted by jsfisher View Post
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.


Originally Posted by jsfisher View Post
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.
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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 4th September 2020, 01:51 PM   #356
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Originally Posted by doronshadmi View Post
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".
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Old 4th September 2020, 01:53 PM   #357
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Originally Posted by doronshadmi View Post
What do you mean by "fully quantified meaning"?
Everyone is better in Comic Sans:
S = { x : P(x) } ⇔ ∀x ( x∈ S ⇔ P(x) )
Quote:
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.)
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Old 4th September 2020, 01:58 PM   #358
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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.

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_..._specification .
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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 4th September 2020, 02:02 PM   #359
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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?

Quote:
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....
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Old 4th September 2020, 02:04 PM   #360
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Originally Posted by jsfisher View Post
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.
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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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