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

[doronshadmi]

You still have not defined your set N. A membership function would be an important step towards that goal.

Definition 1: Any given set that is not of the form (x∪{x}), is called a base set (if included in some set, it is called a base member in that set).

Definition 2: Any given set that is of the form (x∪{x}), is called a successor set (if included in some set, it is called a succesor member in that set).

Definition 3: Given a successor member in some set, it is called largest successor member iff given a base member in that set, it has at least one successor member that does not have its successor member in that set.

EDIT:

The membership function of set N:

Given set N, no one of its successor members has its largest successor member (M(x) returns false about largest successor members).

Given set N, any one of its members is a base member OR a successor member that is not largest successor member (M(x) returns true about any base member OR any successor member that is not largest successor member).

 

[jsfisher]

The membership function of set N:

Given set N, no one of its successor members has its largest successor member.

Not a membership function.

M(x) : true if x is in N, and false if not (plus or minus a constraint or two consistent with the set theory axioms).

 

[doronshadmi]

Not a membership function.

M(x) : true if x is in N, and false if not (plus or minus a constraint or two consistent with the set theory axioms).

I apologize, you replied during my editing of http://www.internationalskeptics.com/forums/showpost.php?p=13202420&postcount=261 .

 

[jsfisher]

I apologize, you replied during my editing of http://www.internationalskeptics.com/forums/showpost.php?p=13202420&postcount=261 .

You have been scolded by multiple people multiple times about your rude post editing practices. Fixing typos and minor clarifications are entirely acceptable; substantial rewrites are not.

Still not a membership function. I note, too, your use of the word "returns"; it emphasizes the depth of your misunderstanding.

 

[doronshadmi]

Still not a membership function. I note, too, your use of the word "returns"; it emphasizes the depth of your misunderstanding.

No problem, without "returns":

The membership function of set N:

Given set N, no one of its successor members has its largest successor member (M(x) is false about the membership of largest successor members).

Given set N, any one of its members is a base member OR a successor member that is not largest successor member (M(x) is true about the membership of any base member OR any successor member that is not largest successor member).

edit:

Please see definitions 1,2,3 in http://www.internationalskeptics.com/forums/showpost.php?p=13202420&postcount=261 .

 

[jsfisher]

No problem, without "returns":

The membership function of set N:
....

Still not a membership function.

 

[doronshadmi]

Still not a membership function.

Well, here is an example of two base members and their successor members in set N:

N = {
∅,                    ---> {{∅}}, ---> ...   <-- base members
{∅},                  ---> {{∅}, {{∅}} }, ---> ...
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...
...                         ...
}

As can be seen, set N members are of the form ~(x∪{x}) (base members) OR (x∪{x}) (successor members) (M(x) is true), but no largest successor member is a member of set N (M(x) is false).

edit:

Also see definitions 1,2,3 in http://www.internationalskeptics.com/forums/showpost.php?p=13202420&postcount=261 .

 

[doronshadmi]

I note, too, your use of the word "returns"; it emphasizes the depth of your misunderstanding.

B.T.W what exactly is the problem of using the word "returns"?

Function M(x) returns the truth value false if x is not a member of set N.

Function M(x) returns the truth value true if x is a member of set N.

 

[jsfisher]

Well, here is an example of two base members and their successor members in set N:

N = {
∅,                    ---> {{∅}}, ---> ...   <-- base members
{∅},                  ---> {{∅}, {{∅}} }, ---> ...
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...
...                         ...
}

How do you know that is actually a set?

You need to decide what it is you mean by this set N of yours, be specific, then provide a definition for it as a set. A membership function would be a convenient way to do that.

 

[jsfisher]

B.T.W what exactly is the problem of using the word "returns"?

It demonstrates the computer sciency procedural perspective you have for things.

Your earlier attempts to describe your set, N, were nothing more than processes to enumerate the membership. You frequently express things as if a sequence of steps were required, and you more than once feigned denial with "it happens all at once". That excuse is not credible for a variety of reasons.

 

[doronshadmi]

How do you know that is actually a set?

You need to decide what it is you mean by this set N of yours, be specific, then provide a definition for it as a set. A membership function would be a convenient way to do that.

EDIT:

Let's put what is needed in one post, in order to define N as a set.

Definition 1: Any given set that is not of the form (x∪{x}), is called a base set.

Definition 2: Any given set that is of the form (x∪{x}), is called a successor set.

Definition 3: Given a successor set in some set, it is called largest successor set iff given a base set in that set, it has at least one successor set that does not have its successor set, in that set.

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

The membership function of set N:

Given set N, no one of its successor sets has its largest successor set (M(x) is false about the membership of largest successor sets).

Given set N, any one of its members is a base set OR a successor set that is not largest successor set (M(x) is true about the membership of any base set OR any successor set that is not largest successor set).

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

Still not a membership function.

Please support your claim, by details.



An example:

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

As can be seen, set N members are of the form ~(x∪{x}) (base sets) OR (x∪{x}) (successor sets) (M(x) is true), but no largest successor set is a member of set N (M(x) is false).

 

[jsfisher]

...
The membership function of set N:
...

Still not a membership function.

M(x) : true if x is in N, and false if not (plus or minus a constraint or two consistent with the set theory axioms).

Stop focusing on the irrelevant (i.e., whether there is a "largest successor"). Focus on membership: Is x in the set N or not?

 

[doronshadmi]

Still not a membership function.

M(x) : true if x is in N, and false if not (plus or minus a constraint or two consistent with the set theory axioms).

Stop focusing on the irrelevant (i.e., whether there is a "largest successor"). Focus on membership: Is x in the set N or not?

It is relevant.

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

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

 

[jsfisher]

It is relevant.

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

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

Whether x is a largest successor isn't the same question as whether x is a member.

 

[doronshadmi]

Whether x is a largest successor isn't the same question as whether x is a member.

x is largest successor is the same as x is not a member.

x is ~largest successor is the same as x is a member.

 

[Little 10 Toes]

Let's make it easy since doronshadmi is not understanding the simple items.

Which, if any, of the following items are in N?
1) The War of 1812
2) 10% GDP
3) Oranges
4) The ratio between the circumference of a circle and it's diameter
5) Beethoven's Fifth
6) {... -2, -1, 0, 1, 2, ...}
7) { {∅} ,{ { { {∅} } } } }
8) Double shot of espresso, 3 soy creams, 1 brown sugar, 97 degree, no whip

Please note, since doronshadmi has used N before, I am confirming that I am not talking about the set of natural numbers (also known/written as N or N).

 

[doronshadmi]

Let's make it easy since doronshadmi is not understanding the simple items.

Which, if any, of the following items are in N?
1) The War of 1812
2) 10% GDP
3) Oranges
4) The ratio between the circumference of a circle and it's diameter
5) Beethoven's Fifth
6) {... -2, -1, 0, 1, 2, ...}
7) { {∅} ,{ { { {∅} } } } }
8) Double shot of espresso, 3 soy creams, 1 brown sugar, 97 degree, no whip

Please note, since doronshadmi has used N before, I am confirming that I am not talking about the set of natural numbers (also known/written as N or N).

Only (7) as a base set.

Any N member is a "pure" set ( represented only by "{" or "}" (where ∅ is {} in terms of "pure" sets) ).

 

[Little 10 Toes]

You miss the point.

They all can be part of set N. Specifically, they all can be elements of set N.

Edit:
--------------------------------------------------------------------------------------------------------------
I did not ask about base set or a "pure" set. And again, you start making up your own definitions.

All I asked was "Which, if any, of the following items are in N?"

 

[jsfisher]

x is largest successor is the same as x is not a member.

x is ~largest successor is the same as x is a member.

You have been outspoken that there is no largest successor. That means that any set I choose is going to be "~largest successor". That means that any set I choose (including N) is going to be a member of N.

 

[doronshadmi]

You have been outspoken that there is no largest successor. That means that any set I choose is going to be "~largest successor". That means that any set I choose (including N) is going to be a member of N.

N does not have even one largest successor, this is exactly why N can't be but an infinite set that includes base sets OR successor sets that are not largest successors sets.

EDIT:
Once again, universal quantification (the term "for all", notated by ∀) is not a property of N.

Please read carefully what I wrote about Russell's Paradox and ∀, in the second part of http://www.internationalskeptics.com/forums/showpost.php?p=13201047&postcount=257 .

As for N being a member of itself, it can be one of its base members, where the term ∀ is not a property of this base member (or any one of its members etc.) and so is the case about its successors, no one of them has its largest successor.

Without ∀, N is its one of its base members without being failed by Russell's Paradox.