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

[jsfisher]

Please demonstrate exactly how one can't determine if a given set is of the form x ∪ {x} or not of the form x ∪ {x}.

I didn't say one couldn't determine if a set has a certain structure; I am asking you to clearly state how can be determined.

 

[jsfisher]

Relation is a noun, related is a verb.

I used "related".

If you have a better word of the connection between ∀ and a set, then please say it.

Not my responsibility.

Be that as it may, how about getting back to the task at hand: Clean up your definitions and define this set N of yours.

 

[doronshadmi]

You said: "Not being largest successor set is...". So, yeah, it is a definition (albeit in the negative).

Please read all of http://www.internationalskeptics.com/forums/showpost.php?p=13206541&postcount=296 about this case, it was corrected according to your criticism, but somehow you skipped on it.

 

[doronshadmi]

I didn't say one couldn't determine if a set has a certain structure; I am asking you to clearly state how can be determined.

Please look at ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]).

Exactly as given in this axiom, any given set that is of the form (x∪{x}), can't be but of the form (x∪{x}).

 

[jsfisher]

Please look at ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]).

Exactly as given in this axiom, any given set that is of the form (x∪{x}), can't be but of the form (x∪{x}).

That is of no use to you. You need a function, let's call it IsSucc, that is true if and only if y is of the required form. Then and only then can you clean up your definitions to be something like this:

Definition: A set X is a successor set iff IsSucc(X).
Definition: A set X is a base set iff it is not a successor set.​

Neither definition is all that important, by the way, since you can just use the IsSucc() function directly as you formulate your membership function for N.

 

[Little 10 Toes]

and remember, these are not computer science functions, but math functions.

 

[doronshadmi]

OK, I see what you mean, for example:

{} ∪ any non-empty set is that non-empty set, so the form (x∪{x}) can't be used in order to distinguish between what I call base set and successor set.

In that case I have to define what is the distinction between base set and a successor set, in the first place.

In other words, I have no framework until I define successor.

Thank you jsfisher.

So, back to work.

 

[doronshadmi]

• Have the common decency to use the quote function properly to refer to the message that you are quoting from. I have done that for you with jsfisher's original message. You're welcome.
• How about you stop shifting the burden. Your claim, your evidence. You show how you would determine "any given set" is of the form x ∪ {x}.

Also:
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

Do you understand that all of these items can be elements of set N?

Since this is the third time that I am asking, I am going to assume that you can't.

Little 10 Toes, the current discussion is only about sets.

For example 1 or 8 can be members of sets, but they are not themselves sets (they are called urelements ( see, for example, in https://en.wikipedia.org/wiki/Urelement )).

 

[Little 10 Toes]

Little 10 Toes, the current discussion is only about sets.

For example 1 or 8 can be members of sets, but they are not themselves sets (they are called urelements ( see, for example, in https://en.wikipedia.org/wiki/Urelement )).

No, the current discussion is that you currently can't define sets.

Here is a list of elements:
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

Do you understand that all of these items can be elements of a set?

Since this is the fourth that I am asking, I am going to assume that you can't.

 

[doronshadmi]

{} ∪ any non-empty set is that non-empty set, so the form (x∪{x}) can't be used in order to distinguish between what I call base set and successor set.

Ok, no more jokes, it is easy to understand that what is written above is nonsense, since in case of x={} the form (x∪{x}) is the particular case {}∪{{}} = {{}} that is the successor of {}.

In other words, any given set of the form (x∪{x}) can't be but the successor set of any given set x.

This exactly the reason why in ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]) the form (x∪{x}) appears whiteout the need to first define it as a successor.

So the following quote

Definition: A set X is a successor set iff IsSucc(X).
Definition: A set X is a base set iff it is not a successor set.​

Is actually

Definition 1: A set X is a successor set iff (x∪{x}).
Definition 2: A set X is a base set iff ~(x∪{x}).
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.
​

Neither definition is all that important, by the way, since you can just use the IsSucc() function directly as you formulate your membership function for N.

Set N members are (x∪{x}) or ~(x∪{x}), such that ~(x∪{x}) is a base set and (x∪{x}) is a successor set, where given any successor set, it is not largest successor set, since unlike definition 3, its successor set is a member of N.

Moreover, if x=N, then N is not the same as (N∪{N}), and we can't claim that N is a member of N, exactly as we can't claim that, for example, {} is a member of itself, since {}∪{{}}={{}} not={}.

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.

 

[jsfisher]

Definition 1: A set X is a successor set iff (x∪{x}).​

(x ∪ {x}) is not a boolean expression; it is a set. Did you mean X = (x ∪ {x})? If so, then what is this lower-case x? How is this lower-case x identified?

 

[jsfisher]

This exactly the reason why in ∃I (∅∈I ∧ ∀x∈I [(x∪{x})∈I]) the form (x∪{x}) appears whiteout the need to first define it as a successor.

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. That is what your first two still-in-need-of-work definitions attempt do: name things unnecessarily.

 

[doronshadmi]

(x ∪ {x}) is not a boolean expression; it is a set. Did you mean X = (x ∪ {x})? If so, then what is this lower-case x? How is this lower-case x identified?

Definition 1: A set X is a successor set iff (x∪{x}).

Definition 2: A set X is a base set iff ~(x∪{x}).

x appears in two forms (~(x ∪ {x}) or (x ∪ {x})) under unification that provides the members of set X, if X is defined by definition 1.

For example:


x={} (it is ~(x ∪ {x}) and therefore it is a base set).

In that case X is the unification of the "members" of {} and the members of {{}}, so X={{}}, which is the successor set of set x.


x={{}} (it is (x ∪ {x}) and therefore it is a successor set).

In that case X is the unification of the members of {{}} and the members of {{{}}}, so X={{},{{}}}, which is the successor set of set x.


x={{},{{}}} (it is (x ∪ {x}) and therefore it is a successor set).

In that case X is the unification of the members of {{},{{}}} and the members of {{{},{{}}}}, so X={{},{{}},{{},{{}}}}, which is the successor set of set x.


x={{},{{}},{{},{{}}}}

etc. ad infinitum ...

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

x={{{}}}

and we have a base set and successor sets that are distinct from the sets in the example above, ad infinitum ...

etc. ad infinitum ...

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


Definition 2: A set X is a base set iff ~(x∪{x}).

In that case X is, for example, {}, {{{}}}, etc. that are not of the form (x∪{x}).

 

[doronshadmi]

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.

This is the reason why you do not distinguish between base sets ( ~(x ∪ {x}) ) and successor sets ( (x ∪ {x}) ).

 

[doronshadmi]

X is ~(x∪{x}) OR (x∪{x}) and so is x.

 

[jsfisher]

Definition 1: A set X is a successor set iff (x∪{x}).

Repeating this doesn't make them into a well-formed formula. The expression, x ∪ {x}, continues to be a set and not logical-valued.

You need something that is true or false on the right-hand side of if-and-only-if.

 

[doronshadmi]

The definitions without iff.

Let's research ∃N (∅∈N ∧ ∀x∈N [(x∪{x})∈N]).

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.

Let M(x) be the membership function of set N.

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

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

Not being largest successor set in N is:

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

An example:

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

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

Set N members are (x∪{x}) or ~(x∪{x}), such that ~(x∪{x}) is a base set and (x∪{x}) is a successor set, where given any successor set, it is not largest successor set, since unlike definition 3, its successor set is a member of N.

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.

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. That is what your first two still-in-need-of-work definitions attempt do: name things unnecessarily.

Without define the formula (x∪{x}) as a successor set, 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 the formula (x∪{x}) as a successor set, is ∀ as the cause of Russell's Paradox, and this time please do not skip on it.


∀ is the cause of Russell's Paradox, whether a given collection of distinct objects is finite, or not.

For example:

U is a set of two distinct members, such that one of the members, called u, shaves ∀ the members of set U that do not shave themselves and only these members of set U (this is supposed to be his property in order to be a member of set U).

Who shaves u?

If u shaves himself, then he must not shave himself (shaves AND ~shaves himself, which is a contradiction) exactly because of the term ∀.

If u does not shave himself, then he must shave himself (~shaves AND shaves himself, which is a contradiction) exactly because of the term ∀.

So, because the term ∀ is used as a part of the terms that define u as a member U, u must be referred to himself, and we get the contradictions that actually prevents to well-define ∀ the members of set U (the term ∀ itself is actually not well-defined in case of U).

The same problem holds also among infinite sets that the term ∀ is one of their properties, therefore the Axiom of Restricted Comprehension was add to ZF in order to avoid Russell's Paradox, but it is done without being aware of the fact that the term ∀ is the cause of any given contradictory self reference, whether it is used among finite or infinite sets.

 

[jsfisher]

Let's research ∃N (∅∈N ∧ ∀x∈N [(x∪{x})∈N]).

That is the Axiom of Infinity. It does not define any set; it simply declares the existance of some set that satisfies two properties. In particular, it does not define your set, N, so let's go with this version of the Axiom

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])​

to avoid any confusion.

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

For any given set, let's call it K, how do you determine if K is of the form x ∪ {x}?

 

[doronshadmi]

That is the Axiom of Infinity. It does not define any set; it simply declares the existance of some set that satisfies two properties. In particular, it does not define your set, N, so let's go with this version of the Axiom

∃I (∅∈I ∧ ∀x∈I [(x ∪ {x})∈I])​

to avoid any confusion.



For any given set, let's call it K, how do you determine if K is of the form x ∪ {x}?

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}).

Definition 2: A set K is a successor set iff K property is (x∪{x}).

Each one of the given definitions simply declares the existence of some set that satisfies a given property.

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

Let M(x) be the membership function of set N.

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

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

Not being largest successor set in N is:

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

An example:

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

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

Set N members are (x∪{x}) or ~(x∪{x}), such that ~(x∪{x}) is a base set and (x∪{x}) is a successor set, where given any successor set, it is not largest successor set, since unlike definition 3, its successor set is a member of N.

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.

 

[doronshadmi]

Some improvement of definition 3:

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