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

[jsfisher]

Now about Russell's Paradox.

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

This is Russel's Paradox: R = { x : x ∉ x }

The "cause" of Russell's Paradox, the cause of just about all paradoxes in logic is self-reference, or circular reference if you prefer to call it that. Self-reference can be very direct, as in:

This statement is false.​

Note the total lack of any quantifiers. It can also be spread out to obscure the self-reference, as in:

The following statement is false.
The preceding statement is true.​

Again, note the total lack of any quantifiers. It can also be decorated up with stories about barbers and facial hair. But at the base, there is a self-reference that triggers the contradiction.

Russell's Paradox relies on a self-reference to create the paradox.

 

[doronshadmi]

This is Russel's Paradox: R = { x : x ∉ x }

The "cause" of Russell's Paradox, the cause of just about all paradoxes in logic is self-reference, or circular reference if you prefer to call it that. Self-reference can be very direct, as in:

This statement is false.​

Note the total lack of any quantifiers. It can also be spread out to obscure the self-reference, as in:

The following statement is false.
The preceding statement is true.​

Again, note the total lack of any quantifiers. It can also be decorated up with stories about barbers and facial hair. But at the base, there is a self-reference that triggers the contradiction.

Russell's Paradox relies on a self-reference to create the paradox.

Well, x is the set of all sets that do not belong to themselves.

Does x belong to itself?

Also please do not miss http://www.internationalskeptics.com/forums/showpost.php?p=13215529&postcount=400 or http://www.internationalskeptics.com/forums/showpost.php?p=13215513&postcount=399 .

 

[jsfisher]

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)}

You left something important out. Your Definition #3 requires two sets be identified. You've identified x as mapping to L, but what maps to that other dummy variable, X? x is a largest successor in what?

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

Why would you set p = N? Nothing in your Definition #1 requires that. Your Definition #1 is this:

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

Since we are applying the definition to N, we can substitute N for the dummy variable S to get:

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

It does not say that p must equal N. The right-hand side of the equivalence is "there exists a set p in N such that ...". Well, it would appear that no such set meeting the requirements exists, so the right-hand side is false.

Therefore, the left-hand side is false. Therefore, N is not a successor set.

Nothing more. Nothing less.


By the way, Doronshadmi, you are racking up the examples demonstrating your lack of understanding of quantification.

 

[doronshadmi]

Why would you set p = N? Nothing in your Definition #1 requires that.

p can be any set including N, by definition 1.

 

[jsfisher]

Well, x is the set of all sets that do not belong to themselves.

You might want to change that. It isn't at all what I wrote.

Be that as it may, apparently you would disallow set builder notation completely. How now will you define your set N?


By the way, do you realize existential quantification and universal quantification are complementary? Either is sufficient to express the other. If you cannot abide universal quantification, plan on eschewing existential quantification as well. In Doronetics, it is impossible to express much of anything.

Luckily, Mathematics isn't required to cater to your odd proclivities. Complain as you will, Mathematics retains its upside down A and backwards E.

 

[jsfisher]

p can be any set including N, by definition 1.

Oh, wow! Quantification escapes you completely.

∃p∈S (S = p∪{p}) is a logical proposition. It can be true or false.
∃p∈S (S = p∪{p}) states that there is a member of S we will call p such that S = p∪{p}.

It does not say you get free choice in picking what you'd like p to be. It does not tell you which set it is, only that it must satisfy the requirement S = p∪{p}.

 

[doronshadmi]

Oh, wow! Quantification escapes you completely.

∃p∈S (S = p∪{p}) is a logical proposition. It can be true or false.
∃p∈S (S = p∪{p}) states that there is a member of S we will call p such that S = p∪{p}.

It does not say you get free choice in picking what you'd like p to be. It does not tell you which set it is, only that it must satisfy the requirement S = p∪{p}.

Given set p, the members of the successor set of set p are the members of set p AND set p.

In order to see this beauty, please look at http://www.internationalskeptics.com/forums/showpost.php?p=13215350&postcount=386 .



Now, let's correct your "corrections" that you have suggested for definition 3 and the membership function of N.

Here there are the right 3 definitions, and the right membership function of N:

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 is a base set) ∨ (x is a successor set ∧ ~largest successor set)}

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

Let R be the set of all sets that are not members of themselves.

If we include R as a member of itself, it is not set R anymore, but it is the successor set of set R, which its members (by definition 1) are the members of set R AND set R.

Since no set is its successor set, we actually avoid Russell's Paradox without any need for special axioms.

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

As for set N.

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:

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

 

[jsfisher]

Given set p, the members of the successor set of set p...

"Successor set of a set" is not a term you have defined.

Definition 3: A successor set L is a largest successor set iff ∃L ∈ X AND L∪{L} ∉ X

So, you've taken it back to being broken. For this to be a definition, X cannot be a free variable. ∃L ∈ X as you have used it is garbage.


You don't understand quantification. You don't understand predicate calculus. Please stop pretending.

 

[doronshadmi]

Why would you set p = N? Nothing in your Definition #1 requires that. Your Definition #1 is this:

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

Since we are applying the definition to N, we can substitute N for the dummy variable S to get:

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

It does not say that p must equal N. The right-hand side of the equivalence is "there exists a set p in N such that ...". Well, it would appear that no such set meeting the requirements exists, so the right-hand side is false.

Therefore, the left-hand side is false. Therefore, N is not a successor set.

Nothing more. Nothing less.

Definition 1 does not define the required set N that we are talking about it all along our last discussion.

Definition 1 defines successor set , Nothing more. Nothing less.

You can put any letter that you like, it does not change anything about definition 1.

Moreover, if p=N and you use N instead of S, the result is false exactly because N is not a member of itself, since no set is its successor set.

 

[jsfisher]

Moreover, if p=N ...

More proof you don't understand quantification. Please stop pretending.

 

[doronshadmi]

"Successor set of a set" is not a term you have defined.

Being a successor set can't be but a successor set of some set.

So, you've taken it back to being broken. For this to be a definition, X cannot be a free variable. ∃L ∈ X as you have used it is garbage.

Since L is a successor set, X can't be less than {{∅}}, so X is definitely not a free variable.

 

[doronshadmi]

More proof you don't understand quantification. Please stop pretending.

Take your time the actually realize what's going in http://www.internationalskeptics.com/forums/showpost.php?p=13216013&postcount=407 .

 

[jsfisher]

Being a successor set can't be but a successor set of some set.

So define it. The successor set of P is.... Of course, then you'll have two definitions for "successor set"; you should rename one to keep them straight.

Since L is a successor set, X can't be less than {{∅}}, so X is definitely not a free variable.

The term, free variable, is not something you grasp, either, then. You have also introduced "less than" in a meaningless way.

In your Definition #3, what is X? Is it a parameter as is the first L or a quantified value like your other L? You need to make X a bound variable in some fashion.

 

[jsfisher]

Take your time the actually realize what's going in http://www.internationalskeptics.com/forums/showpost.php?p=13216013&postcount=407 .

It is obvious what is going on in that post (most of your posts, actually). You piece together words and notation in ways that don't mean at all what you think they mean and often have no meaning at all.

You embark on very long paths of notation oddities, mangled concepts, unnecessary labels (that are seldom used consistently), and muddled reasoning that end up always at dead ends. The current thread arc is no exception.

 

[doronshadmi]

So define it. The successor set of P is.... Of course, then you'll have two definitions for "successor set"; you should rename one to keep them straight.

For example:

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

is the successor set of set

{ ∅, {∅}, {∅, {{∅}}} }

The needed details are already given in http://www.internationalskeptics.com/forums/showpost.php?p=13216013&postcount=407 .

EDIT:

Thank you again jsfisher about set X, definition 3 here is a correction of definition 3 as appears in the link above.

In your Definition #3, what is X?

X is a set that includes any given single successor set, where its successor set is not a member of X.

Here are the needed definitions, which formally define X, as written above:

Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})

Definition 3: A successor set L is a largest successor set iff ∃X such that L ∈ X AND L∪{L} ∉ X

 

[jsfisher]

For example...

Examples are not definitions.

X is a set that includes any given successor set, where its successor set is not a member of X.

Nothing in what you have written requires X to be anything other than a set. If it doesn't meet the criteria you specified, then the logical proposition containing X is false. No big deal; logical propositions are allowed to be false (and true, too).

So, what is X?

 

[doronshadmi]

Examples are not definitions.

This example is without loss of generality since given a successor set, its members can't be but the members of a given set, and that set.

In other words, for any given successor set its members depend on a given set.

Nothing in what you have written requires X to be anything other than a set. If it doesn't meet the criteria you specified, then the logical proposition containing X is false. No big deal; logical propositions are allowed to be false (and true, too).

So, what is X?

Ok, I corrected it to

Definition 3: A successor set L is a largest successor set iff ∃X such that L ∈ X AND L∪{L} ∉ X

But still, I have to check it if it satisfies my goal about set N, so I am working on it.

Thank you jsfisher.

 

[jsfisher]

This example is without loss of generality since given a successor set, its members can't be but the members of a given set, and that set.

In other words, for any given successor set its members depend on a given set.

So, define it.

Ok, I corrected it to

Definition 3: A successor set L is a largest successor set iff ∃X such that L ∈ X AND L∪{L} ∉ X

Then all successor sets are largest successor sets. Not that useful.

 

[doronshadmi]

So, define it.

It is defined:

Definition 1: S is a successor set ⇔ ∃p∈S (S = p∪{p})

S members are exactly the members of set p AND set p itself.

p is a given set and S is its successor set.

Then all successor sets are largest successor sets. Not that useful.

If we look at this example of set N

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

my goal is to deal with any given proper subset of N that "starts" by some base set and "continues" by its successors sets , such that no such proper subset has its last successor (which means that such proper subset is finite "backward" and infinite "forward").

Since such proper subsets are bijective, I think that to reach this goal about one such proper subset, is actually to cover the other proper subsets.

 

[jsfisher]

It is defined

"Is a successor set" is a set property -- that's your Definition #1.
"Successor set of" is a set function -- not defined.

my goal is to deal with any given proper subset of N that "starts" by some base set and "continues" by its successors sets

That cannot be done without a circular reference to the members of set N.