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

[doronshadmi]

opss ...

 

[doronshadmi]

You haven't defined what you mean by "finite set", yet.

By set(s) I mean "pure" set(s).

A finite set is any set that the quantity of its members (which are also sets) is a natural number (including 0) (order is irrelevant).

In this case the sets, as members of the following set, are defined such that if x is a member of that set, then x∪{x} is also a member of that set.

But we can use any set that its members are sets, where the quantity of the members of these sets, is a natural number (including 0) (order is irrelevant), without spiking on any natural number (including 0).

Here is the entire show in one ticket (in case of x and x∪{x}, where this case is without loss of generality) :

{
 0  =   ∅, 
        ↓  [B]No-bijection[/B]
 1  = { ∅ },
        ↓  [B]No-bijection[/B] 
 2  = { ∅, {∅} },
        ↓  [B]No-bijection[/B] 
 3  = { ∅, {∅} , {∅, {∅}} },
        ↓  [B]No-bijection[/B] 
 4  = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} },
        ↓  [B]No-bijection[/B] 
...  

}

x (as a member of that set) is any given natural number (including 0) in terms of a "pure" set.

There is no injection (<=) from x to x∪{x}, no matter where x or x∪{x} are in the set (order is irrelevant).


Now please plug in universal quantification without ignoring that there is no injection (<=) from x to x∪{x} (no matter where x or x∪{x} are in the set (order is irrelevant)) and please share with me your conclusions about the considered set (the set of natural numbers in terms of "pure" sets (including {})).

If injection fails among finite sets, it definitely does not exist among, so called, infinite sets.

 

[Little 10 Toes]

By set(s) I mean "pure" set(s).

A finite set is any set that the quantity of its members (which are also sets) is a natural number (including 0) (order is irrelevant).

Wonderful. By the above definition {2} is a non-finite set.

How about you start defining "pure" and then "pure set"?

 

[jsfisher]

By set(s) I mean "pure" set(s).

A finite set is any set that the quantity of its members (which are also sets) is a natural number (including 0) (order is irrelevant).

"Quantity of its members"? "Natural number"? Sounds like you are trying assume cardinality into existence without definition. No deal. Your only freebie was the minimal set guaranteed by the Axiom of Infinity. Nothing beyond that.


So, how do you define "finite set" (or, equivalently, "non-finite set" or "infinite set")?

 

[doronshadmi]

Sounds like you are trying assume cardinality into existence without definition.

It is very simple, jsfisher.

Any given natural number (including 0) can be taken in terms of quantity (how many?) or in terms of order (first, second, third, etc.).

Since order is irrelevant in case of sets, they are defined only in terms of the quantity of any given natural number (including 0), where such sets can't be but finite (no definition of cardinality, as an extension of the quantity of any given natural number (including 0) is possible, since no-bijection is constant among finite sets, as will be shown later in this post).

In the considered case the sets, which are members of the following set, are defined such that if x (where x is a finite set) is a member of that set, then x∪{x} is also a member of that set.

But we can use any set that its members are sets, where the quantity of the members of these sets, is a natural number (including 0) (only quantity, since order is irrelevant in case of sets), without spiking on any natural number (including 0).

Here is the entire show in one ticket (in case of x and x∪{x}, where this case is without loss of generality) :

{
 0  =   ∅, 
        ↓  [B]No-bijection[/B]
 1  = { ∅ },
        ↓  [B]No-bijection[/B] 
 2  = { ∅, {∅} },
        ↓  [B]No-bijection[/B] 
 3  = { ∅, {∅} , {∅, {∅}} },
        ↓  [B]No-bijection[/B] 
 4  = { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} },
        ↓  [B]No-bijection[/B] 
...  

}

x (as a member of that set) is any given natural number (including 0) in terms of a "pure" set (only quantity is relevant, since order is irrelevant in case of sets).

There is no injection (<=) from x to x∪{x}, no matter where x or x∪{x} are in the set (order is irrelevant).


Now please plug in universal quantification without ignoring that there is no injection (<=) from x to x∪{x} (no matter where x or x∪{x} are in the set (order is irrelevant)) and please share with me your conclusions about the considered set (the set of natural numbers in terms of "pure" sets (including {})).

If injection fails among finite sets, it definitely does not exist among, so called, infinite sets.

 

[doronshadmi]

... (or, equivalently, "non-finite set" or "infinite set")?

The set, which its members are finite sets, is the minimal set, which its members are defined, such that ∅ ( or {}) is in the set, and for any given finite set x in that set, x∪{x} is in that set, such that no-bijection is constant from x to x∪{x}, no matter where x or x∪{x} are in that set, since order is irrelevant.

no-bijection prevents injection among finite sets, and the result is a set that its exact quantity is undefined.

Sets that their exact quantity is undefined, are defined as non-finite sets.

More details are already given in http://www.internationalskeptics.com/forums/showpost.php?p=13219833&postcount=451 and http://www.internationalskeptics.com/forums/showpost.php?p=12671520&postcount=3305.

 

[jsfisher]

It is very simple, jsfisher.

Any given natural number....

Define "natural number".

 

[jsfisher]

In the considered case the sets, which are members of the following set, are defined such that if x (where x is a finite set) is a member of that set, then x∪{x} is also a member of that set.

How do you propose to define that set? You failed miserably in your prior attempt.

 

[jsfisher]

The set, which its members are finite sets...

You have not yet defined "finite set".

 

[jsfisher]

I'm just going to place this here for now:

1. Ø ∈ N
2. ∀a (a∈N ⇒ a ⋃ {a} ∈ N)
3. ∀a (a∈N ⇒ (a = Ø ∨ ∃b (b∈N ∧ (a = b ⋃ {b}))))

At some point it may become entirely relevant...or not.

 

[doronshadmi]

Let's carefully look as a set that containing {} and closed under the successor function S, defined by S(x) = x ∪ {x}.

jsfisher, if you look at, https://en.wikipedia.org/wiki/Set-t...al_numbers#Definition_as_von_Neumann_ordinals , and then carefully read all of http://www.internationalskeptics.com/forums/showpost.php?p=13221666&postcount=465 and http://www.internationalskeptics.com/forums/showpost.php?p=13221676&postcount=466 more than once, before you replay, maybe you will eventually realize that natural numbers are simply abstract notions, which are used to answer the questions "how many?" or "in what order?".

By define a natural number (including 0) in terms of a set, we simply take "in what order?" as an irrelevant question, and we are left only with the "how many?" question, which can be accurately answered by isolating such set from the endless non-bijective mapping among those sets, which prevents an accurate answer to the "how many?" question, in case of a set that includes such endless non-bijective mapping among those sets.

Such a set is exactly a non-finite set.

This set containing {} and closed under the successor function S, defined by S(x) = x ∪ {x}, where x is any set that enables one to provide an exact answer to the "how many?" question (where Cardinality is a forced notion, which its aim is to go beyond the "how many?" question and still be closed under the notion of set).

This forced notion is very simply exposed in http://www.internationalskeptics.com/forums/showpost.php?p=13219833&postcount=451 as the λ trick, which wishes "to eat the cake and save the sets".

 

[jsfisher]

jsfisher, if you look at...

I am fully aware of what the numbers are, and what the ordinals are, and what many other things are.

You, however, haven't been able to show much of any understanding of those things. Finite and infinite sets are fundamental notions. Your inability to define them is telling. If you cannot deal with the most basic of concepts, nothing you can claim in Mathematics can be trusted.

The current topic arc has to do with cardinality of infinite sets having no definable value. If you intend to show that, you'll need to first define what you mean by infinite set and by cardinality and to do it in such a way you don't build the conclusion you seek into the definitions.

If you need much beyond what the basic axiom set for set theory provide to define your idea of cardinality, you'll need to define that. The tools you get are the axioms, ordered sets, mappings, and the minimal set guaranteed by the Axiom of Infinity.

 

[doronshadmi]

I am fully aware of what the numbers are, and what the ordinals are ...

I know that you believe that you are fully aware of what the numbers are, and what the ordinals are.

jsfisher, your belief fails right at the beginning, since natural number is a common notion for at least two questions, which are: "how many?" or "in what order?"

Finite and infinite sets are fundamental notions.

When the notion of natural number is defined in terms of a set the "in what order?" question becomes irrelevant.

This is the fundamental thing about sets, and being finite or non-finite is based on it.

The details are already given in http://www.internationalskeptics.com/forums/showpost.php?p=13222026&postcount=471.

 

[doronshadmi]

For any given n ∈ N (n is a set containing n sets) and (n∪{n} ∈ N) and (there is no bijection from n to n∪{n}).

 

[jsfisher]

I guess we are stuck again, Doronshadmi. You cannot define the term, finite set. You cannot define the term, cardinality. Instead, once again, you seek shelter pretending they don't need to be defined.

Finite set and set cardinality are concepts in Set Theory, not number theory. If you want to be critical of Set Theory, you need to do it at the set-theoretic level.

But you'd rather pretend. Please stop pretending.

 

[doronshadmi]

I guess we are stuck again, Doronshadmi. You cannot define the term, finite set. You cannot define the term, cardinality. Instead, once again, you seek shelter pretending they don't need to be defined.

Finite set and set cardinality are concepts in Set Theory, not number theory. If you want to be critical of Set Theory, you need to do it at the set-theoretic level.

But you'd rather pretend. Please stop pretending.

Well jsfisher, the "border" between a given natural number and a given set exists only in your mind, but not in the minds of mathematicians like von Neumann, which explicitly defined the natural numbers in terms of sets.

For any given n ∈ N (n is a set containing n sets that their order is irrelevant, but only their quantity) and (n∪{n} ∈ N) and (there is no bijection from n to n∪{n} in N, which prevents the exact quantity of the ns in N).

λ as a limit cardinal, was invented in order to establish one framework for both finite and infinite sets, which does not distinguish between finite and infinite sets.

Injection (<=) was the key, as follows:

|A| <= |B| iff there exists an injection from A to B.
|A| = |B| iff |A| <= |B| and |B| <= |A|
|A| < |B| iff |A| <= |B| and not |B| <= |A|

As can be seen set A or B can be finite or infinite under the same framework, where the "how many?" question was replaced by "what size?" question, and quantity was replaced by Cantor's notion of Cardinality, which is relatively defined by functions form a given domain (notate here as A) to a given co-domain (notated here as B).

Moreover, iff enables to define a membership function that determines whether a given set is a member of some set, or not.

It all looks simple and elegant, but it can't eliminate this little thing called natural number (including 0) as the abstract notion that enables one to, at least, provide the answer to the following questions: "How many?" or "In what order?".

When defined in terms of a set, the "In what order?" becomes irrelevant, but the "How many?" question remains.

Furthermore, it can't be eliminated by the "what size?" question, which does not distinguish between finitely many and infinitely many things.

So, here it is again, the simple distinction between finite and infinite sets, right at the fundamental level, which has no border whatsoever between numbers and sets:

For any given n ∈ N (n is a set containing n sets that their order is irrelevant, but only their quantity) and (n∪{n} ∈ N) and (there is no bijection from n to n∪{n} in N, which prevents the exact quantity of the ns in N).

 

[jsfisher]

Well jsfisher, the "border" between a given natural number and a given set exists only in your mind, but not in the minds of mathematicians like von Neumann, which explicitly defined the natural numbers in terms of sets.

"Border" is your word, not mine. And von Neumann's treatment of the natural numbers is right in line with my point: Just about everything in Mathematics is rooted in Set Theory.

Long ago, I provided a meanings for cardinality and finite set. You didn't like them--you didn't understand them, but you didn't like them either. So now it is your turn to give the terms meaning within the bounds of Set Theory and what you can easily construct from Set Theory.

 

[doronshadmi]

Long ago, I provided a meanings for cardinality and finite set.

Where exactly you provided a meaning for finite set?

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

n is a set containing n sets, where the minimal n = {}.

As about cardinality, your meaning does not water since λ is inaccessible by endless succession (there is no bijection from n to n∪{n} in N (no matter where n or n∪{n} are in N, since order is irrelevant) which prevents the exact quantity of the ns in N).

It means that λ=|N|, where |N| can't the measurement value of those that, by definition, are inaccessible to it (injection (<=) does not hold, exactly because non-bijection (the (=) part of (<=)) is not satisfied, right among the finite sets of N).

It means that "N is a set containing N sets" is nonsense.

More about the λ ad hoc nonsense, is found in http://www.internationalskeptics.com/forums/showpost.php?p=13219833&postcount=451.

Finite set and set cardinality are concepts in Set Theory, not number theory. If you want to be critical of Set Theory, you need to do it at the set-theoretic level.

0 = {}
1 = {{}}
2 = {{},{{}}}
...

This "=" between a given natural number and a given set, means that their theories are on the same level.

"Border" is your word, not mine. And von Neumann's treatment of the natural numbers is right in line with my point: Just about everything in Mathematics is rooted in Set Theory.

You have no point, simply because this "=" between a given natural number and a given set, means that their theories are on the same level.

{} = 0
{{}} = 1
{{},{{}}} = 2
...

is the same as

0 = {}
1 = {{}}
2 = {{},{{}}}
...

in case that you have missed the very meaning of = as the "bridge" on the same level between Number theory and Set theory.

Moreover, defining the "bridge" between, so called, "different" mathematical branches, this is the actual foundation of Mathematics.

Please see https://www.youtube.com/watch?v=b8e_HMEwKIY .

 

[jsfisher]

Where exactly you provided a meaning for finite set?

That would be back when I defined a set X to be non-finite when |N| <= |X| (where N is the minimal set satisfying the requirements of the Axiom of Infinity). Since finite and non-finite are complementary terms, defining one automatically defines the other.

So, what is your definition for a finite set? It will likely involve cardinality one way or the other, therefore you will likely need to define cardinality first.

 

[doronshadmi]

That would be back when I defined a set X to be non-finite when |N| <= |X| (where N is the minimal set satisfying the requirements of the Axiom of Infinity). Since finite and non-finite are complementary terms, defining one automatically defines the other.

So N is a general set of infinitely many finite sets, as long as no membership function defines its infinitely many members according to yes/no (iff) rule.

In that case, how the membership function of set V of all the finite sets of von Neumann set of ordinals (where the order of V members is irrelevant) will be look like?

V is the set of all the finite sets of von Neumann set of ordinals iff ...