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

doronshadmi said:
Exactly, and it can't be done unless there is no bijection from p to S
Click to expand...

That may be some conclusion you'd like to reach, but it is nowhere to be found in your definition...even with that "∧ p≠S" nonsense.
 
I note, too, you have abandoned trying to define your set N. That would imply you have also abandoned trying to define your version of cardinality, leaving Mathematics intact, at least with respect to cardinality and infinite sets.


So, for everyone's benefit in following along, what is it are you trying to prove or disprove now?
 
jsfisher said:
I note, too, you have abandoned trying to define your set N.
Click to expand...

It is not needed anymore, all I need is a set with infinitely many finite sets, and this is exactly what set G is, also known as von Neumann set of ordinals that can't be used in order to establish the von Neumann universe, since no-bijection is constant among its members, which means that its cardinality is not satisfied.
 
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jsfisher said:
Some deep truth is there, ...
Click to expand...
No, I have found it in the shallow water among the finite members of von Neumann set of ordinals that define the natural numbers (inducing 0) in terms of "pure" sets.
 
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doronshadmi said:
It is not needed anymore, all I need is a set with infinitely many finite sets
Click to expand...

How is that different from the set N you were unable to define?

...and this is exactly what set G is, also known as von Neumann set of ordinals that can't be used in order to establish the von Neumann universe, since no-bijection is constant among its members, which means that its cardinality is not satisfied.
Click to expand...

Well, since you make all sorts of claims with no basis, you will need to provide a reasonable definition for this set G, so you can show the validity of your claims.

Also, "von Neumann universe" hasn't been defined by you, either.

And somewhere along the line you can show the relevance of this no-bijection claim of yours.
 
You know what, I even do not need the definition, in order to prove that non-bijection is constant among the finite members of von Neumann set of ordinals, which defines the natural numbers (inducing 0) in terms of "pure" sets.

Look at it:
Code: 
{
       	    ∅, 
            ↓  [B]No-bijection[/B]
     	   {∅},
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} },
            ↓  [B]No-bijection[/B] 
...  

}

Also look at this:

Code: 
N = {
 ∅,                    ---> {{∅}}, ---> ...    
 ↓  [B]No-bijection[/B]
{∅},                  ---> {{∅}, {{∅}} }, ---> ...   
 ↓  [B]No-bijection[/B]
{∅, {∅}},             ---> {{∅}, {{∅}}, {{∅}, {{∅}}}}, ---> ...   
 ↓  [B]No-bijection[/B]
{∅, {∅}, {∅, {∅}}},  ---> {{∅}, {{∅}}, {{∅}, {{∅}}}, {{∅}, {{∅}}, {{∅}, {{∅}}}}}, ---> ...   
 ↓  [B]No-bijection[/B]
...                         ...
}

In other words, ω or |N| (the cardinality of the set of natural numbers, including 0 (where set N above is not set N)) are not established.
 
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doronshadmi said:
You know what, I even do not need the definition, in order to prove that non-bijection is constant among the finite members of von Neumann set of ordinals, which defines the natural numbers (inducing 0) in terms of "pure" sets.
Click to expand...

Are you ever going to explain your fascination with bijections?
 
jsfisher said:
Are you ever going to explain your fascination with bijections?
Click to expand...
You can mix them up as much as you like, still they are there, x and x∪{x} (where x is any natural number (including 0) in terms of a "pure" set), such that no-bijection is constant among them, which prevents the existence of ω or |N|.

No bijection among natural numbers (including 0), no ω or |N|.

Classical mathematicians are using an ad hoc trick, and inconstantly establish out of nowhere λ :
( https://en.wikipedia.org/wiki/Limit_ordinal )
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal.
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( https://en.wikipedia.org/wiki/Limit_cardinal )
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.
Click to expand...

This means that one cannot "reach" λ from another cardinal by repeated successor operations.
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This means that λ is beyond collections, as seen, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=12584400&postcount=3285.
 
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jsfisher said:
Are you ever going to explain your fascination with bijections?
Click to expand...


Since non-bijection is constant among the finite members of von Neumann set of ordinals (which defines the natural numbers (inducing 0) in terms of "pure" sets) the (=) of the cardinality of the set of natural numbers is missing, and you can't use your <= trick in order to define |N| as the cardinality of the natural numbers (the best you can get is < |N| = non established λ, as long as λ is a measurement of collections).

This means that λ is beyond collections, as seen, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=12584400&postcount=3285.
 
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doronshadmi said:
jsfisher said:
Are you ever going to explain your fascination with bijections?
Click to expand...
...<snip of stuff that doesn't answer the question>...
Click to expand...


Ok. So where are we then?

You've invested a lot of effort and posts accomplishing very little other than further demonstrating your inability to understand basic mathematical concepts and mathematical notation, and a near total inability to express your own concepts.

So, you are calling for a do-over. Fine. Without back references to the ever shifting sands of your prior posts, let's start back at the beginning of all this with the meanings for cardinality and infinite set. Present them, please, and please present them one at a time. They are far easier to discuss if you do that. I will wait with great expectation to see that "bijection" will surface as important somewhere in there.

It appears now, too, you want to adopt the minimal set that satisfies the Axiom of Infinity as a given. Ok, but it comes at a price. The path to defining the minimal set explicitly uses universal quantification. (So does the axiom itself.) If you want the minimal set as a given you must accept universal quantification without further complaint; otherwise, it becomes your responsibility to show your own pathway to the minimal set that doesn't involve quantification (of either type since the two are complementary).


I now await your definitions for cardinality and for infinite set. If you still reject quantification, then there is a whole lot of work you'll need to do, too, to establish that minimal set.
 
Dear jsfisher, thank you very much for being with me all along our last discussion, you are simply great :thumbsup: :)

http://www.internationalskeptics.com/forums/showpost.php?p=13219983&postcount=452 shows exactly why one must not accept universal quantification without further complaint.

λ is beyond collections, as seen, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=12584400&postcount=3285, and a lot of "non-standard" work (which can't ignore anymore the non-composed (whether it is defined as contradiction or tautology)) has to be done, where by "non-standard" I exactly mean that the mathematician is not an observer-only of any mathematical framework, but not less than a participator of it, which means, being responsible for possible results.
 
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doronshadmi said:
http://www.internationalskeptics.com/forums/showpost.php?p=13219983&postcount=452 shows exactly why one must not accept universal quantification without further complaint.
Click to expand...

First, you tout all of your posts as clearly showing something. And I am sure you believe that, but it is seldom true. The cited post is no exception. No point dissecting it since that will just have us embark on yet another pointless arc on yet another topic with you unable to express any of your concepts unless someone feeds you the answer, and even then you will try to twist it into something else.

Second, if you reject universal quantification, then you also reject existential quantification. They are complementary, so either can be used to express the other.

Third, without quantification, you are disallowing just about all of Mathematics. Set theory is about as fundamental as it gets, and set theory requires quantification for its formalization.

Fourth, you have never adequately explained why quantification is unacceptable. Usually, too, buried in your tortured explanations is a circular argument. Quantification offends you in some way that seems to be completely based in some philosophic view of infinity, but Mathematics isn't restricted by your idiosyncrasies.


So, do you want to return to cardinality or not?

I will predict, however, that you would define cardinality as restricted to what can be represented by whole numbers then claim some meaningful victory when you later "prove" infinite sets fall "outside of cardinality", whatever that means.
 
jsfisher said:
First, you tout all of your posts as clearly showing something. And I am sure you believe that, but it is seldom true. The cited post is no exception. No point dissecting it since that will just have us embark on yet another pointless arc on yet another topic with you unable to express any of your concepts unless someone feeds you the answer, and even then you will try to twist it into something else.

Second, if you reject universal quantification, then you also reject existential quantification. They are complementary, so either can be used to express the other.

Third, without quantification, you are disallowing just about all of Mathematics. Set theory is about as fundamental as it gets, and set theory requires quantification for its formalization.

Fourth, you have never adequately explained why quantification is unacceptable. Usually, too, buried in your tortured explanations is a circular argument. Quantification offends you in some way that seems to be completely based in some philosophic view of infinity, but Mathematics isn't restricted by your idiosyncrasies.


So, do you want to return to cardinality or not?

I will predict, however, that you would define cardinality as restricted to what can be represented by whole numbers then claim some meaningful victory when you later "prove" infinite sets fall "outside of cardinality", whatever that means.
Click to expand...
jsfisher, by second thought, I do not have to reject universal quantification in order to establish my argument.

Since non-bijection is constant among the finite members of von Neumann set (which defines the natural numbers (inducing 0) in terms of "pure" sets) the (=) is missing.

It means that there is no injection (<=) from x to x∪{x}, no matter where x or x∪{x} are in the set (order is irrelevant) (where x is any natural number in terms of a "pure" set (including {})).

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) and please share with me your conclusions about the considered set (the set of natural numbers in terms of "pure" sets (including {})).
 
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doronshadmi said:
Since non-bijection is constant among the finite members of von Neumann set (which defines the natural numbers (inducing 0) in terms of "pure" sets) the (=) is missing.
Click to expand...

As often the case, you jump forward without defining your terms, or you use terms in peculiar or ambiguous ways.

"Non-bijection is constant" -- curious phrase, that. What do you mean?

"Von Neumann set" -- which set is that?

"Finite members" -- do you mean members that are themselves finite sets, or the members that are finite in number? And what do you mean by "finite"?
 
jsfisher said:
As often the case, you jump forward without defining your terms, or you use terms in peculiar or ambiguous ways.

"Non-bijection is constant" -- curious phrase, that. What do you mean?

"Von Neumann set" -- which set is that?

"Finite members" -- do you mean members that are themselves finite sets, or the members that are finite in number? And what do you mean by "finite"?
Click to expand...


This set (such the its members are only finite sets (also known as the set of natural numbers)):
Code: 
{
       	    ∅, 
            ↓  [B]No-bijection[/B]
     	   {∅},
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}} },
            ↓  [B]No-bijection[/B] 
     	   { ∅, {∅} , {∅, {∅}}, {∅, {∅}, {∅, {∅}}} },
            ↓  [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.
 
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