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

jsfisher said:
I skipped any treatment of the natural numbers because I don't need any. You are the one insisting it be there, I have shown it isn't required.



Yes, that is what I wrote and what you already quoted. Repeating it was unnecessary.



No, N is defined by the previous expression.



Restating it again does not add value. And you made an error in the process. Be that as it may, though, I am confident you have no idea how the expression actually achieves the goal. You recognize the individual "words" by you don't comprehend their collective meaning.



Yep, proof you don't comprehend their collective meaning.





Oh? Are you unfamiliar with the phrase, vacuously true? F : ∅ → X is an injective map from the empty set to any set X. A function F is an injective map from the not necessarily distinct sets A to B iff:
∀a∈A (∃b∈B (F(a) = b ∧ ∀c∈A (F(a) = F(c) ⇒ a = c)))​
Nothing there forbids A or B from being empty.
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You have missed my correction in http://www.internationalskeptics.com/forums/showpost.php?p=13234768&postcount=559 and replied to an irrelevant post.
 
doronshadmi said:
It is a new post, where you replied to the older one.
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No, it was not a new post. It was an substantial edit to an existing post.

And I replied to the version of the post in place at the time I hit the Quote button. It is rude for you to dismiss my post in its entirety, too. If it comments on mistakes of yours that have been corrected, point out in response to my post how your mistakes were corrected. If it comments on things you thought were correct in the first place, comment on that.

I await the mitigation to your rudeness.
 
jsfisher said:
No, it was not a new post. It was an substantial edit to an existing post.

And I replied to the version of the post in place at the time I hit the Quote button. It is rude for you to dismiss my post in its entirety, too. If it comments on mistakes of yours that have been corrected, point out in response to my post how your mistakes were corrected. If it comments on things you thought were correct in the first place, comment on that.

I await the mitigation to your rudeness.
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jsfisher your post was created almost two hours after my new post was created.

So next time please refresh your screen before you reply.

Please reply to http://www.internationalskeptics.com/forums/showpost.php?p=13234768&postcount=559.
 
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The relevant details are as follows:

jsfisher, in order to establish the minimal set satisfying the requirements of the Axiom of Infinity, one first has to show that all the natural numbers (which are defined by von Neumann's treatment of the natural numbers in terms of sets) are actually establish a set that satisfying the requirements of the Axiom of Infinity.

So at the first stage the following definitions are restricted only to all the members of this minimal set, such that the initial member is the empty set (v=∅) and the rest of the members are defined as v⋃{v}, which are all finite sets:

Definition 1: |v| = |v| iff there is bijection from v to v.

Definition 2: |v| < |v⋃{v}| iff (there is an injective non-surjective function from v to v⋃{v}) OR (injective function from v to v⋃{v} that is based on empty function) (in case that v=Ø)

Definitions 1 and 2 look trivial at first glance, but these are the functions among the finite sets, which define all the natural numbers by von Neumann's treatment.

Since ∀v ∈ V(v∪{v} ∈ V ∧ |v∪{v}| can't be but < |V|) |V| is not established, simply because it can't be defined as the cardinality of all V members, which are all the finite sets by von Neumann's treatment of all the natural numbers.
 
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Just a reminder:

jsfisher said:
No, it was not a new post. It was an substantial edit to an existing post.

And I replied to the version of the post in place at the time I hit the Quote button. It is rude for you to dismiss my post in its entirety, too. If it comments on mistakes of yours that have been corrected, point out in response to my post how your mistakes were corrected. If it comments on things you thought were correct in the first place, comment on that.

I await the mitigation to your rudeness.
Click to expand...
 
I'm starting this message at 9:13pm local time according to my tablet. I am purposely waiting and stalling a few minutes to have the forum clock to change.

Edit: the system clock is 7-8 minutes behind. This would confirm that the time on the post is not when you start writing it, but when you submit/save/post it.

Edit 2: trying to see what the edit time is based off. Waiting about 30mins. Looks like it's based off the save button.
 
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Little 10 Toes said:
And what the deal about waiting almost two hours before posting the final "edit"?
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From my point of view, I don't care. I responded to an actual post. It matters not that I was interrupted by something more important in real life that delayed my complete response. I responded to an actual post.

For Doronshadmi to then claim I hadn't responded to a post he had edited to become something else and therefore I had to start all over is unacceptable rudeness, especially since my observations about the post I responded to were still valid after all of Doronshadmi's editing.

If Doronshadmi chooses to dismiss my post out of hand, so be it, we are done. The only other option is to address it as is and respond to the comments directly or how he has (yet again) changed is argument in a way that my remarks aren't about what he currently believes.
 
jsfisher said:
... especially since my observations about the post I responded to were still valid after all of Doronshadmi's editing.
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Let's see.

1) He ignores the minimal set that satisfies the requirements of the Axiom of Infinity.

2) This minimal set is exactly the von Neumann's treatment of the natural numbers in terms of sets (let's call this set V (which, of course, can be replaced by any other chosen name)).

3) All the members of set V are finite sets.

4) There are four functions, divided into two types, among V members, as follows:

a) 2 functions from a given member to itself:

a.1) Empty function from Ø to itself.

a.2) Bijective function from v (where v≠Ø) to itself.

a.3) Bijective function from v⋃{v} to itself.

b) 2 functions from a given member to a different member:

b.1) Injective non-surjective function from v to v⋃{v} (in case that v≠Ø).

b.2) Injective function from v to v⋃{v} (in case that v=Ø, where all v⋃{v} in V are finite sets, according to von Neumann's treatment of the natural numbers in terms of sets).

jsfisher said:
Every element of v ⋃ {v} is an element of V. The identity function is a perfectly adequate injection from v ⋃ {v} to V.

Therefore (under my definition): |v ⋃ {v}| ≤ |V|
Click to expand...

But by (1) to (4) there can't be injection from v⋃{v} to V, which means that:

∀v ∈ V(v∪{v} ∈ V) ∧ (|v∪{v}| < |V|).

No wonder that jsfisher ignores "weak limit cardinal" as seen in http://www.internationalskeptics.com/forums/showpost.php?p=13219833&postcount=451.

( 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.
Click to expand...

λ is an ad hoc invention out of nowhere, that all of its purpose is to establish the Cantorian transfinite system by hook or by crook.
 
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jsfisher said:
Not all closed curves are circles, but all circles are closed curves. Not all injections are bijections, but all bijections are injections.
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closed curves = one-to-one functions
circles = one-to-one AND onto functions

So, according to you, not all one-to-one functions are one-to-one AND onto functions (in this case we have |A| < |B|) , but all one-to-one AND onto functions are one-to-one functions (in this case we have |A| = |B|).

jsfisher said:
You can, in fact, use those meanings to show that the expression |A| <= |B| is identical to (|A| < |B|) OR (|A| = |B|)
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By using your analogy:

(not all one-to-one functions are one-to-one AND onto functions)

OR

(all one-to-one AND onto functions are one-to-one functions)

Since all the members of the von Neumann's treatment of the natural numbers in terms of sets are finite sets, only the following part of his analogy holds:

(not all one-to-one functions are one-to-one AND onto functions)

which means that only all one-to-one functions that are not one-to-one AND onto functions, are used in case of |v∪{v}| < |V|, exactly because v∪{v} can't be but a finite set, the von Neumann's treatment of the natural numbers in terms of sets.
 
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On two previous occasions, I posted some cryptic hieroglyphics suggesting they may be useful sometime later. Now that the thread's star celeb is working on his monologue, re-presenting past nonsense, I will close out my participation in this arc of the thread explaining those two posts for anyone who really cares.

First, a reminder. This latest "infinity bad" arc had at its start "proof" based in the cardinality of infinite sets. Since seemingly standard terms and phrases have been used in such novel ways in this thread, it was only reasonable to demand my interlocutor define what he meant by cardinality. This proved impossible, but did show the need to build the desired conclusion on cardinality into the definition of cardinality, creating a natural loop.

Ok, set theory has the Axiom of Infinity. It is necessary in an axiomatic set theory; otherwise you cannot construct an infinite set. Here is the axiom expressed in first order predicate calculus:
∃I (Ø∈I ∧ ∀x (x∈I ⇒ x ⋃ {x} ∈ I))​
Looks ugly, but it expresses a simple idea: There exists a set we will call I (∃I) which contains the empty set as a member (Ø∈I) and (∧) for any set x (∀x) if x is a member of I (x∈I) then (⇒) the "successor" of x is also a member of I (x ⋃ {x} ∈ I).

The Axiom of Infinity guarantees some set, I, exists that has two properties:
Ø ∈ I
∀a (a∈I ⇒ a ⋃ {a} ∈ I)​
We can even give it a name. Any set that has these two properties is an "inductive set." Notice, by the way, that the two properties tell us somethings about what must be in the set, but say nothing about what cannot be in the set. The Axiom of Infinity guarantees one such set, but there could be many.

One way to look at inductive sets is that they all must contain Ø and they must contain an infinite sequence of successors arising from Ø as the starting point. Such a set could also contain other starting points, other than Ø, and the infinite sequence of successors arising from those other starting points.

So many, many inductive sets could exist in set theory, the Axiom of Infinity guarantees there is at least one, but is there a "minimum" such set, one that has only Ø and its sequence of successors and not others? Formally, is there a set which I will call N with this property:
∀a (a∈N ⇒ (a = Ø ∨ ∃b (b∈N ∧ (a = b ⋃ {b})))) ​
What these chicken-scratches express is that every member of N is either the empty set or the successor of some other member of N. I'll leave that as an exercise for the reader to decipher.

So, how can we show that such a set must exist? Well, more chicken-scratches:
N = {x∈I : ∀y ((Ø∈y ∧ ∀z ((z∈y ⇒ z ⋃ {z} ∈ y))) ⇒ x∈y)}​
It certainly doesn't look like it, but all this is stating is that N (my minimal set) is the intersection of all the inductive sets. Obvious, now, right?

Well, the intersection of two sets, A ∩ B, is simply all the members that are common to both sets. More formally,
A ∩ B = {x∈A : x∈B}​
The gibberish, N = {x∈I : ∀y ((Ø∈y ∧ ∀z ((z∈y ⇒ z ⋃ {z} ∈ y))) ⇒ x∈y)}, is just a generalized version of that. The ∀y ((Ø∈y ∧ ∀z ((z∈y ⇒ z ⋃ {z} ∈ y))) part is simply identifying all the inductive sets and intersecting them with the set I.

For completeness, we'd need to prove this set N has that additional property. I won't bore you further, but I will outline the proof: N is the intersection of all inductive sets. If N doesn't satisfy the additional property, then there must be some member besides the empty set in N that isn't the successor of some other member. We can then define a set N' that is N without that other member. N' must be an inductive set, and N ∩ N' = N'. Since N ≠ N', we have a contradiction of N being the intersection of all inductive sets, so N must satisfy the additional property.

Whew.
 
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jsfisher said:
On two previous occasions, I posted some cryptic hieroglyphics suggesting they may be useful sometime later. Now that the thread's star celeb is working on his monologue, re-presenting past nonsense, I will close out my participation in this arc of the thread explaining those two posts for anyone who really cares.
Click to expand...
You close your participation in this arc by a monologue that does not satisfy fundamental things for anyone who really cares:

1) You have proved that N exists, such that Ø is its one and only one member that is not a successor of the rest of N members, where all the members of set N are themselves sets, such that for all x (where x is any arbitrary member of N), x⋃{x} is also a member of N.

2) All along the way you ignored the existence of all the finite sets that are definitely members of N.

3) Whether you like it or not, all these finite sets that are members of set N, satisfying the requirements of the Axiom of Infinity.

4) If you disagree with (3), you actually disagree with the existence of all the natural numbers in terms of sets (as defined by the von Neumann's treatment of the natural numbers, where their initial set is Ø, as written in (1)).

5) By not ignoring all the finite sets of N, one can easily understand http://www.internationalskeptics.com/forums/showpost.php?p=13235103&postcount=567, which very simply addresses the fact that no x⋃{x} has the cardinality of N.

6) Your entire framework is based on injective function (≤), which according to it x⋃{x} (which is a finite member of N, if N is indeed the minimal set that satisfying the requirements of the Axiom of Infinity) has the cardinality of N, but http://www.internationalskeptics.com/forums/showpost.php?p=13238449&postcount=572 very simply shows why |x⋃{x}| = |N| is not established.

7) You take injection as some generalization, which somehow enables bijection to sneak in order to establish |x⋃{x}| = |N|, but http://www.internationalskeptics.com/forums/showpost.php?p=13240783&postcount=574 shows why this generalization has no basis.

8) No wonder that you ignores "weak limit cardinal" as seen in http://www.internationalskeptics.com...&postcount=451.

( 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.
Click to expand...

λ is an ad hoc invention out of nowhere, that all of its purpose is to establish the Cantorian transfinite system by hook or by crook.

====================================

Is there any alternative to jsfisher's framework?

The answer is yes, as can be seen, for example, in https://en.wikipedia.org/wiki/Alternative_set_theory.
 
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|A| ≤ |B| is identical to (|A| < |B|) OR (|A| = |B|).

p="|A| < |B|"

q="|A| = |B|"

Code: 
p OR q
--------
F    F --> F
T    F --> T
F    T --> T
T    T --> T


Logically < can be replaced by ≤, but ≤ can't be replaced by <.

Some example:

1) 1<2 and 1≤2 are both true (T OR F --> T of the truth table above).

2) 1≤1 is true, but it's not true that 1<1 (F OR T --> T of the truth table above).

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

3) T OR T --> T is possible only if |A| is a variable for values that are < or = |B|.

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

Since all the members of the von Neumann's treatment of the natural numbers in terms of sets, satisfying the requirements of the Axiom of Infinity, they can't be ignored by any one that works under ZF(C).

Option (3) above is impossible, since given any arbitrary x (where x is some set by the von Neumann's treatment of the natural numbers in terms of sets) |x⋃{x}| can't be but < |N| (T OR F --> T of the truth table above) if N is claimed to be a set with infinitely many members.

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

One claims: "You can't pick parts from a truth table in order to support your argument, since the whole truth table must be taken".

Well, in this case how, for example, 1<2 and 1≤2 are both true by the whole OR truth table?


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

Maybe I am mixing things up, so please correct my mistakes.
 
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By ZFC, the minimal set satisfying the requirements of the axiom of infinity, is the intersection of all inductive sets.

In case that the axiom of infinity is expressed as

∃I (Ø ∈ I ∧ ∀x (x ∈ I ⇒ x ⋃ {x} ∈ I))

the intersection of all inductive sets (let's call it K) is defined as

set K = {x∈I : ∀y ((Ø ∈ y ∧ ∀z ((z ∈ y ⇒ z ⋃ {z} ∈ y))) ⇒ x ∈ y)}.

The members of set K can be defined by von Neumann's construction of the natural numbers in terms of sets (https://en.wikipedia.org/wiki/Natural_number#Constructions_based_on_set_theory).

So K satisfying the requirements of the axiom of infinity, where all of its members are finite sets, such that only Ø is not a successor of the rest of K's members.

In that case ∀k ∈ K (k ∪ {k} ∈ K), where |k ∪ {k}| can't be but < |K|, if |K| = λ = weak limit cardinal (such that λ is neither a successor cardinal nor zero).

In that case how λ is formally defined as the cardinality of all k in K ?
 
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von Neumann's construction of the natural numbers in terms of sets

No successor of von Neumann's construction of the natural numbers in terms of finite ordinals, is in bijection with its members.

For example: {{}, {{}}} can't be in bijection with {} or {{}} but it is its own cardinal number |{{}, {{}}}|.

Yet {{}, {{}}} is in bijection with {{}, {{}}} (itself) and in this case it is defined as an ordinal number by von Neumann.

ω is the set of all natural numbers (including 0) in terms of sets.

All the members of ω are finite sets and ω can't be in bijection with any one of them.

Therefore ω is the cardinal number of itself and it can't be a finite cardinal number.

The main notions here are mathematical induction and the acceptance of ∀ (for all) also in case of infinite sets.

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

There is also an agreement among mathematicians about elements that are defined by induction, which does not accept the notion of infinitely many elements that are gathered into a set (ω, in this case) as follows:

Suppose P(0) is true. Suppose for every n, if P(n) is true, then P(n+1) is true, for example: {} is the basis of the induction, where the next element {{}} is based on its immediate predecessor {} by a union operation {}U{{}} which is the inductive step.

The important thing here is the invariant form of the inductive step such that for any arbitrary n, which is not the basis of the induction ({} in this case), n is defined by its immediate predecessor n-1 by the union operation n-1 U {n-1} and the defined n is used as the basis in order to define n+1 by the inductive step n U {n}.

So we get n elements without stopping the inductive steps and without the need to gather the defined elements into a set (ω, in this case).

Moreover, such unbounded collection can be in bijection with some unbounded collection of it, which has less elements than it (it is equivalent to Dedekind Infinite without the need to be gathered into a set) simply because this is a fundamental property of being unbounded.
 
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doronshadmi said:
Moreover, such unbounded collection can be in bijection with some unbounded collection of it, which has less elements than it (it is equivalent to Dedekind Infinite without the need to be gathered into a set) simply because this is a fundamental property of being unbounded.
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Let's rewrite it more carefully, unbounded collection A can be in bijection with unbouded collection B even if any B object is also A object, but not any A object is also B object (it is equivalent to Dedekind Infinite without the need to be gathered into a set) simply because this is a fundamental property of being unbounded (∀ quatifier is logically valid only in case of finite collections).
 
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