Cont: Deeper than primes - Continuation 2

[doronshadmi]

And the set of natural numbers does exist.

There is a finite or infinite set of natural numbers, yet the set of natural numbers as a complete infinite set, does not exist exactly because {n}$n.

 

[jsfisher]

I already wrote about this set in my previous post, but I was wrong and unclear about my claims, so here is my corrected and clearer reply about "if X is a member of the set, then {X} is a member of the set".

1) By using such set, the set of natural numbers can't be defined, since {0, 1, 1, 1, ...} is the best you can get.

You continue to be wrong on multiple counts. The set is still { {}, {{}}, {{{}}}, {{{{}}}}, ...}, and that is clearly an infinite set. Also, if one wanted to associate the set members to natural numbers, the obvious assignment is {} = 0, {{}} = 1, {{{}}} = 2, {{{{}}}} = 3, ....

 

[jsfisher]

...mercifully snipped...

Congratulations. You found an alternate version of the axiom of infinity thus proving you were wrong about Mathematics being stuck on a particular successor relationship.

You were also wrong about the standard Axiom of Infinity being flawed because it used Von Neumann's construct instead of Zermelo's.


You entire argument about infinite sets being incomplete boils down to because you said so. That's not how Mathematics works, and it would be worthless if it did.

 

[doronshadmi]

Congratulations. You found an alternate version of the axiom of infinity thus proving you were wrong about Mathematics being stuck on a particular successor relationship.

It is a general successor relationship.

If {X} is a successor of X, the considered set is infinite AND (externally OR internally) incomplete.

In order to realize it all is needed is to understand the concept of set, as follows:

The void between the outer "{" and "}" is weaker than any amount of existing members, where the outer "{" and "}" is stronger than any amount of existing members.

By using this notion {||}=0 where |{}|= ∞

By using {X} as a successor of X, any amount of members approaches but not reaches ∞, therefore such an amount can't be determined and therefore it is incomplete.

By not using {X} as a successor of X, no amount of members approaches ∞, therefore such an amount can be determined and therefore it is complete.

So no matter if a given non-empty set is "fed" externally or internally by {X}, in both cases the amount of its existing members can't be determined and therefore it is incomplete.

The following example in http://www.internationalskeptics.com/forums/showpost.php?p=11279727&postcount=1485 holds for both external or internal options.

 

[jsfisher]

It is a general successor relationship.

If {X} is a successor of X, the considered set is infinite AND (externally OR internally) incomplete.

You made up that part.

In order to realize it all is needed is to understand the concept of set, as follows:

the void between the outer "{" and "}" is weaker than any amount of existing members, where the outer "{" and "}" is stronger than any amount of existing members.

You made up this part, too.

By using this notion...

...you abandon Mathematics in favor of things you made up.

 

[doronshadmi]

You were also wrong about the standard Axiom of Infinity being flawed because it used Von Neumann's construct instead of Zermelo's.

In both cases {x} as a successor of x is involved, so no matter if Zermelo's or Von Neumann's constructions are used, the result is an inductive AND incomplete set.

You entire argument about infinite sets being incomplete boils down to because you said so. That's not how Mathematics works, and it would be worthless if it did.

Mathematics is not some object independent of the mathematicians that discover or invent it.

So nor me neither you have the privilege the determine Mathematics as some object.

That's how Mathematics works, and it would be worthless if it didn't.

 

[jsfisher]

In both cases {x} as a successor of x is involved

Both? Wrong again. Only Zermelo and not Von Neumann incorporated "x in I => {x} in I".

...so no matter if Zermelo's or Von Neumann's constructions are used, the result is an inductive AND incomplete set.

That part you made up.

 

[doronshadmi]

You made up that part.



You made up this part, too.


...you abandon Mathematics in favor of things you made up.

Well, this is a reflexive reply of a person that gets Mathematics as an object independent of mathematicians.

 

[doronshadmi]

Both? Wrong again. Only Zermelo and not Von Neumann incorporated "x in I => {x} in I".

Wrong, as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374 Von Neumann's construction is the external form of the incompleteness, where Zermelo's construction is the internal form of the incompleteness.

Generally, you are still failing to comprehend http://www.internationalskeptics.com/forums/showpost.php?p=11337788&postcount=1723.

 

[jsfisher]

Both? Wrong again. Only Zermelo and not Von Neumann incorporated "x in I => {x} in I".

Wrong

How do you figure that? Zermelo's construction uses "if x in I then {x} in I" and von Neumann's uses "if x in I then x U {x} in I". Surely even you can see that those two are not the same.

...
Von Neumann's construction is the external form of the incompleteness, where Zermelo's construction is the internal form of the incompleteness.

More things you made up. You cannot disprove Mathematics by making up stuff to fit your argument.

 

[doronshadmi]

How do you figure that? Zermelo's construction uses "if x in I then {x} in I" and von Neumann's uses "if x in I then x U {x} in I". Surely even you can see that those two are not the same.

Again, the incompleteness of a given infinite set is the result of {x} as a successor of x, whether it is used internally by Zermelo's construction or externally by Von Neumann's construction.

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=11341233&postcount=1784.

Generally, you are still failing to comprehend http://www.internationalskeptics.com/forums/showpost.php?p=11337788&postcount=1723 and also http://www.internationalskeptics.com/forums/showpost.php?p=11269502&postcount=1344.

More things you made up. You cannot disprove Mathematics by making up stuff to fit your argument.

http://www.internationalskeptics.com/forums/showpost.php?p=11341251&postcount=1786
http://www.internationalskeptics.com/forums/showpost.php?p=11341267&postcount=1788

 

[jsfisher]

Again, the incompleteness of a given infinite set is the result of {x} as a successor of x, whether it is used internally by Zermelo's construction or externally by Von Neumann's construction.

No, "incompleteness of a set" is a concept you made up but cannot define in any operational way.

No, "successor of x" is also a concept you cannot define, although it seems related to the successor function you have borrowed from the Peano axioms.

No, any relationship between incompleteness and successor is something you simply imagined to be true and have failed to substantiate in any way.

No, your latest vocabulary extension to include "external" and "internal" is a feeble attempt to cover up a whole sequence of blunders you made regarding Zermelo set theory.


Got any real math? Your fantasies are boring.

 

[doronshadmi]

No, "incompleteness of a set" is a concept you made up but cannot define in any operational way.

No, "successor of x" is also a concept you cannot define, although it seems related to the successor function you have borrowed from the Peano axioms.

No, any relationship between incompleteness and successor is something you simply imagined to be true and have failed to substantiate in any way.

No, your latest vocabulary extension to include "external" and "internal" is a feeble attempt to cover up a whole sequence of blunders you made regarding Zermelo set theory.


Got any real math? Your fantasies are boring.

Hands waiving of a person that real math for him is some object that is independent of mathematicians.

What a boring approach about real math.

Got any real math? Your approach is boring, so boring until you can't comprehend http://www.internationalskeptics.com/forums/showpost.php?p=11341356&postcount=1791.

 

[jsfisher]

Hands waiving of a person that real math for him is some object that is independent of mathematicians.

What a boring approach about real math.

Got any real math? Your approach is boring, so boring until you can't comprehend http://www.internationalskeptics.com/forums/showpost.php?p=11341356&postcount=1791.

Prove me wrong, then. Show us where these made-up terms of yours are defined in set theory. Show us how in real Mathematics your completeness concept and your successor concept are related to set theory.

Proof would be far more impressive than your Peewee Herman routine.

 

[doronshadmi]

Once again jsfisher, I wish to thank you from the bottom of my heart, because I find time after time that your boring approach and your trenchant criticism is the best fertilizer for my mathematical notions and developments.

 

[doronshadmi]

Prove me wrong, then.

Already done in http://www.internationalskeptics.com/forums/showpost.php?p=11341356&postcount=1791, but your boring view of real math prevents any comprehension of it.

But as I have said in http://www.internationalskeptics.com/forums/showpost.php?p=11341407&postcount=1795 it probably the best situation between us, in case of further discussion about Mathematics.

 

[doronshadmi]

I have found that there are at least 3 options of {X}$X : external, intermediate, internal.

The external is the case of X as the set of infinitely many forms of singleton sets.

The intermediate is the Von Neumann's construction of an inductive set (seen in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374).

The internal is Zermelo's construction of an inductive set ( { {}, {{}}, {{{}}}, {{{{}}}}, ...} ).

In all cases no infinite set is complete (http://www.internationalskeptics.com/forums/showpost.php?p=11341233&postcount=1784 talks about intermediate (called external in this link, because I did not count the external case there, as mentioned here) and internal cases, but it can be extended to all three cases).

So, jsfisher, thank you once again for mention also Zermelo's construction of an inductive set.

 

[Little 10 Toes]

Again, the incompleteness of a given infinite set is the result of {x} as a successor of x, whether it is used internally by Zermelo's construction or externally by Von Neumann's construction.

Define successor that you claim is being used by Von Neumann and Zemelo.

And how can they be using your idea if their ideas have been accepted long before you started posting here?

 

[doronshadmi]

Define successor that you claim is being used by Von Neumann and Zemelo.

http://www.internationalskeptics.com/forums/showpost.php?p=11341451&postcount=1797 is all you need in order to fully know my ideas about the issue at hand.

And how can they be using your idea if their ideas have been accepted long before you started posting here?

Their ideas are based on the notion that in order to understand the issue at hand, it first must be frozen (stand still).

By my ideas infinite collections are defined as ever developed things (can't be frozen).

Think about ever moving cars.

They may look frozen w.r.t each other, or not and this is exactly the flexibility among infinite collections that can't found among standing cars, which are actually frozen.

Moreover, a request like "please show me missing members of the set of natural numbers" is simply based on the attempt to freeze the ever developed.

My notions enable infinite numbers and infinitesimals, where the current accepted notions can't deal with them.

 

[Little 10 Toes]

Still waiting for you to define successor.

No links to other posts. Just simple words please.