Cont: Deeper than primes - Continuation 2

[doronshadmi]

Could you be more specific? Which of the set of natural numbers is not present in the set of natural numbers?

Your attempt to define some n that is not included in the collection of n's, still uses the standard notion of successor, which is:

Successor(n) = n u {n}

In order to understand my answer, you first have to understand the notion of infinite numbers (http://www.internationalskeptics.com/forums/showpost.php?p=11259463&postcount=1185) that are used to define the sizes of sets with infinitely many members.

Infinite numbers are the result of {n} as a successor of n, and since {n} is permanently not in the range of the collection of n's (expect the case of the successor of the empty set), such collection is incomplete. A concrete example about sets is given in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374 and http://www.internationalskeptics.com/forums/showpost.php?p=11274453&postcount=1427.

Also the analogies in http://www.internationalskeptics.com/forums/showpost.php?p=11271494&postcount=1388 or http://www.internationalskeptics.com/forums/showpost.php?p=11279727&postcount=1485 are very helpful to understand the issue at hand.

Again, by my non-standard definition of successor, Successor(n) is {n} only if {n} is used as a successor of n (which means that also the option that {n} is not used as a successor of n holds).

If {n} is used as a successor of n, then the collection of n's is incomplete, such that there are infinitely many sizes of this collection, that are bijective with each other, or non-bijective with each other.

Generally, you are still define Successor(n) = n u {n}, and as a result you get {n} as irrelevant at the issue at hand.

 

[doronshadmi]

No, it isn't. As you keep telling us, it is or is not the successor, that being an optional property which, and with a wave of our hands we can decide it isn't.

There are two options ({n} is a successor of n, and as a result the collection of n's is incomplete) OR ({n} is not a successor of n, and as a result the collection of n's is complete).

{n} and n are the must-have primitives, where being a successor or being incomplete, are the result of the optional (non-must-have) relations between {n} and n.

For any integer n, {n} may or may not be one of those doron-successor things, but so what?

Once again your current used reasoning can't distinguish between {n} and n as must-have primitives, and the result of the optional (non-must-have) relations between {n} and n.

{n} isn't an integer. Its non-appearance in the set of integers is neither surprising nor relevant.

Since by your current used reasoning Successor(n) = n u {n}, there is no wonder that you do not understand the result of {n} as a successor of a collection of n's.

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More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=11314932&postcount=1661.

 

[jsfisher]

Since by your current used reasoning Successor(n) = n u {n}

Cool story, bro!, but I do no such thing. Successor is not a set theoretic construct. It doesn't enter into the Axiom of Infinity at all, and the axiom itself guarantees an infinite set complete in every regard.

Now, if one wanted to develop a basis for arithmetic within ZFC, then relating the successor function, S, from the Peano axioms to the union of n and {n} would be very useful.

Your choice for successor function, if that is what you had meant right along but were never really able to express, isn't particularly useful with respect to the Peano Axioms, and it results in the set of integers being non-existent.

Great job, Doron. Arithmetic doesn't work in doronetics.

 

[doronshadmi]

Cool story, bro!, but I do no such thing. Successor is not a set theoretic construct. It doesn't enter into the Axiom of Infinity at all, and the axiom itself guarantees an infinite set complete in every regard.

You are wrong, the concept of successor is in the heart of ZF Axiom Of Infinity ( https://en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement ):

Formal statement

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.


...


This construction forms the natural numbers. However, the other axioms are insufficient to prove the existence of the set of all natural numbers. Therefore its existence is taken as an axiom—the axiom of infinity. This axiom asserts that there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each element of I, the successor of that element is also in I.

By this quote Successor(x) = x u {x}, and this is the standard definition of the concept of successor.

By my non-standard definition of successor, Successor(x) is {x} only if {x} is used as a successor of x (which means that also the option that {x} is not used as a successor of x holds, as already explained in http://www.internationalskeptics.com/forums/showpost.php?p=11314944&postcount=1662 (and all the related links)).

Your choice for successor function, if that is what you had meant right along but were never really able to express, isn't particularly useful with respect to the Peano Axioms, and it results in the set of integers being non-existent.

Wrong again.

It results in the set of integers being incomplete, if {n} is used as a successor of n, as already explained in http://www.internationalskeptics.com/forums/showpost.php?p=11314944&postcount=1662 (and all the related links)).

 

[jsfisher]

By this quote Successor(x) = x u {x}, and this is the standard definition of the concept of successor.

When you learn to read with comprehension, we can talk. Cherry picking remarks without understanding their relationship to the whole of Mathematics isn't that.

...the set of integers being incomplete, if {n} is used as a successor of n

Fortunately, Mathematics under ZFC doesn't do that, so your point is moot.

 

[zooterkin]

Your attempt to define some n that is not included in the collection of n's

Not my attempt. You're the one who said:

...
I do reject the existence of the non-finite set of all natural numbers.

But you won't say which numbers are missing.


(You are still hung up on the misapprehension that the set needs to be constructed in some mechanical way by enumerating all the members. It does not, the members of the set are in the set when the set is defined.)

 

[Little 10 Toes]

Speaking of definitions, still can't come up with successor can you doronshadmi?

 

[jsfisher]

Speaking of definitions, still can't come up with successor can you doronshadmi?

He has backed into the same definition, now, a second time, except it is fully obfuscated behind a screen of arbitrary.

No matter, because it isn't important. All this stems from Doron's latest confused thread arc. Doron has claimed all infinite sets are defective by virtue of incompleteness (whatever that means), but to get there he has to add his own constructs and terminology he cannot define, none of which exist in Mathematics.

So, Mathematics once again emerges unscathed by Doronshadmi's current assault.

The set of integers, for example, exists and continues to contain all the integers.

 

[doronshadmi]

But you won't say which numbers are missing.

The incompleteness of a set of infinitely many natural numbers is the result of {n} as a successor of n, exactly as the incompleteness of a set of set S of infinitely many singleton sets is the result of {S} as a successor of S.

In both cases there is a singleton set, which is not in the range of these sets (except in the case that {x} is the successor of {} if a set of infinitely many natural number is defined, as can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374) that constantly provides further members to such sets, which prevent their completeness.

It does not, the members of the set are in the set when the set is defined.

This is your consequence exactly because by your definition Successor(x) = x u {x}.

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I show in http://www.internationalskeptics.com/forums/showpost.php?p=11314932&postcount=1661 that if {x} is used as a successor of x, any given infinite set is incomplete.

More generally, as long as you do not understand how infinite numbers are defined and how they are used to define the sizes of infinite sets, you do not understand the issue at hand, which is not based on Successor(x) = x u {x} that is your current used definition of Successor.

 

[Little 10 Toes]

So define successor. You keep defining the "function", but don't define the word.

 

[zooterkin]

The incompleteness of a set of infinitely many natural numbers is the result of {n} as a successor of n, exactly as the incompleteness of a set of set S of infinitely many singleton sets is the result of {S} as a successor of S.

In both cases there is a singleton set, which is not in the range of these sets (except in the case that {x} is the successor of {} if a set of infinitely many natural number is defined, as can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374) that constantly provides further members to such sets, which prevent their completeness.


This is your consequence exactly because by your definition Successor(x) = x u {x}.

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

I show in http://www.internationalskeptics.com/forums/showpost.php?p=11314932&postcount=1661 that if {x} is used as a successor of x, any given infinite set is incomplete.

More generally, as long as you do not understand how infinite numbers are defined and how they are used to define the sizes of infinite sets, you do not understand the issue at hand, which is not based on Successor(x) = x u {x} that is your current used definition of Successor.

Um, no, you're the one insisting on using the notion of 'Successor', though you've not defined it. It has no place in defining sets in real maths.

Now, try again; in the set of natural numbers, which ones are missing from the set?

 

[doronshadmi]

Um, no, you're the one insisting on using the notion of 'Successor',

Wrong, this notion is used also by Standard Mathematics, as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=11315376&postcount=1664.

though you've not defined it.

I defined it as an optional property of a set, such that ({X} is a successor of X) OR ({X} is not a successor of X)

It has no place in defining sets in real maths.

What you call real Math is based on Successor(X) = X U {X}

Now, try again; in the set of natural numbers, which ones are missing from the set?

Now try again to use {X} as a successor of X, and you immediately understand that X is an incomplete set, exactly as very simply given in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374, http://www.internationalskeptics.com/forums/showpost.php?p=11279727&postcount=1485 or http://www.internationalskeptics.com/forums/showpost.php?p=11274453&postcount=1427.

Generally, you simply reject http://www.internationalskeptics.com/forums/showpost.php?p=11315876&postcount=1669 by ignoring the fact that you are using Successor(x) = x u {x} as the basis of your notion about inductive sets (where a set of infinitely many natural numbers is some particular case of an inductive set).

 

[Little 10 Toes]

Still waiting doronshadmi. What is the simple definition of successor? No links to other posts please.

 

[doronshadmi]

Still waiting doronshadmi. What is the simple definition of successor? No links to other posts please.

({X}$X) OR ({X}~$X) ($ is an optional property of {X}).

 

[MetalPig]

({X}$X) OR ({X}~$X) ($ is an optional property of {X}).

Yes, we know a car is either blue or not blue. That doesn't define what blue is though.

 

[doronshadmi]

Yes, we know a car is either blue or not blue. That doesn't define what blue is though.

Wrong analogy.

The right analogy is that a given color is an optional property of a given thing (more details are seen in http://www.internationalskeptics.com/forums/showpost.php?p=11284073&postcount=1525).

Another mistake of yours is that you are using the adverb either that is actually XOR logical connective, where I use OR logical connective (more details are given in http://www.internationalskeptics.com/forums/showpost.php?p=11294564&postcount=1579).

 

[jsfisher]

Another mistake of yours is that you are using the adverb either that is actually XOR logical connective, where I use OR logical connective

Yet another sterling example of Doron's logic processes wherein the propositions A and ~A are independent of each other.

 

[doronshadmi]

Yet another sterling example of Doron's logic processes wherein the propositions A and ~A are independent of each other.

Well, it is an example of jsfisher's logic that forces OR to be XOR.

In logic, or by itself means the inclusive or, distinguished from an exclusive or, which is false when both of its arguments are true, while an "or" is true in that case.

(https://en.wikipedia.org/wiki/Logical_disjunction)

 

[MetalPig]

Are you saying {X} can be a successor of X and also not be a successor of X?

Which still doesn't define 'successor', of course.

 

[doronshadmi]

Are you saying {X} can be a successor of X and also not be a successor of X?

No. I am saying that {X} can be a successor of X OR also not be a successor of X.

Which still doesn't define 'successor', of course.

It fully defines successor as an optional property of {X}.