Cont: Deeper than primes - Continuation 2

[doronshadmi]

The task at hand is to define the IsComplete operator. Showing us miscellaneous expressions the include the IsComplete operator does not define it.

You need to provide an expression that shows to what IsComplete S is equivalent. Something of the form:

IsComplete S <=> ...​

EDIT:

There are two possible states for set y, it can be complete OR incomplete, depending on its relations with its singleton set.

If {y} is not used as a successor of y, then set y is complete.

If {y} is used as a successor of y, then set y is incomplete.

It can also be written as follows:

"If {y}~$y, then y is complete", which is the same as "y is complete if {y}~$y".

"If {y}$y, then y is incomplete", which is the same as "y is incomplete if {y}$y".

Since y can be complete OR incomplete, depending on its relations with its singleton set, the following expression is wff:

(y is complete if {y}~$y) OR (y is incomplete if {y}$y)

That's all we need to know about y, according to its relations with {y} (which is its singleton set).

The following link is a concrete example of what is written above:

http://www.internationalskeptics.com/forums/showpost.php?p=11268767&postcount=1334.

 

[jsfisher]

Please look at this:

No, let's get those ten twelve eleven^* lines down to just one of the form:

IsComplete S <=> ...​

...concisely telling us what IsComplete S is equivalent to.


^*Count adjusted to reflect Doron's continual editing.

 

[doronshadmi]

No, let's get those ten twelve eleven^* lines down to just one of the form:

IsComplete S <=> ...​

...concisely telling us what IsComplete S is equivalent to.


^*Count adjusted to reflect Doron's continual editing.

IsComplete is one of two states of S, where the other state is IsnotComplete, such that:

(S IsComplete) OR (S IsnotComplete), depending on its relations with its singleton set (notated as {S}).

As long as this knowledge is not in your mind, you wrongly interpret IsComplete as an operator on S, instead of as S state (out of two possible states, with OR logical condition between them).

 

[jsfisher]

IsComplete is one of two states of S, where the other state is IsnotComplete

No. Complete is a property for some set, S; a state if you must. Not complete (or incomplete) is its complement. IsComplete, on the other hand, is a boolean monadic operator on a set, as in IsComplete S. IsComplete S is true-valued when S is a complete set, and IsComplete S is false-valued when S is not a complete set.

IsComplete S <=> ...​

If you, Doron, truly understand this complete-set property you've been harping on for so long, then you should not be having all this difficulty defining the IsComplete operator. Not all that long ago, you told us that

IsComplete X <=> NOT({X} IsSuccessorOf X)​

What changed to make it not that now?

 

[doronshadmi]

No. Complete is a property for some set,

Some set (is Complete) OR (is not-Complete) so your maneuvers are not must-have determinations for consistent reasoning.

If your maneuvers are used, then (IsComplete X <=> NOT({X} IsSuccessorOf X)) OR (NOT(IsComplete X) <=> {X} IsSuccessorOf X).

The result is the same.

 

[Nonpareil]

So an incomplete set is any set which mentions in its formal description "element X is the successor of element Y", where successor is... rather nebulously defined, but whatever.

Poor definitions aside, what about this prevents the existence of the set of all natural numbers?

 

[doronshadmi]

, where successor is... rather nebulously defined,

"is a member of" is a very clearly-defined idea: the element in question belongs to the set. It is simple, but fully defined.

"is a succesor of" is a very clearly-defined idea: the element in question is next to the set. It is simple, but fully defined.

Moreover:

http://www.internationalskeptics.com/forums/showpost.php?p=11263283&postcount=1235

http://www.internationalskeptics.com/forums/showpost.php?p=11266069&postcount=1275

http://www.internationalskeptics.com/forums/showpost.php?p=11266617&postcount=1277

http://www.internationalskeptics.com/forums/showpost.php?p=11266665&postcount=1279

http://www.internationalskeptics.com/forums/showpost.php?p=11269502&postcount=1344

http://www.internationalskeptics.com/forums/showpost.php?p=11270371&postcount=1361

http://www.internationalskeptics.com/forums/showpost.php?p=11270904&postcount=1365

, what about this prevents the existence of the set of all natural numbers?

http://www.internationalskeptics.com/forums/showpost.php?p=11264035&postcount=1249

http://www.internationalskeptics.com/forums/showpost.php?p=11266617&postcount=1277

 

[Nonpareil]

"is a succesor of" is a very clearly-defined idea: the element in question is next to the set.

I believe you meant next in.

http://www.internationalskeptics.com/forums/showpost.php?p=11264035&postcount=1249

http://www.internationalskeptics.com/forums/showpost.php?p=11266617&postcount=1277

None of this actually answers my question.

You say that a set is "incomplete" if it contains any element that is defined as the successor to a preceding element. Let us say that, for the moment, I accept this as writ.

I fail to see how this demonstrates that the set of all natural numbers cannot exist, even assuming that the set of all natural numbers is defined in terms of successors. In this context, "incomplete" just means "uses successor relationships in its definition". You seem to want to make the word "incomplete" pull double duty, and simultaneously imply that some element is necessarily missing - but you have not established this. You haven't even attempted to establish it.

All that your proposed concept of successors means is that, even if it is accepted, the set of all natural numbers becomes defined as "the set of all natural numbers and their successors". Slapping a meaningless "incomplete" on top of that doesn't actually change anything, since incomplete does not mean what you want it to here.

 

[doronshadmi]

"the set of all natural numbers and their successors".

What your wrote above is contradictory, since all does not hold if successor holds, and vice versa.

More about it is found, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=11266617&postcount=1277.

 

[Nonpareil]

What your wrote above is contradictory, since all does not hold if successor holds, and vice versa.

More about it is found, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=11266617&postcount=1277.

Yyyyyes it does.

All the post you have linked does is reiterate upon your definition of "incomplete", which is "makes use of the successor relationship". Which changes precisely nothing. "Incompleteness" still does nothing to say that a given member would be missing from the set of all natural elements, or that such a set cannot be formulated.

Again, you seem to want "incomplete" to mean something different, in this context, than what it has been defined to mean.

 

[doronshadmi]

"Incompleteness" still does nothing to say that a given member would be missing from the set of all natural elements, or that such a set cannot be formulated.

A set of the form {{y}$y} is inherently and permanently under construction.

A set of the form {{y}~$y} is inherently and permanently not under construction.

Any inductive set (where the set of natural number is a particular case of an inductive set) is of the form {{y}$y}.

 

[doronshadmi]

When you understand your avater (as found in http://www.internationalskeptics.com/forums/showpost.php?p=11264137&postcount=1251), you immediately understand {{y}$y} and how, for example, infinite numbers of the form 1... or 10... (as seen in http://www.internationalskeptics.com/forums/showpost.php?p=11259463&postcount=1185) can be used in Mathematics.

 

[doronshadmi]

, or that such a set cannot be formulated.

Please read vary carefully http://www.internationalskeptics.com/forums/showpost.php?p=11266069&postcount=1275, in order to understand the impact of Godel's incompleteness theorems on ZF(C), by using my non-standard definition of successor.

 

[doronshadmi]

{ {} }not-${ } provides completeness.


{ {} }${ },
{ {{}} }${ {} },
{ {{}, {{}}} }${ {}, {{}} },
{ {{}, {{}}, {{}, {{}}}} }${ {}, {{}}, {{}, {{}}} }

.. etc. ad infinitum, provides incompleteness.

Please be aware of the fact that any singleton set that is not a successor of the empty set, is not a member of the set of natural numbers (as constructed, for example, by Von Neumann method).

Order is insignificant since

{ {{}, {{}}, {{}, {{}}}} }${ {}, {{}}, {{}, {{}}} }
{ {} }${ },
{ {{}, {{}}} }${ {}, {{}} },
{ {{}} }${ {} },

... etc. ad infinitum, holds.

By the standard definition of successor, Successor = n u {n}

By my non-standard definition of successor, Successor is {n} only if {n} is used as a successor of n (which means that also {n}~$n holds).

 

[jsfisher]

Some set (is Complete) OR (is not-Complete) so your maneuvers are not must-have determinations for consistent reasoning.

If your maneuvers are used, then (IsComplete X <=> NOT({X} IsSuccessorOf X)) OR (NOT(IsComplete X) <=> {X} IsSuccessorOf X).

The result is the same.

Why do you insist on treating IsComplete X as somehow independent of NOT (IsComplete X)? Since NOT X <=> Y is equivalent to X <=> NOT Y, what you provided above reduces to

IsComplete X <=> NOT({X} IsSuccessorOf X​

You've posted quite a bit denying this, yet, once again, you confirm it to be the same thing you said before about the meaning of IsComplete X.

Will you continue to do the same with the IsSuccessorOf relation? It would save some time and trouble if you'd just confirm it is still

X IsSuccessorOf Y <=> X = {Y}​

 

[doronshadmi]

Why do you insist on treating IsComplete X as somehow independent of NOT (IsComplete X)? Since NOT X <=> Y is equivalent to X <=> NOT Y, what you provided above reduces to

IsComplete X <=> NOT({X} IsSuccessorOf X​

You've posted quite a bit denying this, yet, once again, you confirm it to be the same thing you said before about the meaning of IsComplete X.

Will you continue to do the same with the IsSuccessorOf relation? It would save some time and trouble if you'd just confirm it is still

X IsSuccessorOf Y <=> X = {Y}​

Why do you insist to avoid the reasoning, which according to it ({y} is a successor of y (and in that case y is incomplete)) OR ({y} is not a successor of y (and in that case y is complete))?

As can you see your reduction can't be used in order to define it.

In other words, your "NOT X <=> Y is equivalent to X <=> NOT Y" has nothing to do with the reasoning that enables the example given in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374.

 

[jsfisher]

Why do you insist to avoid the reasoning, which according to it ({y} is a successor of y (and in that case y is incomplete)) OR ({y} is not a successor of y (and in that case y is complete))?

The rules of boolean algebra do not support your reasoning.

 

[doronshadmi]

In other words, I was wrong when I have said that

(IsComplete X <=> NOT({X} IsSuccessorOf X)) OR (NOT(IsComplete X) <=> {X} IsSuccessorOf X) is the same as "Some set (is Complete) OR (is not-Complete)", where "Some set (is Complete) OR (is not-Complete)" is the used reasoning in my framework.

 

[Little 10 Toes]

Why do you insist on treating IsComplete X as somehow independent of NOT (IsComplete X)? Since NOT X <=> Y is equivalent to X <=> NOT Y, what you provided above reduces to

IsComplete X <=> NOT({X} IsSuccessorOf X​

You've posted quite a bit denying this, yet, once again, you confirm it to be the same thing you said before about the meaning of IsComplete X.

Will you continue to do the same with the IsSuccessorOf relation? It would save some time and trouble if you'd just confirm it is still

X IsSuccessorOf Y <=> X = {Y}​

Why do you insist to avoid the reasoning, which according to it ({y} is a successor of y (and in that case y is incomplete)) OR ({y} is not a successor of y (and in that case y is complete))?

As can you see your reduction can't be used in order to define it.

In other words, your "NOT X <=> Y is equivalent to X <=> NOT Y" has nothing to do with the reasoning that enables the example given in http://www.internationalskeptics.com/forums/showpost.php?p=11271084&postcount=1374.

Is there a reason why you don't answer the last part of jsfisher'a post? You know, the part about defining "is a successor of".

 

[jsfisher]

In other words, I was wrong when I have said that

(IsComplete X <=> NOT({X} IsSuccessorOf X)) OR (NOT(IsComplete X) <=> {X} IsSuccessorOf X) is the same as "Some set (is Complete) OR (is not-Complete)", where "Some set (is Complete) OR (is not-Complete)" is the used reasoning in my framework.

That may sound good in your head when you say it, but it is not math.