Cont: Deeper than primes - Continuation 2

[jsfisher]

What is written in http://www.internationalskeptics.com/forums/showpost.php?p=11264035&postcount=1249 has nothing to do with you wrote above.

In other words, you wrongly interpret http://www.internationalskeptics.com/forums/showpost.php?p=11266069&postcount=1275.

No problem. It was merely an attempt to extract some sense from your word salad.

It remains your task to give meaning to "is the successor of".

 

[doronshadmi]

It remains your task to give meaning to "is the successor of".

Already done in http://www.internationalskeptics.com/forums/showpost.php?p=11266617&postcount=1277 and http://www.internationalskeptics.com/forums/showpost.php?p=11265946&postcount=1273.

 

[jsfisher]

Already done in ...

Mere appearance in an attempt at an axiom does not a definition make. Nothing you've provided revolves whether something is or is not your variation on "a successor of" something else.

Is {A} a successor of A?
Is {A, B} a successor of A?
Is A a successor of {}?
Is {B} a successor of {A, B, C}?

How can we decide these questions based on the axiom you have proffered?

 

[doronshadmi]

Mere appearance in an attempt at an axiom does not a definition make. Nothing you've provided revolves whether something is or is not your variation on "a successor of" something else.

Is {A} a successor of A?
Is {A, B} a successor of A?
Is A a successor of {}?
Is {B} a successor of {A, B, C}?

Is {A} a successor of A? yes.
Is {A, B} a successor of A? no, {{A, B}} is a successor of {A, B}.
Is A a successor of {}? only if A is {{}}.
Is {B} a successor of {A, B, C}? no, {{A, B, C}} is a successor of {A, B, C}.

How can we decide these questions based on the axiom you have proffered?

{y} is a successor of y.

Once again, jsfisher, please handle with http://www.internationalskeptics.com/forums/showpost.php?p=11266696&postcount=1280 if you really wish to understand {y}$y.

 

[Little 10 Toes]

Is A a successor of {}? only if A is {{}}

So the empty set is a successor of the empty set.

 

[doronshadmi]

So the empty set is a successor of the empty set.

No, the singleton set {{}} is not the empty set {}.

 

[Little 10 Toes]

So when you say "is a successor of" you really mean, "is the singleton set of".

 

[jsfisher]

So when you say "is a successor of" you really mean, "is the singleton set of".

More simply still, "is the set of"...but Doron has denied this. I suspect it lacks the mandatory obfuscation.

 

[jsfisher]

...
{y} is a successor of y.
...

For arbitrary Y, are you stating that a successor of Y is {Y}? Are you also saying that that is the only thing that is the successor of Y? (Knowing what is a successor is not sufficient. We also need to identify what is not a successor.)

You denied this to be the case before. Has your position now reversed?

 

[doronshadmi]

So when you say "is a successor of" you really mean, "is the singleton set of".

Please define set y as the set of all singleton sets, in order to understand why {y} the a successor of y and not the singleton set of y.

 

[jsfisher]

Please define set y as the set of all singleton sets, in order to understand why {y} the a successor of y and not the singleton set of y.

No such set.

 

[doronshadmi]

For arbitrary Y, are you stating that a successor of Y is {Y}? Are you also saying that that is the only thing that is the successor of Y?

Yes.

(Knowing what is a successor is not sufficient. We also need to identify what is not a successor.)

Any member of y is not a successor of y.

You are invited to define {y} as a member of y, and see what happens by yourself.

 

[doronshadmi]

No such set.

Exactly because {y} as a successor of y prevents the completeness of y.

 

[jsfisher]

[I inadvertently sent this to Doronshadmi as an PM. My apologies, Doron. It was meant for the thread.]

For arbitrary Y, are you stating that a successor of Y is {Y}? Are you also saying that that is the only thing that is the successor of Y?

Yes.

Then, just as I stated before, your added conjunction term is a tautology. Your change accomplishes nothing.

 

[jsfisher]

Exactly because {y} as a successor of y prevents the completeness of y.

No, that bunch of gibberish has nothing to do with it. The "set of all singleton sets" is not a set because it is not permitted under the set theory axioms.

 

[doronshadmi]

Then, just as I stated before, your added conjunction term is a tautology. Your change accomplishes nothing.

On the contrary, jsfisher, what you recognize as a tautology, prevents the completeness of y.

This time please deal with the rest of post http://www.internationalskeptics.com/forums/showpost.php?p=11267561&postcount=1292.

 

[doronshadmi]

No, that bunch of gibberish has nothing to do with it. The "set of all singleton sets" is not a set because it is not permitted under the set theory axioms.

Yes I know.

ZF(C) is a weak set theory that can't deals with notions like "{y} is a successor of y".

 

[jsfisher]

Yes I know.

ZF(C) is a weak set theory that can't deals with notions like "{y} is a successor of y".

It deals with it just fine, as shown previously. Irrelevant to the issue at hand, though, which was the non-existence of a set of all singleton sets.

 

[jsfisher]

On the contrary, jsfisher, what you recognize as a tautology, prevents the completeness of y.

Fascinating. Perhaps it is time for you to define what you mean by completeness. If a tautology "prevents it", then incompleteness must be a tautology, too.

 

[doronshadmi]

It deals with it just fine,

By preventing the existence of incomplete sets, exactly because {y}$y is not defined by ZF(C).