Cont: Deeper than primes - Continuation 2

[doronshadmi]

Still haven't defined it. Only described it.

Try again.

Little 10 Toes, it seems that you are able to distinguish between a description and a definition of a given wff that is at least (objects) AND (possible relations among objects).

In that case please use your distinction abilities in order to define successor.

The stage is yours

 

[doronshadmi]

Meanwhile please look at https://www.youtube.com/watch?v=sGCtMKthRh4 and http://math.mit.edu/~bush/wordpress/wp-content/uploads/2014/09/Bush-ARFM-2015.pdf.

In that movie drops of liquid is what I call objects, where the waves around them are their possible relations with other objects.

 

[doronshadmi]

A formula of propositional logic is a tautology if the formula itself is always true regardless of which valuation is used for the propositional variables.

There are infinitely many tautologies. Examples include:

(A ∨ ¬A) ("A or not A"), the law of the excluded middle. This formula has only one propositional variable, A. Any valuation for this formula must, by definition, assign A one of the truth values true or false, and assign ¬A the other truth value.

( https://en.wikipedia.org/wiki/Tautology_(logic)#Definition_and_examples )

So (A ∨ ¬A) is about a variable called A, where by using this variable no distinction between being an object and being a possible relation of this object w.r.t objects, is actually done.

Here is an example of a reasoning that distinguishes between being an object and being a possible relation of this object w.r.t objects:

There is an inexhaustible energy source, notated as object {X}.

There is a car, notated as object X.

If {X} is related to X, this relation is notated as {X}$X, and as a result X location is constantly changed (X location is indeterminable).

If {X} is not related to X in terms of an inexhaustible energy source, this relation is notated as {X}~$X, and as a result X location is not constantly changed (X location is determinable).

(X location is indeterminable) is equivalent to (X is incomplete).

(X location is determinable) is equivalent to (X is complete).

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=11351017&postcount=1859 and http://www.internationalskeptics.com/forums/showpost.php?p=11350375&postcount=1857.

 

[doronshadmi]

By standard or currently alternative set theories, the outer "{" and "}" are used but not mathematically defined.

By my set theory framework, the outer "{" and "}" are used and mathematically defined.

http://www.internationalskeptics.com/forums/showpost.php?p=11342271&postcount=1804 define {||} as NOthing (contradiction) and |{}| as YESthing (tautology), where being a member is irreducible into NOthing and not expandable into YESthing.

By analyzing the intermediate level of members such that the outer "{" and "}" is not ignored, being a successor is exactly the outer "{" and "}" of any member, for example:

{ {}=0, {{}}=1, {{{}}}=2, {{{{}}}}=3,... }

By understanding all what is written above, one immediately realizes that an inductive set of natural numbers is incomplete.

 

[jsfisher]

By standard or currently alternative set theories, the outer "{" and "}" are used but not mathematically defined.

The braces are not part of set theory at all. They are a notational convenience. They are, however, defined.

 

[doronshadmi]

The braces are not part of set theory at all. They are a notational convenience. They are, however, defined.

Are they defined not only in terms of notational convenience, by standard or alternative set theories?

If yes, then please write such definition.

 

[jsfisher]

Are they defined not only in terms of notational convenience, by standard or alternative set theories?

If yes, then please write such definition.

They are not part of set theory.

 

[doronshadmi]

They are not part of set theory.

The difference between {X} and X clearly demonstrates that outer "{" and "}" are parts of set theory.

Here is another correction of what I wrote before:

By using outer "{" and "}" as a successor, any amount of members that approaches but not reaches "{" and "}", is defined as incomplete.

By using outer "{" and "}" as a successor, any amount of members that does approach "{" and "}", is defined as complete.

{ {}=0, {{}}=1, {{{}}}=2, {{{{}}}}=3,... } is some example of an incomplete set, where each member is complete.

 

[MetalPig]

In that case A and ~A are both true

And therefore not eachother's negation. Thank you.

 

[doronshadmi]

And therefore not eachother's negation. Thank you.

Wrong, since by your own words you "wasn't writing a logic statement" (as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=11349782&postcount=1854) in case of "A and ~A" you can't conclude logically anything about A,~A.

Generally, you do not comprehend http://www.internationalskeptics.com/forums/showpost.php?p=11350375&postcount=1857, exactly because you do not use logic (ant kind of logic) in case of "A and ~A".

By my reasoning (that can't comprehended by your reasoning) A is at least (objects) AND (possible relations among objects), and by following this determination http://www.internationalskeptics.com/forums/showpost.php?p=11350375&postcount=1857 is logically "crystal clear".

 

[MetalPig]

Wrong, since by your own words you "wasn't writing a logic statement"

I used your words this time.

 

[doronshadmi]

I used your words this time.

It means that you still do not get A as at least (objects) AND (possible relations among objects), such that A OR ~A is not simultaneously taken (yet each option holds).

Your response in http://www.internationalskeptics.com/forums/showpost.php?p=11337861&postcount=1726 to http://www.internationalskeptics.com/forums/showpost.php?p=11337788&postcount=1723 clearly demonstrates that you can't comprehend that A OR ~A is not simultaneously taken (yet each option holds exactly because unlike AND logical connective that forced A,~A to be taken simultaneously, OR logical connective does not force A,~A to be taken simultaneously, and this is exactly the reason of why the last row of A OR ~A truth table is T OR T --> T).

 

[MetalPig]

It means that you still do not get A as at least (objects) AND (possible relations among objects), such that A OR ~A is not simultaneously taken (yet each option holds).

Explain how that makes contradictions true:

In that case A and ~A are both true

 

[jsfisher]

The difference between {X} and X clearly demonstrates that outer "{" and "}" are parts of set theory.

More proof by assertion, Doron? You continue to conflate set theory and notation.

 

[doronshadmi]

More proof by assertion, Doron? You continue to conflate set theory and notation.

Wrong, the outer "{" and "}" is defined as a tautology that is inaccessible to members, and it is the complement of contradiction, which is the void between "{" and "}" that members are irreducible to it.

 

[doronshadmi]

Explain how that makes contradictions true:

~A and A, such that and is not the logical connective AND, is equivalent to A OR ~A that logically defines that A has two optional properties (for example: being OR not being a successor) that simply are not simultaneously taken.

Again, being OR not being a successor are true options of A, and this fact is defined by A OR ~A, such that the two options are not simultaneously taken, unlike in the case of ~A AND A, where the two options are simultaneously taken and the result is contradiction since A can't be successor AND not successor, but it defiantly can be successor OR not successor, and this is exactly the logical meaning of A OR ~A that is determined by its optional properties that, again, are not simultaneously taken.

Please read the rest of http://www.internationalskeptics.com/forums/showpost.php?p=11355310&postcount=1872 for better understanding of the difference between AND and OR four rows truth tables.

 

[jsfisher]

Wrong, the outer "{" and "}" is defined as a tautology that is inaccessible to members

If that were the case, then it would be no problem whatsoever for you to prove it. Please show where in Mathematics, (1) the outer "{" and "}" are defined as a tautology, (2) that they are inaccessible to members, and (3) what "inaccessible to members" even means.

...and it is the complement of contradiction, which is the void between "{" and "}" that members are irreducible to it.

Ditto.

 

[doronshadmi]

If that were the case, then it would be no problem whatsoever for you to prove it. Please show where in Mathematics, (1) the outer "{" and "}" are defined as a tautology, (2) that they are inaccessible to members, and (3) what "inaccessible to members" even means.



Ditto.

As much as I know, the void between the outer "{" and "}" is defined as "always false" (contradiction) but currently no mathematician defined the outer "{" and "}" as "always true" (tautology), so you are probably right that I am that first person that defines the outer "{" and "}" as "always true" (tautology).

By doing this, the void between the outer "{" and "}" and the outer "{" and "}" are logically defined (the outer "{" and "}" are not a notational convenience anymore since they are logically defined).

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=11342271&postcount=1804.

Some example:

By using the outer "{" and "}" as a successor, any amount of members that approaches but not reaches "{" and "}", is defined as incomplete.

By using the outer "{" and "}" as a successor, any amount of members that does not approach "{" and "}", is defined as complete.

{ {}=0, {{}}=1, {{{}}}=2, {{{{}}}}=3,... } is some example of an incomplete set, where each member is a complete set.

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

By looking again on what I wrote above I have realized that I do not have to define successor as an optional property of the concept of set, so from now on, all what I wrote about successor as an optional property, is not considered anymore.

 

[MetalPig]

and the result is contradiction

I know that. Explain why in your system cantradictions can be true.

Or just admit that in my car example your ~A is not the negation of A.

 

[doronshadmi]

I know that. Explain why in your system cantradictions can be true.

1) My system does not need the optional anymore in order to logically be defined, please see http://www.internationalskeptics.com/forums/showpost.php?p=11356821&postcount=1878 for more details.

2) If the optional is considered by a given system, then A is at least (objects) AND (possible relations among objects).

In that case A is defined also by its optional properties, where one of the cases of being optional is that an optional property is used OR not used, where being used OR not being used are true options of A that are not simultaneously taken.

~A is the case that A's optional property is not used.

A is the case that A's optional property is used.

Both cases are true options of A, but since they are not simultaneously taken (by using OR logical connective) A OR ~A --> True.

On the contrary, if both cases are true options of A, but they are simultaneously taken (by using AND logical connective) A AND ~A --> False.

Or just admit that in my car example your ~A is not the negation of A.

Since your car example does not deal with the optional logically

Let's say SPQ is used and my car moves. Now A and ~A are both true. Therefore ~A is not the negation of A.

I used 'and', not AND. If I meant AND, I would have written something like
(A AND ~A) is True
but I didn't. I wasn't writing a logic statement.

it can't logically comprehend A in terms of the optional.