Cont: Deeper than primes - Continuation 2

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jsfisher said:
Mathematics supports me in this.
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Indeed your Mathematics supports you in this.

Since by my mathematics X is not a free variable, X is simply at least two options: X, ~X.

On the basis of X two options, binary logical connectives are used.
 
Take for example T.

It is not a free variable, yet T has at least two options, which are T,~T.

Any binary logical connective between the two options, is defined by at least four rows.
 
doronshadmi said:
Indeed your Mathematics supports you in this.

Since by my mathematics X is not a free variable, X is simply at least two options: X, ~X.
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Making up your own meanings of terms again? I won't ask you for your definition because (1) history bears out that you will be unable to provide a definition and (2) since it isn't Mathematics, I do not care.
 
I know it isn't the definition everyone has been waiting on but I thought I'd at least give you troopers something, How you've all kept your sanity is admirable.

The successor operation is a primitive recursive function S such that S(n) = n+1 for each natural number n. For example, S(1) = 2 and S(2) = 3.
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Nay_Sayer said:
I know it isn't the definition everyone has been waiting on but I thought I'd at least give you troopers something, How you've all kept your sanity is admirable.

The successor operation is a primitive recursive function S such that S(n) = n+1 for each natural number n. For example, S(1) = 2 and S(2) = 3.
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Yes, that is the conventional one. It comes up in the Peano Axioms for the natural numbers.

John von Neumann modeled the natural numbers in set theory by defining S(X) to be X U {X}. The definition is well chosen because it has useful properties I won't belabor here. In what is now the most widely accepted formalization for set theory (ZFC), von Neumann's successor function is reflected in the Axiom of Infinity. Other versions of the axiom are built on other functions, but the von Neumann function provides a natural foundation to build the natural numbers on top of set theory, that is, the Peano Axioms.

The issue for this thread, though, is Doronshadmi's poorly expressed claim of a deficiency in infinite sets because of The Successor FunctionTM. The one included in ZFC (but not all others) isn't set theory's successor function, per se, but just a very useful tool for the Axiom of Infinity that simplifies the bridge to the natural numbers.

It does not, in any way, impart features into sets that make them complete or incomplete (whatever that term is supposed to mean).
 
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jsfisher said:
Yes, that is the conventional one. It comes up in the Peano Axioms for the natural numbers.

John von Neumann modeled the natural numbers in set theory by defining S(X) to be X U {X}. The definition is well chosen because it has useful properties I won't belabor here. In what is now the most widely accepted formalization for set theory (ZFC), von Neumann's successor function is reflected in the Axiom of Infinity. Other versions of the axiom are built on other functions, but the von Neumann function provides a natural foundation to build the natural numbers on top of set theory, that is, the Peano Axioms.
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Some examples:

Von Neumann model: {{}=0, {{}}=1 ,{{},{{}}}=2, {{},{{}},{{},{{}}}}=3, ...}

Zeremelo model: {{}=0, {{}}=1, {{{}}}=2, {{{{}}}}=3, ...}

jsfisher said:
The issue for this thread, though, is Doronshadmi's poorly expressed claim of a deficiency in infinite sets because of The Successor FunctionTM. The one included in ZFC (but not all others) isn't set theory's successor function, per se, but just a very useful tool for the Axiom of Infinity that simplifies the bridge to the natural numbers.
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In both models, outer braces are used but not logically addressed.

As a result any given set theory that uses outer braces as a notational convenience, is not entirely logically addressed.

jsfisher said:
It does not, in any way, impart features into sets that make them complete or incomplete (whatever that term is supposed to mean).
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jfisher airs his view about terms even if he does not know what they mean.
 
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By not using free variables I logically address outer braces as tautology (always true) and the void between outer braces as contradiction (always false).

Members are logically addressed as ~contradiction AND ~tautology.

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

Some analogy: If successor is the horizon, then any attempt to actually reach it, is not satisfied.

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

Some analogy: If successor is the horizon, then any attempt not to actually reach it, is satisfied.

Examples:

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

or

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

are two examples of an incomplete set, where each member is a complete set.

http://www.internationalskeptics.com/forums/showpost.php?p=11342271&postcount=1804 is an example of 2-valued logic that without loss of generality is consistently expandable into multi-valued logic.
 
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jsfisher said:
You denying they are free variables doesn't change the fact they are free variables. Mathematics does not submit to your arbitrary rule changes.
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By your mathematical framework, free variables are used to define truth tables.

By my mathematical framework, free variables are not used to define truth tables.

The arbitrary rule is entirely your arbitrary axiom not to address outer braces logically (no need of free variables).
 
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By logically address outer braces, also Fuzzy logic is logically addressed as follows:

1 = outer braces that are logically addressed as tautology = {} or {}

0 = the void between two pairs of outer braces, which is logically addressed as contradiction = {}{}

~0 AND ~1 is logically ~contradiction AND ~tautology = {{}}

Again, no free variables are used, as follows:

Tautology = {} or {}

Contradiction = {}{}

~({} or {}) AND ~({}{}) = {{}}
 
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doronshadmi said:
By your mathematical framework, free variables are used to define truth tables.
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Ignoring your improper use of the word, define, that would coincide with the definition of free variables. Truth tables are, of course, not the only place they arise, though.

By my...framework, free variables are not used to define truth tables.
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You do this both in ignorance of Mathematics and the meaning of words.
 
doronshadmi said:
They are used but not logically addressed
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Nope. Braces are not a formal part of set theory. If they were, they'd appear in the axioms in some meaningful way and be defined either explicitly or implicitly by them. They do not.
 
jsfisher said:
Ignoring your improper use of the word, define,
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Your maneuvers around improper use of the words are not changing the fact that your framework is not entirely logically addressed, exactly because outer braces are not logically addressed (and again, no free variables are need in order to logically address them).

Moreover, also the void is not logically addressed by your mathematical framework.
 
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jsfisher said:
Nope. Braces are not a formal part of set theory.
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This is the realm of your mathematical framework, which is not entirely logically addressed, exactly because outer braces are not logically addressed (and again, no free variables are need in order to logically address them).

Moreover, also the void is not logically addressed by your mathematical framework.

Furthermore, you avoid http://www.internationalskeptics.com/forums/showpost.php?p=11362427&postcount=1943.

Why is that, jsfisher? Is it because I wrote
doronshadmi said:
Take for example T.

It is not a free variable, yet T has at least two options, which are T,~T.

Any binary logical connective between the two options, is defined by at least four rows.
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?
 
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Little 10 Toes said:
That's so funny. You do circular definitions, make up words and then can't define them. And then afterwards, claim that you dont need the words you make up.
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http://www.internationalskeptics.com/forums/showpost.php?p=11362427&postcount=1943 is logically straightforward, and this is exactly what is done in http://www.internationalskeptics.com/forums/showpost.php?p=11363027&postcount=1948 or http://www.internationalskeptics.com/forums/showpost.php?p=11363119&postcount=1953.

As for circularity, please show it in the 3 links above.
 
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