Eliminating Undecidability and Incompleteness in Formal Systems

[PeteOlcott]

The elementary statements which belong to T are called the elementary theorems of T and said to be true.

Right. They are SAID to be true. Like in chess, the fact that the knight can jump over other pieces is SAID to be true. It's not a law of nature. It's colloquially said to be true to get the game off the ground.

Just like in Euclidean geometry, the parallel postulate is said to be true. And in non-Euclidean geometry, it's said to be false.

None of these things are true or false in an absolute sense. They are SAID to be true in order to get a deductive system or formal game going.

You misunderstood this simple point 22 years ago and you've been deluding yourself all this time. And now you can't even see that your own sources refute your claims. Just like when you insisted that I read your link earlier, and when I did I found it directly refuted your own point.

{Axiom} can retain its original definition and it can acquire a new sense meaning.
Axiom(1) Expressions of language that are hypothetically thought of as true so see where the reasoning leads.

Axiom(2) Expressions of language that are defined to have the semantic value of True to form the basis of the foundation of the whole notion of truth.

When we say that a {dog} is an {animal} specifying a relation between two conceptual classes this is not merely a hypothetical possibility it is a necessary truth that is entirely derived from the meaning of the terms {dog} and {animal}.

All of semantic meaning can be fully specified in a single language expressing relations between finite strings.

Example: “a cat is an animal”.
Formalized as: ("cat" ◁ "animal")
where ◁ is the [is_a_type_of] operator adapted from UML Inheritance relation.
The only reason that we know that “a cat is an animal” is that it is defined to be True.

[PeteOlcott]

You misunderstood this simple point 22 years ago and you've been deluding yourself all this time. And now you can't even see that your own sources refute your claims. Just like when you insisted that I read your link earlier, and when I did I found it directly refuted your own point.

https://en.wikipedia.org/wiki/Stipulative_definition

Stipulative Definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

It is only by axioms of this type that a cat is an animal.

[Logik]

You misunderstood this simple point 22 years ago and you've been deluding yourself all this time. And now you can't even see that your own sources refute your claims. Just like when you insisted that I read your link earlier, and when I did I found it directly refuted your own point.

https://en.wikipedia.org/wiki/Stipulative_definition

Stipulative Definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

It is only by axioms of this type that a cat is an animal.

From your own link:

A stipulative definition is a type of definition in which a new or currently-existing term is given a new specific meaning for the purposes of argument or discussion in a given context. When the term already exists, this definition may, but does not necessarily, contradict the dictionary (lexical) definition of the term. Because of this, a stipulative definition cannot be "correct" or "incorrect"

Which is exactly what wtf has been saying. Without an interpretative framework you can't DECIDE whether a definition is "correct" or "incorrect".

If it is "only by axioms of this type that a cat is an animal" and I can CHOOSE my axioms...
The logical conclusion is that you can CHOOSE your "Truth".

What is an "incorrect choice" ?

it can only differ from other definitions, but it can be useful for its intended purpose

Pete has not stated any "intended purpose".
It leads to teleology

[PeteOlcott]

You misunderstood this simple point 22 years ago and you've been deluding yourself all this time. And now you can't even see that your own sources refute your claims. Just like when you insisted that I read your link earlier, and when I did I found it directly refuted your own point.

https://en.wikipedia.org/wiki/Stipulative_definition

Stipulative Definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

It is only by axioms of this type that a cat is an animal.

From your own link:

A stipulative definition is a type of definition in which a new or currently-existing term is given a new specific meaning for the purposes of argument or discussion in a given context. When the term already exists, this definition may, but does not necessarily, contradict the dictionary (lexical) definition of the term. Because of this, a stipulative definition cannot be "correct" or "incorrect"

Which is exactly what wtf has been saying. Without an interpretative framework you can't DECIDE whether a definition is "correct" or "incorrect".

If it is "only by axioms of this type that a cat is an animal" and I can CHOOSE my axioms...
The logical conclusion is that you can CHOOSE your "Truth".

What is an "incorrect choice" ?

it can only differ from other definitions, but it can be useful for its intended purpose

Pete has not stated any "intended purpose".
It leads to teleology

I really should not have to stipulate that: ¬(True ↔ False) that would be nuts.

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

Stipulating these specifications of True and False
(1) True(F, x) ↔ (F ⊢ x)
(2) False(F, x) ↔ (F ⊢ ¬x)

Then formal proofs to theorem consequences specify sound deduction from true premises to true conclusions preventing truth and provability from diverging.

An argument is unsound unless there is a connected chain of deductive inference steps from true premises to a true conclusion, truth and provability cannot diverge. In other words sound deduction species Axiom(1). When sound deduction is applied to the negation of a conclusion then it species Axiom(2).

Applying the law of the excluded middle (P ∨ ¬P) to the above we derive:
(3) Sound(F, x) ↔ (True(F, x) ∨ False(F, x))

[Logik]

I really should not have to stipulate that: ¬(True ↔ False) that wold be nuts.

This doesn't make ANY sense to you, does it?

It depends on what you mean by True and False. We have many theories of truth
The above is not even a complete sentence! You haven't stipulated anything. You have simply expressed: ¬(True ↔ False)

Depending on the framework in which your expression is evaluated it could result in true, false, 1, 0 or 42.

Stipulating this definition of Axiom:
An expression of language defined to have the semantic value of Boolean True.

It seems to me that you have defaulted to Boolean semantics. Why?
Why not many-valued semantics?

Personally, I hate Boolean logic. It leads to black-and-white thinking. It robs you of nuance.

[PeteOlcott]

This doesn't make ANY sense to you, does it?

It depends on what you mean by True and False. We have many theories of truth
The above is not even a complete sentence! You haven't stipulated anything. You have simply expressed: ¬(True ↔ False)

Depending on the framework in which your expression is evaluated it could result in true, false, 1, 0 or 42.

That is just not the way that truth actually works.

There is no possible world where that actual concept of an integer is the type
of animal known as a dog.

In some possible worlds the integer {five} could be encoded as "dog" yet it
still remains the integer {five}.

I use curly braces to represent the pure concept apart from its symbolic
representation and quoted strings to indicate the symbolic representation.
{True} ↔ "True"

¬({True} ↔ {False}) remains {True} even if {True} is encoded as "a box of rocks".

[Logik]

That is just not the way that truth actually works.

There is no possible world where that actual concept of an integer is the type
of animal known as a dog.

In some possible worlds the integer {five} could be encoded as "dog" yet it
still remains the integer {five}.

I use curly braces to represent the pure concept apart from its symbolic
representation and quoted strings to indicate the symbolic representation.
{True} ↔ "True"

¬({True} ↔ {False}) remains {True} even if {True} is encoded as "a box of rocks".

I just invented the realm you claim does not exist.

It's a concept in my head. Meet Dog The FIve; and even his friend: Dog The Six.
Also meet the Twins. Dog Seven and Dog Seven.

https://repl.it/repls/OldCriminalMultithreading

Code: Select all

class Dog(object):
  def __init__(self,name):
    self.name=name
  
  def __eq__(self, other):
    if type(other) == int:
      return self.name == other
    else:
      return False

print(Dog(5) == 5)
=> True
print(Dog(6) == 6)
=> True
print(Dog(7) == Dog(7))
=> False

I think you are a very long way off from telling how "how Truth works"...

[PeteOlcott]

That is just not the way that truth actually works.

There is no possible world where that actual concept of an integer is the type
of animal known as a dog.

In some possible worlds the integer {five} could be encoded as "dog" yet it
still remains the integer {five}.

I use curly braces to represent the pure concept apart from its symbolic
representation and quoted strings to indicate the symbolic representation.
{True} ↔ "True"

¬({True} ↔ {False}) remains {True} even if {True} is encoded as "a box of rocks".

I just invented the realm you claim does not exist.

It's a concept in my head. Meet Dog The FIve; and even his friend: Dog The Six:

There is no {living animal} that IS an {integer} because they are mutually exclusive
classes. The concepts will not fit together coherently in the same way that a {square
circle} cannot possibly be anything more than a misconception.

I tried and tried and tried to make a square circle to test this concept.
The closest possible thing to a {square circle} was a thing that looked
like a square from one two-dimensional perspective and looked like a circle
from another different two dimensional perspective. In actual 3D reality
it was neither a square nor a circle.

Saying that {True} ↔ {False} is the same thing as saying that {square circles}
exist, the only difference is that the impossibility of square circles is easier
to understand than the impossibility of {True} ↔ {False}.

This is the general principle:
Nothing with simultaneous mutually exclusive properties exists.

[Logik]

There is no {living animal} that IS an {integer} because they are mutually exclusive
classes. The concepts will not fit together coherently in the same way that a {square
circle} cannot possibly be anything more than a misconception.

Misconception?
When did you get to tell me how and what to think?

Every heard of multiple inheritance

It works like this: https://repl.it/repls/UpbeatScientificBots

Code: Select all

class LivingAnimal(object):
  pass

class Integer(object):
  pass

class Dog(LivingAnimal, Integer):
  pass

I tried and tried and tried to make a square circle to test this concept.
The closest possible thing to a {square circle} was a thing that looked
like a square from one two-dimensional perspective and looked like a circle
from another different two dimensional perspective. In actual 3D reality
it was neither a square nor a circle.

Can you even tell the difference between a Type 1 and Type 2 Chomsky grammars?

Square and circle are context-sensitive. They imply 2 dimensions. They imply an Euclidian space.

Something which this Universe isn't. You do know that squares and circles don't actually exist, right?

[wtf]

I tried and tried and tried to make a square circle to test this concept.

The unit circle is a square in the taxicab metric. The Wiki page has a picture of it.

https://en.wikipedia.org/wiki/Taxicab_geometry

[PeteOlcott]

I tried and tried and tried to make a square circle to test this concept.

The unit circle is a square in the taxicab metric. The Wiki page has a picture of it.

https://en.wikipedia.org/wiki/Taxicab_geometry

I am referring to each individual sense meaning as a unique unit.
The fact that the same finite string is associated with different meanings
does not coherently combine the [geometric object of a circle] with the
{geometric object of a square} such that a [geometric square circle] is formed.
This is TOTALLY IMPOSSIBLE because they are mutually exclusive classes.

It is a single general principle that can be ignored yet not refuted
[no object requiring simultaneous mutually exclusive properties exists].

[wtf]

The fact that the same finite string is associated with different meanings
does not coherently combine the [geometric object of a circle] with the
{geometric object of a square} such that a [geometric square circle] is formed.
This is TOTALLY IMPOSSIBLE because they are mutually exclusive classes.

I just showed you the counterexample.

It is a single general principle that can be ignored yet not refuted
[no object requiring simultaneous mutually exclusive properties exists].

Then it follows that being a circle and being a square are not mutually exclusive properties. The unit circle in the taxicab metric is a square.

[PeteOlcott]

The fact that the same finite string is associated with different meanings
does not coherently combine the [geometric object of a circle] with the
{geometric object of a square} such that a [geometric square circle] is formed.
This is TOTALLY IMPOSSIBLE because they are mutually exclusive classes.

I just showed you the counterexample.

It is a single general principle that can be ignored yet not refuted
[no object requiring simultaneous mutually exclusive properties exists].

Then it follows that being a circle and being a square are not mutually exclusive properties. The unit circle in the taxicab metric is a square.

You are only giving the same word another meaning, the meanings themselves remain distinct.

[wtf]

You are only giving the same word another meaning, the meanings themselves remain distinct.

No, I used the original meanings. A circle is a set of points equidistant from a given point. You agree, right? And a square is a 4-sided quadrilateral with all sides equal and all angles equal. Right? Right.

And with those definitions -- the original, normal, everyday definitions -- the unit circle in the taxicab metric is a square.

Let me add that a perfectly sensible response on your part would be: "Wow that's an interesting example. I didn't know about it because I don't study math. But now that you pointed it out, I'll stop using that example and instead talk about a married bachelor. That conveys the same idea that an object can't have contradictory properties; without being subject to this particular counterexample that shows that being a circle and being a square are not in fact contradictory."

Wouldn't that be a sensible response? Instead, you are trying to claim that I'm changing definitions. But in fact I'm not. I'm using the perfectly standard definitions of circles and squares. It's a mathematical curiosity that the unit circle is a square in the taxicab metric. Once again those dastardly fiends at Wikipedia are in on the conspiracy.

[Logik]

I'll stop using that example and instead talk about a married bachelor. That conveys the same idea that an object can't have contradictory properties; without being subject to this particular counterexample that shows that being a circle and being a square are not in fact contradictory."

The trivial counter-example is the married bachelor of sciences.

That pesky semantic overload...

Here is a function which decides whether any particular person is a bachelor: https://repl.it/repls/CruelHiddenTechnician

Code: Select all

def is_bachelor(person):
  if person.sex == 'Male' and not person.marital_status:
    return True
  elif person.bachelor_degree:
    return True
  else:
    return False