Eliminating Undecidability and Incompleteness in Formal Systems

[PeteOlcott]

You keep saying that knowing that is nothing was standard this communication
(using standard English) would be impossible. Why do you keep contradicting yourself?

Strawman.

English doesn't have inflexible rules like the ones you are trying to impose. And you have dodged the point over and over. How does one inject new meaning into a closed system with fixed/deterministic truth?

The constraints are for the purpose of communicating with math people,
they are not arbitrary.

Higher order predicate logic with types would be Chomsky level-2.
The semantics of its operators can be defined in a tiny subset of "c".

Now that is irony of epic proportions.

Observe what is happening here:

1. You are unable to communicate with Math people because you are missing:
* Operators like Recursion (:=) and WellFormulatedFormula()
* Predicates like Provable()

You are missing adequate tools to express yourself.

2. In order to communicate better with Math people ..... you are ..... INVENTING ..... NEW OPERATORS and NEW PREDICATES.

You are EVOLVING the language because the the one you have been given is... what? INSUFFICIENT to express... what you MEAN?

So you are trying to improve communication, by allowing yourself to express yourself better. And you have done that by? INVENTING....NEW....LANGUAGE.

Communication is transfer of meaning from one person to another via any medium (English, pictures, art, music, mathematics, video etc. etc. etc)
What you are trying to devise is an inflexible protocol.

Even mathematicians are moving into the 21st century with tools like Coq and going towards automated theorem proving: https://en.wikipedia.org/wiki/Coq

Methinks you are suffering from a bad case of the sunk cost fallacy.

Everything about predicate logic and the notion of formal system stays the same except:

This is stipulated as the formalized notion of True and False
(1) True x is a theorem of F (F ⊢ x)
(2) False ¬x is a theorem of F (F ⊢ ¬x)

From the law of the excluded middle (P ∨ ¬P) we derive:
(3) Boolean x is True or False in F True(F, x) ∨ False(F, x)

¬Boolean means that an expression lacks a Boolean property
just like non-declarative sentences have no truth value.
Every WFF containing a ¬Boolean term evaluates to ¬Boolean

0=False
1=True
2=¬Boolean

--A B A ∨ B
0 0 0
0 1 1
0 2 2
1 0 1
1 1 1
1 2 2
2 0 2
2 1 2
2 2 2

With just the above slight changes to the foundation of logic incompleteness is eliminated.

Since ¬Boolean is ¬True we can use this ¬True value when deciding expressions such as the
third step of the Tarski Undefinability proof:

Truth Predicate Axioms (Tarski notation)
(1) x ∈ Tr ↔ x ∈ Pr // True(x) ↔ (⊢ x)
(2) ¬x ∈ Tr ↔ ¬x ∈ Pr // False(x) ↔ (⊢ ¬x)
(3) x ∈ Tr ∧ ¬x ∈ Tr // Boolean(x) ↔ (True(x) ∨ False(x))
Theorem
(1) x ∉ Tr ↔ x ∉ Pr // ¬True(x) ↔ (⊬ x)

Anyone truly understanding the Tarski Undefinability proof would know that the whole proof would fail as soon as its third step would be proven false: (3) x ∉ Pr ↔ x ∈ Tr

Applying Truth Theorem (1) decides that Tarski's step(3) is false:
Swap the LHS of Tarski(3) [x ∉ Pr] that matches RHS of Theorem(1) [x ∉ Pr] with the LHS of Theorem(1) and we derive x ∉ Tr ↔ x ∈ Tr, which is clearly false, thus decidable.

[PeteOlcott]

You seem to be just playing head games. I really don't have time for that.

A physical system (called a computer) has DECIDED (via an algorithm that you can examine) that True ⇔ False

Weasel wording head games.

[Logik]

Weasel wording head games.

No. Just satisfying this bar for "proof" as established by Curry-Howard: https://en.wikipedia.org/wiki/Curry%E2% ... espondence

[PeteOlcott]

Weasel wording head games.

No. Just satisfying this bar for "proof" as established by Curry-Howard: https://en.wikipedia.org/wiki/Curry%E2% ... espondence

So essentially you are denying the foundation of all logic that 1 != 0.
You are saying that we can not take the fact that 1 does not equal 0 as a given
or define it as a truth.

[wtf]

So essentially you are denying the foundation of all logic that 1 != 0.
You are saying that we can not take the fact that 1 does not equal 0 as a given
or define it as a truth.

I should know better than to get involved here, but can you explain to me why 1 can't equal 0?

For example in the zero ring, 1 does equal 0. The zero ring is the number system consisting of a single element 0, with addition 0 + 0 = 0, and multiplication 0 x 0 = 0.

In this system, 0 has the property that if x is any element of the system, 0 times x = x. That's the definition of 1 in any ring. For example in the integers, 1 is the integer with the property that if x is any other integer, 1 times x equals x. (And x times 1 also equals x).

So in this system, 0 = 1. Moreover, any mathematician would instantly provide this example to show that 0 can be 1. I thought of it immediately when I saw your comment, and was not surprised to find a nice Wiki writeup on the subject. Well actually I was surprised because Wiki math articles can be good or bad, but this particular one is good and I recommend reading it.

There is no logical reason why it can't be; and in fact here is a model -- that is, a set with some operations on it -- in which 0 = 1.

By the way in the zero ring, you can divide by zero. That is, zero is its own multiplicative inverse. That's because

0 x 0 = 0 = 1.

You see that the above chain of equalities has the truth value TRUE in the zero ring. That's a model in which the statement becomes true.

You've bragged that you don't care about models, but you should. Some statements are tautologies, true in every model (or interpretation). And other statements, such as 0 = 1, are false in some models and true in others. I thought they explained all this in logic class.

[PeteOlcott]

So essentially you are denying the foundation of all logic that 1 != 0.
You are saying that we can not take the fact that 1 does not equal 0 as a given
or define it as a truth.

I should know better than to get involved here, but can you explain to me why 1 can't equal 0?

semantic tautology

For the same reason that no animal can be a airliner.

[wtf]

For the same reason that no animal can be a airliner.

How did you manage to not understand a word I wrote?

And how did you use the word "semantic" without the slightest clue what it means?

[PeteOlcott]

For the same reason that no animal can be a airliner.

How did you manage to not understand a word I wrote?

And how did you use the word "semantic" without the slightest clue what it means?

At the purely conceptual level one does not equal zero no matter what you call them.

We could attach the semantic meaning: {existence exists} to the finite
string: "one equals zero" yet at the purely conceptual semantic level
nothing is changed: {one does not equal zero} remains true.

[wtf]

At the purely conceptual level one does not equal zero no matter what you call them.

The string of symbols "0 = 1" has no truth value in isolation. When we supply an interpretation, it does in fact acquire a truth value. Some interpretations render it false; and others render it true. I gave an interpretation in which it's true.

I can't do anything about the fact that you can't or won't grapple with this in your mind and come to terms with it. But perhaps my posts will help clarify things for some other reader.

Syntax is just marks on paper. Semantics are what you get when you supply an interpretation for your symbols. An interpretation, or model, is a set of objects, along with a mapping from each symbol to some object or relation in the set. In this case 0 is the additive identity in the zero ring; 1 is the multiplicative identity; and = is usual equality.

I have therefore supplied an interpretation under which 0 = 1.

This example is a perfectly standard example in mathematics; so standard that it has not only a Wiki page, but an exceptionally clear one. I quote from the article:

The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide.

If this is something I made up, then at the very least Wikipedia is in on the conspiracy.

[PeteOlcott]

At the purely conceptual level one does not equal zero no matter what you call them.

The string of symbols "0 = 1" has no truth value in isolation. When we supply an interpretation, it does in fact acquire a truth value. Some interpretations render it false; and others render it true. I gave an interpretation in which it's true.

We can bypass all of this whole separate interpretation step and specify the semantic
meaning directly in the formal system as relations between and among finite strings.

Tarski never seemed to understand this with his separate object language and metalanguage.

[wtf]

We can bypass all of this whole separate interpretation step and specify the semantic
meaning directly in the formal system as relations between and among finite strings.

Can you please explain this point to me? How can marks drawn in sand or on paper or in a web browser's edit window have inherent meaning, without anyone assigning an interpretation to them?

Just consider natural language. How could a Martian tell, just by looking at the symbols, that "cat" and "el gato" and "le chat" refer to the same species of cute furry domesticated mammal?

Or that in a different context, or intepretation, the string "cat" refers to the symbolic operation of concatenating the contents of two files in a Unix-like shell? Or that when someone wrote a book called The Joy of Cats they were talking about the mathematical discipline of category theory? Or for that matter that they might be talking about one or those cute furry domesticated mammals, or they might be talking about a large and dangerous wild animal as you might find on safari in Africa?

How could anyone possibly know just by looking at a string of symbols, what meaning is intended, purely by analyzing the symbols? I really want to understand your point of view on this.

Tarski never seemed to understand this with his separate object language and metalanguage.

What a dummy. ‎Gödel and Turing too. A Pole, an Austro-Hungarian, and a Brit walk into a bar.

[PeteOlcott]

We can bypass all of this whole separate interpretation step and specify the semantic
meaning directly in the formal system as relations between and among finite strings.

Can you please explain this point to me? How can marks drawn in sand or on paper or in a web browser's edit window have inherent meaning, without anyone assigning an interpretation to them?

Just consider natural language. How could a Martian tell, just by looking at the symbols, that "cat" and "el gato" and "le chat" refer to the same species of cute furry domesticated mammal?

Or that in a different context, or intepretation, the string "cat" refers to the symbolic operation of concatenating the contents of two files in a Unix-like shell? Or that when someone wrote a book called The Joy of Cats they were talking about the mathematical discipline of category theory? Or for that matter that they might be talking about one or those cute furry domesticated mammals, or they might be talking about a large and dangerous wild animal as you might find on safari in Africa?

How could anyone possibly know just by looking at a string of symbols, what meaning is intended, purely by analyzing the symbols? I really want to understand your point of view on this.

Tarski never seemed to understand this with his separate object language and metalanguage.

What a dummy. ‎Gödel and Turing too. A Pole, an Austro-Hungarian, and a Brit walk into a bar.

They aren't just marks drawn. They are relations between finite strings specified
in a computer that can process these relations as if they were thoughts in a mind.

[Logik]

They aren't just marks drawn. They are relations between finite strings specified
in a computer that can process these relations as if they were thoughts in a mind.

I don't have the energy or patience anymore. Languages are more than just strings and relations.
They are open systems. They function as a whole: strings, grammar, operators.

The whole is greater than the sum of its parts

The fundamental holistic characters as a unity of parts which is so close and intense as to be more than the sum of its parts; which not only gives a particular conformation or structure to the parts, but so relates and determines them in their synthesis that their functions are altered; the synthesis affects and determines the parts, so that they function towards the whole; and the whole and the parts, therefore reciprocally influence and determine each other, and appear more or less to merge their individual characters: the whole is in the parts and the parts are in the whole, and this synthesis of whole and parts is reflected in the holistic character of the functions of the parts as well as of the whole.

You are yet to join the dots. The computer is modeled after the human mind. Not the other way around.

[PeteOlcott]

They aren't just marks drawn. They are relations between finite strings specified
in a computer that can process these relations as if they were thoughts in a mind.

I don't have the energy or patience anymore. Languages are more than just strings and relations.
They are open systems. They function as a whole: strings, grammar, operators.

The whole is greater than the sum of its parts

The fundamental holistic characters as a unity of parts which is so close and intense as to be more than the sum of its parts; which not only gives a particular conformation or structure to the parts, but so relates and determines them in their synthesis that their functions are altered; the synthesis affects and determines the parts, so that they function towards the whole; and the whole and the parts, therefore reciprocally influence and determine each other, and appear more or less to merge their individual characters: the whole is in the parts and the parts are in the whole, and this synthesis of whole and parts is reflected in the holistic character of the functions of the parts as well as of the whole.

You are yet to join the dots. The computer is modeled after the human mind. Not the other way around.

We stipulate this Haskell Curry notion of axiom:
The elementary statements which belong to T are called the elementary theorems of T and said to be true. In this way, a theory is a way of designating a subset of E which consists entirely of true statements. (Curry 2010)

Making Formal Systems correspond to Sound Deductive Inference
When we stipulate the Haskell Curry notion of axiom then formal proofs to theorem consequences simply correspond to sound deductive inference. By restricting the formal notion of Truth to sound deduction we eliminate incompleteness and inconsistency in formal systems.

[wtf]

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.