Refuting Incompleteness and Undefinability

[PeteOlcott]

I can easily assign types to variables, assigning them to numeric digits would not be required
because they are already defined by a regular expression of the lexical analyzer.

Why do you take the digits for granted?

I insist that you define everything. From first principles. In Lambda calculus.

Alphabet.
Digits.
Arithmetic.
Operators.

That is a the very best practice and I greatly commend you for suggesting it.
I would too but could not find anyone that knew how to specify arithmetic
using ASCII digits in lambda calculus.

[Logik]

I would too but could not find anyone that knew how to specify arithmetic
using ASCII digits in lambda calculus.

The ASCII table is just a hash map between integer and character types.

Neither of which exist in a Lambda universe. Until you define them.

You can start with the CHARACTERS (**NOT** digits) [ 1,2,3,4,5,6,7,8,9, 0 ].

Then convert them to digits.

Here's a Proof-of-concept: https://repl.it/@LogikLogicus/INTEGERS

Proof that digit(1) != integer(1)

[PeteOlcott]

I would too but could not find anyone that knew how to specify arithmetic
using ASCII digits in lambda calculus.

The ASCII table is just a hash map between integer and character types.

Neither of which exist in a Lambda universe. Until you define them.

You can start with the CHARACTERS (**NOT** digits) [ 1,2,3,4,5,6,7,8,9, 0 ].

Then convert them to digits.

Here's a Proof-of-concept: https://repl.it/@LogikLogicus/INTEGERS

Proof that digit(1) != integer(1)

It has to be actual UTF-8 characters because this is the system that I am assuming:
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings.

[Logik]

It has to be actual UTF-8 characters because this is the system that I am assuming:
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings.

Correct!

Which is fundamentally what Regular languages are ( https://en.wikipedia.org/wiki/Regular_language ) as based on regular expressions.

From the operations that are legal for strings - derive arithmetic.

[PeteOlcott]

It has to be actual UTF-8 characters because this is the system that I am assuming:
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings.

Correct!

Which is fundamentally what Regular languages are ( https://en.wikipedia.org/wiki/Regular_language ) as based on regular expressions.

From the operations that are legal for strings - derive arithmetic.

The intent is to fully formalize any thought that can be conceived in the mind, so
a regular language may not be enough. https://en.wikipedia.org/wiki/Chomsky_hierarchy

I envision that MTT would be enough, yet I do not see how MTT as defined could do string processing.
https://www.researchgate.net/publicatio ... y_YACC_BNF

[Logik]

The intent is to fully formalize any thought that can be conceived in the mind, so
a regular language may not be enough. https://en.wikipedia.org/wiki/Chomsky_hierarchy

Then you are shit out of luck. The Chomsky hierarchy is the set of ALL formal languages. RE + co-RE

https://en.wikipedia.org/wiki/RE_(complexity)

I envision that MTT would be enough, yet I do not see how MTT as defined could do string processing.
https://www.researchgate.net/publicatio ... y_YACC_BNF

MTT is Turing-complete. It's a Type-3 Chomsky grammar. https://en.wikipedia.org/wiki/Chomsky_h ... 3_grammars

If it's Turing-complete you are in control. You cam make the computer DO anything you tell it to. 1+1 = 3? Sure! No problem.

The architecture might hold you back thought... leaky abstractions.

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

[PeteOlcott]

The intent is to fully formalize any thought that can be conceived in the mind, so
a regular language may not be enough. https://en.wikipedia.org/wiki/Chomsky_hierarchy

Then you are shit out of luck. The Chomsky hierarchy is the set of ALL formal languages. RE + co-RE

https://en.wikipedia.org/wiki/RE_(complexity)

I envision that MTT would be enough, yet I do not see how MTT as defined could do string processing.
https://www.researchgate.net/publicatio ... y_YACC_BNF

MTT is Turing-complete. It's a Type-3 Chomsky grammar. https://en.wikipedia.org/wiki/Chomsky_h ... 3_grammars

If it's Turing-complete you are in control. You cam make the computer DO anything you tell it to. 1+1 = 3? Sure! No problem.

The architecture might hold you back thought... leaky abstractions.

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

The current set of all general knowledge can be expressed in a language with no more expressive
power than Chomsky Type-1.

The architecture should not hold me back because I am modeling the exact same connections
between ideas that the human mind references. The set of knowledge is organized in an inheritance
hierarchy in a very specific way.

[Logik]

The current set of all general knowledge can be expressed in a language with no more expressive
power than Chomsky Type-1.

Yes. Because all of our Turing machines (minds, mechanical computers etc.) have finite spacetime (memory).

If you had infinite space OR time - you would have a Type-0 grammar.

Any monkey with a typewriter could produce it. https://en.wikipedia.org/wiki/Infinite_monkey_theorem

The architecture should not hold me back because I am modeling the exact same connections
between ideas that the human mind references.

Yeah, but you are neglecting a fundamental issue. The human mind references itself.

It describes its own contents ALL the time.
In fact - you should think of language (formal or otherwise) as the mind describing itself. Self-expression.

Self-reference is recursion. Recursion is computation. https://en.wikipedia.org/wiki/Computability_theory
Lambda calculus is a formal language for expressing computation e.g a formal model of the mind

And thought (imagination) follows no rules except the ones you impose on yourself. Which is why Lambda calculus has no axioms.

Lambda calculus is architecture-agnostic. It's conceptual/abstract first.

Digital computers impose their implementation details on you. By virtue of the laws of physics.

[PeteOlcott]

The current set of all general knowledge can be expressed in a language with no more expressive
power than Chomsky Type-1.

Yes. Because all of our Turing machines (minds, mechanical computers etc.) have finite spacetime (memory).

If you had infinite space OR time - you would have a Type-0 grammar.

Any monkey with a typewriter could produce it. https://en.wikipedia.org/wiki/Infinite_monkey_theorem

The architecture should not hold me back because I am modeling the exact same connections
between ideas that the human mind references.

Yeah, but you are neglecting a fundamental issue. The human mind references itself.

It describes its own contents ALL the time.
In fact - you should think of language (formal or otherwise) as the mind describing itself. Self-expression.

Self-reference is recursion. Recursion is computation. https://en.wikipedia.org/wiki/Computability_theory
Lambda calculus is a formal language for expressing computation e.g a formal model of the mind

And thought (imagination) follows no rules except the ones you impose on yourself. Which is why Lambda calculus has no axioms.

Lambda calculus is architecture-agnostic. It's conceptual/abstract first.

Digital computers impose their implementation details on you. By virtue of the laws of physics.

Sure and I can agree with all that, yet none of it refutes anything that I said.
Even if Lambda Calculus has no actual axioms it still has some starting point
that acts as a proxy for axioms. So when you specify the relation that a {dog}
is and {animal} and later specify that a {dog} is not an {animal} it can detect
and report the contradiction. If we think of the initial things that it was told
as comparable to axioms, then on this basis it can detect the non truth of
expressions that is evaluates later.

[Logik]

So when you specify the relation that a {dog}
is and {animal} and later specify that a {dog} is not an {animal} it can detect
and report the contradiction.

But it's not a contradiction? It's just a redefinition of inheritance.
https://en.wikipedia.org/wiki/Inheritan ... ogramming)

In one context {dog} is {animal}. In another context {dog} is {pet}.
This is the philosophical/ontological error. A pet is not a thing. It's a type of thing. A dog is not a thing. It's a type of thing.

There is the type-definition. And then there is the individual instances of a particular type. https://en.wikipedia.org/wiki/Instance_ ... r_science)


The problem (as you pointed out somewhere else) is to define what an "error" is.
You need to define what a "contradiction" is....

You say 1 = 0 is a contradiction. OK. In one language it may be. If you so choose - construct a language in which it's not.

If we think of the initial things that it was told
as comparable to axioms, then on this basis it can detect the non truth of
expressions that is evaluates later.

Sure. But recognize that all axioms are arbitrary. You could simply choose a different set of axioms and evaluate different consequences.

[PeteOlcott]

Sure. But recognize that all axioms are arbitrary. You could simply choose a different set of axioms and evaluate different consequences.

So maybe according to some valid opinions when someone says that the category of {animal} of {dog} is a kind of {dump truck} then they would be correct from their frame of reference until they try to haul 3 tons of gravel with a {dog}.

[Logik]

So maybe according to some valid opinions when someone says that the category of {animal} of {dog} is a kind of {dump truck} then they would be correct from their frame of reference until they try to haul 3 tons of gravel with a {dog}.

Do you recognize that the way you are using the phrase {dump truck} is based on its fitness for a purpose e.g teleology.

Do you recognize that no formal system of truth takes teleology/instrumentalism into account?

Go ahead and formalize {dump truck} based on its teleology/instrumentality.

A dog is perfectly fine {dump truck} for disposing small volumes of left-over food

[PeteOlcott]

So maybe according to some valid opinions when someone says that the category of {animal} of {dog} is a kind of {dump truck} then they would be correct from their frame of reference until they try to haul 3 tons of gravel with a {dog}.

Do you recognize that the way you are using the phrase {dump truck} is based on its fitness for a purpose e.g teleology.

Do you recognize that no formal system of truth takes teleology/instrumentalism into account?

Go ahead and formalize {dump truck} based on its teleology/instrumentality.

A dog is perfectly fine {dump truck} for disposing small volumes of left-over food

Dump truck formalized using Rudolf Carnap (1952) Meaning Postulates
to have the property of moving three tons of gravel at once.
http://liarparadox.org/Meaning_Postulat ... p_1952.pdf