The following is based on the material from the above link:

It combines everything into a single Turing Machine definition

and **corrects, clarifies and simplifies** the original text.

The second start state of Ĥ has been renamed to qx.

**Definition of Turing Machine Ĥ (state transition sequence)**

**q0 ŵ ⊢* Ĥ qx ŵ ŵ ⊢* Ĥ qy ∞ **

**q0 ŵ ⊢* Ĥ qx ŵ ŵ ⊢* Ĥ qn**

**Turing Machine Ĥ is applied to a Turing Machine description of itself ŵ.**

**Would Ĥ transition to its state (qy) or ((qn)) on ŵ ?**

**To answer this question we perform an execution trace on Ĥ**

(1) At q0 * Ĥ makes a copy* of its input, for clarity we will call this copy ŵ2.

(2) At qx

**Execution trace of Ĥ analysis of finite string ŵ**

(3) At q0 * ŵ would make a copy* of its input, we will call this copy ŵ3.

(4) At qx

The end result of all of this is that The Halting Problem proof suffers from the exact same Pathological Self-Reference (PSR) error as the Liar Paradox.

We must first formalize expressions using Higher Order Logic (HOL) with Self-Reference Semantics (SRS). Then we translate these expressions intro their corresponding Directed Acyclic Graph (DAG).

PSR is detected when the translation of a HOL expression into a DAG requires the insertion of infinite cycles into this otherwise acyclic graph.

**(New insight: cycles may be OK infinite cycles must be rejected.)**

**Pathological-Self-Reference(X) means that expression X specifies an infinite evaluation sequence that never resolves to True or False.**

**LP ≡ ~True(LP) // Liar Paradox**

** Ĥ ≡ Halts(⌈Ĥ⌉) // ⌈Ĥ⌉ is TMD of Ĥ**

** G ≡ ~Provable(⌈G⌉) // ⌈G⌉ is Gödel Number of G**

Copyright 2004, 2016, 2017, 2018 Pete Olcott

]]>An axiom is a proposition regarded as self-evidently true without proof.

Wikipedia: Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

Wolfram Mathworld: Theorem

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments.

Wikipedia: Theorem

In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.

**Axiom(MTT)**

This differs from the conventional meaning of Axiom(Math) in that it a formal expression of language that has been tautologically defined to be necessarily True.

These conventional terms of the art are defined in terms of other conventional terms of the art and these terms also have fundamental misconceptions built in to their definitions, on and on.

My system analyzes specific instances the generic notion of finite strings of characters to determine if these finite strings correspond to the generic notion of {**Truth**}.

The generic notion of {**formal provability**} simply examines whether or not a specific preexisting set of finite string transformation rules allow one finite string to be derived from another. In the case where the first finite string has the Boolean property of True, then the derived finite string is also known to have this same Boolean property.

The set of finite string transformation rules includes a set of finite strings that have been defined to have the Boolean property of True. This corresponds to notion of Prolog Facts. From the Fact that “a can is made of tin” we can derive the fact that “a can is made of metal”.

The above systematic reconstruction of the notions of {**Truth**} and {**formal provability**} are applied to the following of specific concrete examples. The essence of these examples are reformalized and analyzed for logical coherence within a new formal system.

The Liar Paradox from antiquity

David Hilbert’s 1928 Entscheidungsproblem

Kurt Gödel’s 1931 incompleteness Theorem

Alfred Tarski’s 1936 Undefinability Theorem

Alan Turing’s 1937 Halting Problem

“The first incompleteness theorem states that in any consistent

formal system F within which a certain amount of arithmetic can

be carried out, there are statements of the language of F which can

neither be proved nor disproved in F.”

**Formalized as: Sentence _{F}(G) ∧ ~Provable_{F}(G) ∧ ~Refutable_{F}(G)**

If the above formalism succinctly states the accurate conclusion

of the 1931 GIT and this conclusion can be proven self-contradictory

then all of the details of the 1931 GIT can be ignored as extraneous

and the 1931 GIT can be refuted entirely on the basis that its conclusion

is self-contradictory.

**copyright 2017, 2018 Pete Olcott**

(1) X

(2) Y

(3) (X ∧ Y) → Z

———–

(4) Z

**Here is what the Propositional Logic proof means:**

(1) X is True

(2) Y is True

(3) If (X and Y) are True then Z is True

(4) Therefore Z is True

]]>

Aristotle Syllogism (slightly enhanced):

…..Premise (a) “It is raining outside.”

…..Premise (b) “You go outside unprotected from the rain.”

Conclusion (c) “You get wet.”

Partial translation to Propositional Logic:

if (a) and (b) then (c)

Full translation to Propositional Logic:

(a ∧ b) → c

Propositional Logic Symbols and their English meanings

P …………. P is true

~P ……….. P is not true

P ∧ P ……. P and Q are both true

P ∨ Q ……. either P or Q (or both) are true

P → Q ….. if P is true then Q is true

…………….. if Q is false then P is false

…………….. if P is false then (Q ∨ ~Q)

If you understand the above then you know your P’s and Q’s

Syntactic versus Semantic Logical Consequence

Meaning postulates specify semantic logical entailment syntactically.

The best two examples from his paper: Bachelor(x) and Warmer(x,y):

Bachelor(x) → ~Married(x)

For example, let ‘W’ be a primitive predicate designating the relation Warmer. Then ‘W’ is transitive, irreflexive, and hence asymmetric in virtue of its meaning.

In the previous example of the predicate ‘W’, we could lay down the following postulates (a) for transitivity and (b) for irreflexivity; then the statement (c) of asymmetry:

(a) ∀(x,y,z) Warmer(x,y) ∧ Warmer(y,z) → Warmer(x,z)

(b) ∀(x) ~Warmer(x,x)

(c) ∀(x,y) Warmer(x,y) → ~( Warmer(y,x) )

Meaning Postulates Rudolf Carnap (1952)

Copyright 2017 Pete Olcott

]]>∀

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

A formula * A* is a

of a set

The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are

https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)

In mathematical logic, **a sentence of a predicate logic**** is a boolean-valued well-formed formula with no free variables**. A sentence can be viewed as expressing a proposition, **something that must be true or false**.

**G @ ∃X ~Provable(X, G)**** // Minimal Type Theory**

(1) The above * expression* says that it is not

(2) The above

(3) However the above

**Expression translated into a Directed Acyclic Graph by the MTT compiler**

[01] G (02)(04)

[02] THERE_EXISTS (03)

[03] X

[04] NOT (05)

[05] Provable (03)(01) // **cycle indicate infinite evaluation loop**

**Cycles in directed graphs of logical expressions are erroneous:**

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

A naive omission of the occurs check leads to the creation of cyclic structures and may cause unification to loop forever.

**Programming in Prolog (2003) by Clocksin and Mellish page 254**

equal(X, X).

?- equal(foo(Y), Y).

… **match a term against an uninstantiated subterm of itself**. In this example, foo(Y) is matched against Y, which appears within it. As a result, Y will stand for foo(Y), which is foo(foo(Y)) (because of what Y stands for), which is foo(foo(foo(Y))), and so on. So Y ends up standing for some kind of infinite structure.

https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom

**(G) F ⊢ G**_{F}** ↔ ¬Prov**_{F}**(⌈G**_{F}**⌉)**

**Match a term against an uninstantiated subterm of itself: **

Equivalent( GF, ¬ProvF( Godel_Number(GF) )

Equivalent( GF, ¬ProvF(GF) )

Equivalent( ¬ProvF(GF), GF )

Equivalent( ProvF(GF), GF )

?- equal(foo(Y), Y).

**Can you see how the above two are the same?**

In order to mathematically formalize every aspect of the Liar Paradox as predicate logic, I had to enhance the way that predicate logic works.

**Defining Self-Reference Semantics <assign alias name> operator**

LHS is assigned as an alias name for the RHS

**LHS ≡ RHS**

The LHS is logically equivalent to the RHS * only Because *the LHS is merely an alias name for the RHS

**LiarParadox ≡ ~True(LiarParadox). **

The name and alias name operator are merely notational conventions for translating the meaning: **{This sentence is not true}** into predicate logic.

**Translation into directed acyclic graph**

**01 LiarParadox (02)
02 Not (03)
03 True (01) **// infinite evaluation loop

**Here is how {alias name} works in computer science:**

In computer science variable names to refer to memory addresses. Variable names are considered one and the same thing as the memory address. When a computer program is compiled the names are translated into these memory addresses and discarded.

**// The Liar Paradox in C++ **

Test the value of a variable before this variable has a value

then assign this non-existent value to this variable.

**bool Liar_Paradox = (Liar_Paradox == false);**

This statement is semantically analogous to the one above

in that the quotient of division by zero does not exist.

**double BadNumber = 1.0 / 0.0;**

Copyright 2016, 2017 2018 Pete Olcott

]]>2.2 The fixed point theorem

GL ⊢ B ↔ A(B)

GL ⊢ LiarParadox ↔ ~True(LiarParadox)

Do you see the similarity? Prolog rejects both of these as erroneous.

http://liarparadox.org/Prolog_Detects_Pathological_Self_Reference.pdf

**Here is the error that Prolog detects: (see above link)**

**match a term against an uninstantiated subterm of itself.**

The exact same thing as this C++

bool LiarParadox = LiarParadox == false;

Copyright 2017 Pete Olcott

Copyright 2016, 2017 Pete Olcott

]]>A formula

of a set

https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)

In mathematical logic, **a sentence of a predicate logic** is a boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that **must be true or false.**

https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom

The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out,

**there are statements of the language of F which can neither be proved nor disproved in F:** **Sentence _{F}(G) ∧ ~Provable_{F}(G) ∧ ~Refutable_{F}(G)**

**G @ ~∃Γ ⊆ F Provable(Γ, G)** // Minimal Type Theory

(1) The above * expression* says that it is not

(2) The above

(3) However the above

**Sentence translated into a Directed Acyclic Graph by the MTT compiler:**

[01] G (02)(07)

[02] NOT (03)

[03] THERE_EXISTS (04)

[04] SUBSET_OF (05)(06)

[05] Γ

[06] F

[07] Provable (05)(01) **// cycle indicates infinite evaluation loop**

The YELLOW highlighted portions form the foundational basis for what I mean by Provable(X).

The paragraph following the YELLOW highlighted paragraph directly states what I mean by Provable.

The remaining material establishes common notational conventions that are essentially the same as I have been using.

]]>Logical_consequence#Syntactic_consequence

**Direct quote from above Wikipedia link:**

A formula ** A** is a syntactic consequence within some formal system

**In other words:** Formula A is Provable within Formal System FS if there exists an inference chain (connected sequence of WFF) from a set of Formula Γ to Formula A.

Furthermore if Γ are axioms of FS, then Provable(Γ, A) ≡ True(A) in FS:

**True _{FS}(A) ≡ Provable_{FS} (Γ, A) **

Last paragraph copyright 2017 Pete Olcott

]]>The first incompleteness theorem states that in any consistent formal system

**When Godel said that he found a sentence G of language F such that:
~(Provable_{F}(G) ∨ Refutable_{F}(G) ) he was wrong.**

https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom

**(G) F ⊢ G _{F} ↔ ¬Prov_{F}(⌈G_{F}⌉) **

1931 GIT entirely summed up as a single line of Minimal Type Theory:

**G @ ∃X ~Provable(X, G)**

(1) The above * expression* says that it is not

(2) The above

(3) However the above

Sentence translated into a Directed Acyclic Graph by the MTT compiler:

[01] G (02)(04)

[02] THERE_EXISTS (03)

[03] X

[04] NOT (05)

[05] Provable (03)(01) // **cycle indicate infinite evaluation loop**

If G is an alias (sentential variable) for ∃X (X ⊢ Y), then G is either True or False, even if both X and Y are gibberish.

If G is an alias (sentential variable) for ~∃X (X ⊢ G) we have pathological self-reference creating infinite recursion. This prevents G from ever evaluating to either True or False, thus making G semantically ill-formed.

copyright 2017 Pete Olcott (Created Friday October 13, 2017)

**Originally posted material is shown below**

If the 1931 Incompleteness Theorem is correct then the much simpler expression at the end of this post equally proves that there are some sentences of predicate logic that are neither provable nor refutable.

Syntactic_Logical_Consequence (Γ proves A)

A formula A is a syntactic consequence within some formal system FS of a set Γ of formulas if there is a formal proof in FS of A from the set Γ:

**Γ ⊢ _{FS} A**

Translation to Minimal Type Theory notational conventions

**Γ ⊢ _{FS} A ≡ ( Γ ⊂ FS (Γ ⊢ A) )**

Named predicate (of predicate logic PL) impossible to prove and impossible to refute:** G( ~∃Γ ⊂ PL (Γ ⊢ G) )**

The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F.

As long as the above sentence accurately sums up the GIT no greater understanding of the GIT is required to totally refute it:

(1) Every WFF of all formal systems must be a truth bearer.

(2) If no formal proof exists within a formal system to show that an expression evaluates to exactly one of the set: {true, false}, then this expression is not a truth bearer, and not a WFF in this formal system.

(3) Therefore there are no WFF in any formal system which can be neither proved nor disproved within this formal system.

]]>**The yellow highlighted pages of the provided book show two things:**

(1) Solving the Liar Paradox is crucially important to fixing the currently inconsistent understanding of the concept of Truth.

(2) Prior to the solution provided on this website it was only understood that the Liar Paradox is defective the precise nature of this defect was unknown.

]]>