On the Tender Line Separating Generalizations and Boundary-Case Exceptions for the Second Incompleteness Theorem Under Semantic Tableaux Deduction

Abstract

Our previous research has studied the semantic tableaux deductive methodology, of Fitting and Smullyan, and observed that it permits boundary-case exceptions to the Second Incompleteness Theorem, when multiplication is viewed as a 3-way relation (rather than as a total function). It is known that tableaux methodologies do prove a schema of theorems, verifying all instances of the Law of the Excluded Middle. But yet we show that if one promotes this schema of theorems into formalized logical axioms, then the meaning of the pronoun “I” in our self-referencing engine changes, and our partial evasions of the Second Incompleteness Theorem come to a complete halt.

Appendices

Appendix A: Formal Definition of a Tableaux Proof

Our definition of a semantic tableaux proof is similar to analogs in the textbooks by Fitting and Smullyan [15, 39]. A tableaux proof of a theorem $\varPsi$ from a set of proper axioms, denoted as $~\alpha ~$, will be a tree structure whose root contains the temporary contradictory assumption of $~\lnot \, \varPsi ~$ and whose every descending root-to-leaf branch affirms a contradiction by containing both some sentence $\phi$ and its negation $\lnot \, \phi$. Each internal node in this tree will be either a proper axiom of $~\alpha ~$ or a deduction from a higher ancestor in this tree using one of the following six elimination rules for the logical connective symbols of $\wedge$, $\vee$, $\rightarrow$, $\lnot$, $\forall$ and $\exists$. These rules use a notation where “A $\Longrightarrow$ B” is an abbreviation for a sentence B being an allowed deduction from its ancestor of A.

• 1. $\varUpsilon \wedge \varGamma \Longrightarrow \varUpsilon$ and $\varUpsilon \wedge \varGamma ~\Longrightarrow ~\varGamma$.

• 2. $\lnot \,\lnot \,\varUpsilon \,\Longrightarrow \,\varUpsilon .$ Other rules for the “$\, \lnot \,$” symbol are: $\lnot (\varUpsilon \vee \varGamma ) \Longrightarrow \lnot \varUpsilon \wedge \lnot \varGamma$, $\lnot (\varUpsilon \rightarrow \varGamma )\, \Longrightarrow \, \varUpsilon \wedge \lnot \varGamma$, $\lnot (\varUpsilon \wedge \varGamma ) \Longrightarrow \lnot \varUpsilon \vee \lnot \varGamma$, $\lnot \, \exists v \, \varUpsilon (v) \Longrightarrow \, \forall v \lnot \, \varUpsilon (v) \,$ and $\lnot \, \forall v \, \varUpsilon (v) \, \Longrightarrow \, \exists v \, \lnot \varUpsilon (v)$

• 3. A pair of sibling nodes $\varUpsilon$ and $\varGamma$ is allowed when their ancestor is $\varUpsilon \, \vee \, \varGamma .$

• 4. A pair of sibling nodes $\lnot \varUpsilon$ and $\varGamma$ is allowed when their ancestor is $\varUpsilon \, \rightarrow \, \varGamma$.

• 5. $\forall v \, \varUpsilon (v)\,\Longrightarrow \,\varUpsilon (t)$ where t may denote any term.

• 6. $\exists v\,\varUpsilon (v)~\Longrightarrow \,\varUpsilon (p)$ where p is a newly introduced parameter symbol.

A minor additional comment about our notation is that we treat “$\forall ~v \le s~\varPhi (v)$” as an abbreviation for $\forall v~\{v \le s\,\rightarrow \,\varPhi (v)\}$ and likewise “$\exists ~v \le s~\varPhi (v)$” as an abbreviation for $~ \exists v ~~ \{ ~ v \le s~~ \wedge ~~\varPhi (v)~ \}$. In our year-2005 article [49], we thus applied Rules 5 and 6 to derive the following further hybrid rules for processing the bounded universal and also the bounded existential quantifiers:

• a. $\forall v \le s\,\varUpsilon (v)\,\Longrightarrow \,t \le s \rightarrow \,\varUpsilon (t)$ where $\,t \,$ may be any arithmetic term.

• b. $~ \exists v \le s~\varUpsilon (v)\,\Longrightarrow \, p \le s~\wedge ~\varUpsilon (p)$ where p is a new parameter symbol.

Appendix B: More Details About Theorem 2’s Proof

The most surprising aspect of Theorem 2 is the sharp contrast between its result with the opposing property of Theorem 1. Our goal in this appendix will be to intuitively explain why the Invariant ++ (from Remark 1) ushers in a machinery that applies only to Theorem 2.

During our discussion, we will employ our U-Grounding language $L^*$ that treats multiplication as a 3-way relation (rather than as a functional operation). Its 3-way predicate Mult(x,y,z), for formalizing multiplication, is defined as follows:

$$\begin{aligned}{}[(x=0 \vee y=0) \Rightarrow z=0]~\wedge [(x \ne 0 \wedge y \ne 0) \Rightarrow ~(\frac{z}{x}=y ~\wedge ~ \frac{z-1}{x}<y)] \end{aligned}$$

(9)

We will say that an axiom basis $\alpha$ is Regular iff

• 1. It presumes all the U-Grounding operations are total functions (including the Addition and Doubling primitives).

• 2. It can prove all true $\varDelta _0^*$ sentences, and $\alpha$ is also consistent.

• 3. It can prove a $\varPi _1^*$ theorem showing addition and multiplication, viewed as 3-way relations, satisfy their usual associative, commutative, distributive and identity-operator properties.

Also in this appendix, we will employ a notation where for any $j \ge 0$, the symbol $\omega _j(x)$ will be recursively defined by the following rules:

• 1. $\omega _0(x)~=~x^2$.

• 2. $\omega _{j+1}(x)~=~2^{\omega _j(2\,\cdot \,Log_{2}(x+1))}$

These two rules imply that $\omega _{j+1}(x)~>~\omega _{j}(x)$ and $\omega _{1}(x)~\ge ~x^x$.

Clarification About Notation: Since our U-Grounding language $L^*$ does not permit using any function symbols to grow as fast as multiplication, it does not technically allow us to use any of the $\omega _j$ primitive symbols. One can, however, use techniques from [25]’s textbook to construct a $\varDelta _0^*$ formula $\psi _j(x,y)$ that satisfies (10)’s invariant for all standard numbers. It will, thus, capture most of $\omega _j$’s salient features.

$$\begin{aligned} \forall ~x~\forall ~y~~ ~~~~\psi _j(x,y)~~ \Leftrightarrow ~~\omega _j(x)=y \end{aligned}$$

(10)

Definition 4

A formula $\varPhi (x)$ will be called Locally-J-Closed relative to the axiom basis $\alpha$ iff $\alpha$ can prove the following three assertions about $\varPhi (x)~$:

• A. All of $\varPhi (0)$, $\varPhi (1)$ and $\varPhi (2)$ are true.

• B. The predicate $\varPhi (x)$ is operationally closed under the growth operation $\omega _j$. (Line (11) formally encodes this closure condition, using the preceding paragraph’s notation.)

  $$\begin{aligned} \forall x~\forall ~y~~~~\{[\psi _j(x,y) ~\wedge ~\varPhi (x)]~~\Rightarrow ~~\varPhi (y)\} \end{aligned}$$

  (11)

• C. The predicate $\varPhi (x)$ is also closed under (12)’s decrement operation.

  $$\begin{aligned} \forall ~x~\forall ~y<x~~\{\varPhi (x)~~~ \Rightarrow ~~~\varPhi (y)\} \end{aligned}$$

  (12)

Theorem 3

For each regular axiom basis $\alpha$ (that is consistent) and for each fixed integer $J\,\ge \,1$, there exists a corresponding formula $\varPhi (x)$ where $\alpha$ can prove that $\varPhi (x)$ is Locally-J-Closed.

Due to a lack of page space, a formal proof of Theorem 3 will be postponed until a longer version of this article. Theorem  3 is related to various intermediate results that were used to establish Remark 1’s Invariant ++ and [10, 16, 23, 25, 31, 35, 41, 42, 44, 46, 48]’s closely related results.

The fascinating feature of Theorem 3 is that it can explain why Theorems 1 and 2 display nearly opposite effects with regards to Hilbert’s Second Open Question. This is because the needed diagonalization for producing Theorem 2’s variations of the Second Incompleteness Effect become feasible only⁷ when $\text {IS}_{Xtab}(\beta )$’s Linear-Sum Effect is applied to the intermediate results produced by its possible derived theorems (which include the formalisms that are illustrated by lines (11) and (12)). On the other hand, no such similar types of nicely compressed constructed proofs are available under Theorem 1’s $\text {IS}_{Tab-1}(\beta )$ formalism (because all instances of the Law of Excluded Middle are excluded by it from becoming logical axioms). This is the intuitive reason that Theorems 1 and 2 display such sharply contrasting results.