Information-Complete and Paradox-Free Finite Ring Calculus

1. Introduction

The classical limits of formal logic, exemplified by the seminal arguments of Russell’s set paradox [1], Gödel’s incompletness [2], and Turing’s halting problem [3], all rely on an unbounded supply of natural-number codes. If however arithmetic is embedded into a finite algebra, the usual diagonal fixed-point method fails to escape the code-space, and no undecidable sentence is produced. In other words, if the universe of discourse is collapsed to a finite algebraic structure, the fixed-point machinery that fuels those paradoxes can no longer produce objects “outside the list”.

This idea dates back to early ultrafinitist proposals [4,5,6], but is now developed into a coherent multi-layered framework: the Finite Ring Continuum (FRC) of [7,8,9]. The aim of this article is to make this intuition formal, yielding a fully decidable calculus that remains expressive enough for algebraic-numeric reasoning and, prospectively, for physics. The present paper is intended as a self-contained logical core for the FRC programme.

Remark 1.

Throughout the extent of the FRC programme, we keep the “FRC” acronym intentionally polyvalent, referring to any of the notions of Finite Relativistic/Relational/Ring Continuum/Cosmology/Calculus interchangeably depending on the context. Our intent is to emphasise how these deeply interlocked notions are all just different aspects of the exact same conceptual framework.

Goal. 

Formulate a minimal first-order theory

$${\mathsf{FRC}}_q={\mathsf{Ring}}_q+“\mathrm{there}\mathrm{are}\mathrm{exactly}q\mathrm{elements}”$$

whose sole model is a fixed finite ring ${\mathbb{Z}}_q$, of some composite cardinality $q={\prod }_n{p}_n$, with $p_n$ odd primes, and prove:

Consistency by explicit model;

Completeness because every sentence can be brute-forced on $F_q$;

Decidability of the theory and its proof relation;

Collapse of diagonalisation: any Gödel sentence is already in the bounded code space, so no paradox arises.

2. Relation to the Finite Ring Continuum Framework

The FRC programme proceeds in three layers (Algebra → Geometry → Composition) as summarised in Fig. 1 of [8]. We briefly recall its main notation.

Definition 1

(FRC notation summary [9]). Let $p\equiv 1(mod4)$.

(a) 

${\mathbb{Z}}_q$ - framed finite ring; ${\mathbb{R}}_q$ - ultrapower pseudo-reals; ${\mathbb{C}}_q={\mathbb{R}}_q\left[i_q\right]$.

(b) 

Canonical invarients: $1,{\pi }_q,e_q,i_q$.

Our logical calculus lives entirely inside the algebraic layer ${\mathbb{Z}}_q$ but is consistent with the smooth-geometry layer carried by ${\mathcal{S}}_q^2\subset {\mathbb{R}}^4$[8,9]. Completeness of ${\mathsf{FRC}}_q$ therefore does not contradict the geometric richness exploited later in the continuum programme; it merely shows that paradox-free arithmetic is possible at the foundational stratum.

3. Language and Axioms of ${\mathsf{FRC}}_{\mathit{q}}$

3.1. Signature

$$\mathcal{L}=〈0,1,{\pi }_q,e_q,i_q,+,-,·,{\left\{c_k\right\}}_{k=0}^{q-1}〉$$

extends the usual ring language by a constant for each field element, matching the FRC convention that all numerals are concrete tokens ([7], §2.1).

3.2. Finite Axiom Set

(A1)

Ring axioms (associativity, distributivity, ...).

(A2)

Characteristic p (or q for non-prime moduli).

(A3)

Frobenius clause $x^q=x$ (cf. ([7], Eq. (2.3))).

(A4)

At least q distinct constants $c_k$.

(A5)

No more than q elements (completed by a finite scheme).

Exactly as in ([8], Thm. 2.3), these axioms single out the intended finite ring.

4. Gödel Coding and Its Modular Collapse

Let $g:\mathrm{Sentences}\twoheadrightarrow \mathbb{N}$ be any primitive-recursive Gödel numbering.

$$\overline{g}:{\mathbb{Z}}_q⟵\mathrm{Sentences},\overline{g}\left(\phi \right)=g\left(\phi \right)modq.$$

Because $\mathrm{range}\left(\overline{g}\right)\subseteq \{0,\dots ,q-1\}$, the fixed-point lemma yields a sentence G with $\overline{g}\left(G\right)=\overline{g}\left(\psi \right)$ for some earlier $\psi$; hence G is not new. This implements the “wrap-around defeats diagonal” slogan of ([7], §4) inside the present proof system.

Lemma 1

(Incompleteness blocked). For every recursive Gödel numbering g there isnosentence that is provably undecidable in ${\mathsf{FRC}}_q$.

Proof. 

Any candidate Gödel sentence duplicates a prior code and is therefore syntactically decidable in the finite proof search. □

Compare with the geometric reading where the code space sits as a hyperfinite lattice in $C_q$ ([8], §2.2); Lemma 1 is the algebraic counterpart of constant curvature $K=1$ preventing “run-away” self-reference on ${\mathcal{S}}_q$.

5. Consistency, Completeness and Decidability

Lemma 2.

${\mathsf{FRC}}_q$ is consistent, complete and decidable.

Proof. 

Consistency: ${\mathbb{Z}}_q$ is a model. Completeness: brute-force truth-table on ${\mathbb{Z}}_q$. Decidability: enumerate proofs or evaluate truth; the two coincide. □

This result supplies the logical keystone required for the higher-level geometry, harmonic analysis and Seifert topology developed in [8,9].

6. Discussion

The Finite Ring Calculus provides the “paradox-free arithmetic engine” hypothesised in ([7], §1). Its compatibility with the canonical constants $(1,i_p,{\pi }_p,e_p)$ ensures smooth interoperability with the analytic structures (Fourier kernel, discrete $S^2$ of constant curvature) that power subsequent layers of the FRC stack.