Idealizing Useful Fictions in Omega Grounded Arithmetic

Abstract: Grounded arithmetic is a family of formal systems for reasoning about computation in which a statement may be asserted only when a terminating computation backs it; the logics are paracomplete - for a sentence whose backing computation never settles, neither the sentence nor its negation is derivable, so paradoxes like the Liar are harmless rather than explosive. The reflective member of the family, RGA, can quantify over its own computations, but cannot certify that its own unbounded searches have definite yes-or-no answers. This paper studies what happens when that openness is closed by exactly one rule - ATI, the $ω$-grounded universal: if every numeric instance of a universal sentence is certified decided, the universal is certified decided. The resulting system, OGA, shares RGA's syntax and rules symbol-for-symbol otherwise, and every consequence is developed as a machine-checked theorem. Decidedness certificates become abundant - every totality question about a computable function is certified to have an answer, whether or not anyone can produce it - and this is exactly the provable separation between the two systems. OGA is complete for its own semantics; certified-but-unresolved sentences receive values built from the system's own open questions. Provability remains recursively enumerable, with a primitive-recursive certificate checker, while $ω$-truth deliberately is not. Within that asymmetry, incompleteness takes a new form. The Gödel sentence is classified, unconditionally, as a genuine fiction: neither provable nor refutable, yet valued, and carrying a computable pedigree recording exactly what adopting it as an axiom commits one to. The adoption is itself a theorem suite: extending OGA by any finite stock of true fictions is consistent, and independently certified adoptions can never collide.

• The paper extends Reflective Grounded Arithmetic (RGA) by adding the rule ATI, which introduces decidability of universally quantified sentences.
• The main result shows that OGA proability aligns with super-truth under admissible valuations while maintaining semantic consistency, creating usefully valued fictions.
• Machine-checking in Isabelle/HOL confirms every finding, with potential useability intimations exceeding classic formal proofs, inviting follow-up theoretical exploration.

Bryan Ford's "Idealizing Useful Fictions in Omega Grounded Arithmetic" (2608.17862) extends the grounded-arithmetic program by a single proof rule and develops the consequences of that extension to the point where Gödel incompleteness acquires a new semantic classification. The paper's central object is OGA, obtained from Reflective Grounded Arithmetic (RGA) by adding one rule, ATI, which certifies that universally quantified sentences have definite yes-or-no answers whenever every numeric instance is certified decided. All results are machine-checked in Isabelle/HOL, roughly 6,800 lines beyond the RGA base, with no unproven assumptions or added axioms.

Background: grounded arithmetic and RGA

Grounded arithmetic replaces classical bivalence with an earned-truth discipline: a sentence holds only when a terminating semantic process grounds it, and sentences whose backing computation never settles carry no value, harmlessly. The logics are paracomplete rather than paraconsistent — the Liar simply gaps. RGA, the reflective tier of the family, makes quantification computational by representing universal sentences as search sentences over compiled step functions, so that the system reasons about its own unbounded searches as ordinary arithmetic objects.

RGA carries a provable limitation that motivates this paper: its search sentences are not provably decided. Certifying an undecided search as decided would collapse the discipline that provability coincides with groundedness, since a never-settling search grounds no verdict about itself. This is a theorem of the prior formalization, not an open question, and OGA is defined as the deliberate closure of exactly this gap.

The system OGA and the single rule

OGA shares RGA's syntax symbol-for-symbol and its forty-two inference rules rule-for-rule; the entire difference is the rule ATI ($\omega$-grounded universal decidedness introduction): from an internal universal over pointwise decidedness claims about a body $u$, conclude that $\forall\cdot u$ is decided. Three design points structure everything downstream:

• The premise is internal and finitary. OGA does not adopt the infinitary $\omega$-rule; it reflects one specific $\omega$-fact — decidedness — into the object logic.
• One premise serves both quantifiers, since deciding an existential requires surveying every instance just as deciding a universal does.
• The conclusion is status, never truth. ATI concludes a decidedness judgment, never an assertion of the universal. Unfolding abbreviations shows the conclusion is literally excluded middle for the quantified sentence, admitted under certificate: OGA is RGA plus classical excluded middle at the quantifier, granted only when every instance is already decided, and still without supplying a witness for either disjunct.

Semantics: facts, fictions, and completeness

OGA has two semantics. The $\omega$-completion extends RGA's grounded semantics with the clause ATI reflects, delivering soundness and consistency. The native fafi ("fact/fiction") semantics is a certificate-gated supervaluation: values live in a Boolean algebra over presumption atoms, one per certified-but-unresolved question, introduced only when an OGA-derivation certifies the corresponding universal decided. Because occurrences of the same atom stay correlated, $(\forall\cdot u)\lor\neg(\forall\cdot u)$ evaluates to truth even when neither disjunct is valued — precisely what truth-functional Kleene-style schemes cannot deliver, and exactly ATI's soundness obligation. The paper reports that an earlier truth-functional design was refuted on paper against this obligation before any formalization began.

The main semantic theorem is a characterization: OGA-provability coincides exactly with super-truth under coherent admissible valuations, so OGA is sound and complete for its own reflective-supervaluational semantics. This extends the family pattern in which each grounded system is complete for the semantics reflecting its own characteristic infinitary move. Notably, the machine refuted the first, pointwise notion of admissibility during development; the corrected joint condition on valuations' commitments is a repair of the definition, not merely of the proof.

Every well-formed sentence falls into exactly one of three classes: Fact (grounded), Fict (valued but neither provable nor refutable, via the gate), or no value (the Liar paradigm). A value's support is finite, and its pedigree — the fictitious part of that support — records computably what adopting it commits one to. This pedigree apparatus is the paper's formal content for its title: fictions are useful because they are valued, consistent to adopt, and priced.

Decidedness of the reflective ground and the separation

A derived ladder of decidedness principles climbs from pointwise $\Sigma_1$ decidedness through schematic forms to the working top: for every primitive-recursive step function $f$, whether or not its search is total, OGA proves the totality question decided — unconditionally in $f$. This is exactly the separation between the systems. RGA provably cannot decide its search sentences (witnessed by the halting diagonal), while OGA decides them all; measured against a generic ladder of quantifier tiers maintained as locales in the formalization, OGA registers at the $u$0-decidedness tier and RGA provably cannot. The separation consists entirely of status judgments: ATI adds no grounded truths, yet its status judgments are new theorems.

A subtlety worth noting: fictionality of search universals admits an exact recursion-theoretic description — such a sentence is a genuine fiction exactly when it lies in the $u$1-minus-r.e. difference. Meanwhile, pointwise instances at diagonal points may themselves gap, because they are redexes rather than universals and the evaluation relation takes no conversion steps; their contracta, however, are always valued. Gaps of presentation and gaps of content both exist, and only the second kind is beyond ATI's reach in principle.

Metatheory: enumerability, revenge, and fictional status

The metatheory splits along a designed asymmetry. Provability remains recursively enumerable, with a primitive-recursive certificate checker accepting exactly the codes of derivable judgments; $u$2-truth deliberately is not r.e., since enumerating it would enumerate the complement of the halting diagonal. Within this asymmetry, OGA's internal provability predicate satisfies D1 together with its converse (admissibility of the provability rule), though the paper is careful to claim nothing about the remaining derivability conditions, and proves outright that the completeness half does not internalize: there is a $u$3 with $u$4 but not $u$5.

Two further results deserve emphasis. First, because $u$6 is itself a search sentence, its membership formula is decided uniformly — and hence the predicate $u$7 ("$u$8 is settled") is decided for every $u$9, flatly under iteration. The paper argues carefully that this does not defeat revenge: $\forall\cdot u$0 reports on the proof system, not on semantic valuedness, which OGA declines to internalize; the Tarskian retreat survives, relocated. The authors also concede that the proper revenge test — making $\forall\cdot u$1 an internally computed function of the Gödel code — has not been run, so the flatness result should be read as evidence rather than proof that ascent is unnecessary.

Second, the sharpened $\forall\cdot u$2-incompleteness: there is a family with every numeric instance provable whose universal closure is certified decided yet neither provable nor refutable. The termination-provability classes stratify strictly below actual termination on both sides of the RGA/OGA inclusion, though the paper explicitly concedes that strictness of the inclusion itself is open, and frames the analysis without ordinals.

The centerpiece is the unconditional classification of the Gödel sentence: it is a genuine fiction — valued, irrefutable, unprovable, with pedigree equal to its own singleton — with no consistency or $\forall\cdot u$3-consistency hypothesis. The paper is precise about what the fiction covers: not the underlying computational fact, which the metatheorist knows, but the absence of an internal certificate. It also blocks overgeneralization: the Liar has no outer answer either, and nothing generalizes from the diagonal witnesses to ungroundedness at large.

The adoption calculus

The final metatheoretic section turns adoption into a theorem suite. Extending OGA by any finite stock of $\forall\cdot u$4-true sentences is consistent; certified adoptions merge without conflict (confluence), because per-sentence $\forall\cdot u$5-truth composes within a single fixed semantics. The safety gradient closes on both sides: adopting the anti-fiction $\forall\cdot u$6 is consistent but $\forall\cdot u$7-unsound — a formerly open tier, made inhabited precisely because OGA's decidedness unlocks the deduction theorem for the reductio, where the corresponding RGA construction stalls. The assembled bundle theorem states that under the diagonal hypotheses, $\forall\cdot u$8 is simultaneously certified decided, irrefutable, unprovable, fictional, and safe to adopt. The warrant, however, is honest about being external: adoption does not certify itself from inside, and status alone cannot distinguish $\forall\cdot u$9 from $\omega$0.

Formalization and constructive audit

The Isabelle/HOL development comprises twenty theories and about 370 named results, with case-aligned rule sets enabling metatheoretic inductions to replay between systems. A constructive audit localizes non-constructivity: Zorn's lemma appears twice, both removable in principle via countable Lindenbaum; the diagonal results were refactored from proofs by contradiction into witnessed instances of a single constructive diagonal lemma, using weaker hypotheses than the originals. One genuine residue remains — a Hilbert choice in indexing primitive-recursive functions keeps gap points concrete in principle but opaque in practice. Three recorded incidents show the assistant changing the paper's content, including the refutation of the naive admissibility definition. The AI contribution statement discloses substantial mechanization and drafting assistance under human direction.

Limitations and open questions

The paper is explicit about what it does not establish. Whether the derivability conditions D2 and D3 hold internally — and hence whether Löb's theorem is available in OGA — is open, with the visible constraint that internal soundness cannot coexist with both. Strictness of the termination-provability inclusion is open, with no candidate separating function. The revenge analysis rests on an unexecuted internalization of $\omega$1. The certification discussion is deliberately deflationary: the paper concedes that the proof-carrying-code pattern requires nothing specific to grounding, that OGA internally is no better placed than PA regarding self-certification, and that against constructive type theory no advantage can currently be identified. The self-verification inversion — evidence that RGA self-verifies while OGA plausibly loses self-verification by gaining decidedness — remains conjectural pending companion work, as does the graded set theory built on certified stocks.

Conclusion

The paper demonstrates that closing RGA's single openness with one finitary rule yields a system that is complete for its own richer semantics, decides its entire reflective ground, preserves r.e. provability, and reclassifies incompleteness: the Gödel sentence becomes an unconditionally certified, priced, adoptable fiction. Every result is machine-checked, and the paper consistently marks the boundary between what the machinery delivers and what it merely suggests — leaving the Löb obstruction's reactivation, the strictness of the termination ladder, and the reflection tower as precisely stated open questions for companion work.