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Fedor Pakhomov

Publications and source records attributed to Fedor Pakhomov.

At least 19 recordsLinked to original sources
arXiv2026

Lévy-Montague reflection is $Π^1_1$-conservative over $\mathsf{WKL}_0$

We study a Lévy-Montague reflection scheme $\mathsf{Rfn}$ in second-order arithmetic: for each formula $φ$, the scheme asserts that every set belongs to a countable coded $ω$-model such that $φ$ is absolute, at all parameters from the model, between the model and the universe. Our central result is a model extension construction: every countable model of $\mathsf{RCA}_0$ can be extended, without changing its first-order part, to a model of $\mathsf{WKL}_0$ together with the full scheme $\mathsf{Rfn}$. It follows at once that $\mathsf{WKL}_0+\mathsf{Rfn}$ is $Π^1_1$-conservative over both $\mathsf{WKL}_0$ and $\mathsf{RCA}_0$, that its first-order part is exactly $\mathrm{I}Σ_1$, and that it is $Π^0_2$-conservative over $\mathsf{PRA}$. The result opens an avenue for adopting, within a theory conservative over $\mathsf{PRA}$, Feferman's $\mathsf{ZFC}$-formalization of universe-based category-theoretic arguments that was achieved using Lévy-Montague reflection. The conservation proof itself, however, is non-finitary. The extension is the union of an $ω_1$-tower of forcing extensions, and its uncountable cofinality is what secures reflection. We are only able to prove the conservation in $\mathsf{PRA}+\text{1-Con}(\mathsf{Z}_2)$. The results were obtained with extensive use of Anthropic's large language model Fable 5.

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arXiv2026

Ranking theories via encoded $β$-models

Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $β$-models: $T\prec_βU$ if every $β$-model of $U$ contains a countable coded $β$-model of $T$. The restriction of $\prec_β$ to theories with $β$-models is well-founded. We establish fundamental properties of the attendant ranking. First, though there are continuum-many theories, every theory has countable $\prec_β$-rank. Second, the $\prec_β$-ranks of $\mathcal{L}_\in$ theories are cofinal in $ω_1$. Third, assuming $V=L$, the $\prec_β$-ranks of $\mathcal{L}_2$ theories are cofinal in $ω_1$. Finally, $δ^1_2$ is the supremum of the $\prec_β$-ranks of finitely axiomatized theories.

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arXiv2026

Linear Orders in Presburger Arithmetic

We prove the linear orders first-order definable in the standard model $(\ZZ;<,+)$ of Presburger arithmetic are exactly those that are $(\ZZ;<,+)$-definably embeddable into the lexicographic ordering on $\ZZ^n$ for some $n$.

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arXiv2026

Speedups for Presburger Arithmetic and Real Closed Fields

In the present paper, we consider Presburger arithmetic PrA and the theory of real closed fields RCF. Due to quantifier elimination in these theories, there are two kinds of natural ways to axiomatize them. Namely, on one hand, PrA can be axiomatized with the full schema of first-order induction, and RCF with the full schema of the first-order least upper bound principle. At the same time, there are natural axiomatizations of these theories that avoid the use of formulas of unbounded quantifier depth. In the present paper, we compare these two groups of axiomatizations from the perspective of proof lengths. We show that the first group of axiomatizations enjoys at least a double exponential speedup.

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arXiv2026

Well-quasi-orders on finite trees and transfinite sequences

We study the well-quasi-order (wqo) consisting of the set of finite trees with leaf labels coming from an arbitrary wqo $Q$, ordered by tree homomorphisms which respect the order on the labels. This is a variant of the usual Kruskal tree ordering without infima preservation. We calculate the precise maximal order types of this class of wqos as a function of the maximal order type of the labels $Q$. In the process, we sharpen some recent results of Friedman and Weiermann. Furthermore, we show a correspondence with indecomposable transfinite sequences with finite range, over elements of the wqo $Q$, of length less than $ω^ω$. Nash-Williams proved that arbitrary transfinite sequences with finite range are also well-quasi-ordered, but there are no known methods to extract bounds on the maximal order type from the proof. More concrete proofs for sequences of length less than $α$ for some $α< ω^ω$ were given by Erdős and Rado. Using the correspondence, we obtain precise bounds for the entire collection of transfinite sequences with finite range of length less than $ω^ω$.

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arXiv2025

Generalized Higman's Theorem and iterated ideals

Generalized Higman's Theorem is the direct counterpart of Higman's Theorem that asserts the closure of the class of \emph{better} quasi-orders, instead of the class of \emph{well} quasi-orders, under the construction $P\mapsto P^{<ω}$ of the embeddability order on finite sequences. Traditionally, this result is obtained as a consequence of very powerful and general techniques of Nash-Williams. In this paper, we propose a new proof of this result that is based on an explicit characterization of the underlying orders. In particular, this new technique allows us to formalize the proof of the result in the formal theory $\mathsf{atr}_0$, thus resolving a long-standing open problem in the field of reverse mathematics. The main ingredient of our proof is the introduction of a transfinite hierarchy of orders $\dot I^*_α(P)$ starting with $\dot I^*_0(P)=P$ and $\dot I^*_{α+1}(P)$ being the inclusion order on ideals of $\dot I^*_α(P)$. On one hand, we show that a quasi-order $P$ is a bqo if and only if all $\dot I^*_α(P)$ are wqos. On the other hand, under the assumption that $P$ is a bqo, we show that the $\dot I^*_α(P^{<ω})$ are wqos, and furthermore give a characterization of their structure in terms of a transfinite iteration of a Higman-like construction. The sufficiently explicit character of this proof allows us to formalize it in a rather straightforward manner.

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arXiv2024

Feferman's completeness theorem

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman's completeness theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.

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arXiv2024

On Nash-Williams' Theorem regarding sequences with finite range

The famous theorem of Higman states that for any well-quasi-order (wqo) $Q$ the embeddability order on finite sequences over $Q$ is also wqo. In his celebrated 1965 paper, Nash-Williams established that the same conclusion holds even for all the transfinite sequences with finite range, thus proving a far reaching generalization of Higman's theorem. In the present paper we show that Nash-Williams' Theorem is provable in the system $\mathsf{ATR}_0$ of second-order arithmetic, thus solving an open problem by Antonio Montalbán and proving the reverse-mathematical equivalence of Nash-Williams' Theorem and $\mathsf{ATR}_0$. In order to accomplish this, we establish equivalent characterization of transfinite Higman's order and an order on the cumulative hierarchy with urelements from the starting wqo $Q$, and find some new connection that can be of purely order-theoretic interest. Moreover, in this paper we present a new setup that allows us to develop the theory of $α$-wqo's in a way that is formalizable within primitive-recursive set theory with urelements, in a smooth and code-free fashion.

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arXiv2024

The Logic of Correct Models

For each $n\in\mathbb{N}$, let $[n]ϕ$ mean "the sentence $ϕ$ is true in all $Σ_{n+1}$-correct transitive sets." Assuming Gödel's axiom $V = L$, we prove the following graded variant of Solovay's completeness theorem: the set of formulas valid under this interpretation is precisely the set of theorems of the linear provability logic GLP.3. We also show that this result is not provable in ZFC, so the hypothesis V = L cannot be removed. As part of the proof, we derive (in ZFC) the following purely modal-logical results which are of independent interest: the logic GLP.3 coincides with the logic of closed substitutions of GLP, and is the maximal non-degenerate, normal extension of GLP.

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arXiv2023

Generalized fusible numbers and their ordinals

Erickson defined the fusible numbers as a set $\mathcal F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $\mathcal F$ is well ordered, with order type $\varepsilon_0$. They also investigated a recursively defined function $M\colon \mathbb{R}\to\mathbb{R}$. They showed that the set of points of discontinuity of $M$ is a subset of $\mathcal F$ of order type $\varepsilon_0$. They also showed that, although $M$ is a total function on $\mathbb R$, the fact that the restriction of $M$ to $\mathbb{Q}$ is total is not provable in first-order Peano arithmetic $\mathsf{PA}$. In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets $\mathcal F$ of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function $g:\mathbb R^n\to\mathbb R$. The most straightforward generalization of $\frac{x+y+1}{2}$ to an $n$-ary function is the function $\frac{x_1+\cdots+x_n+1}{n}$. We show that this function generates a set $\mathcal F_n$ whose order type is just $φ_{n-1}(0)$. For this, we develop recursively defined functions $M_n\colon \mathbb{R}\to\mathbb{R}$ naturally generalizing the function $M$. Furthermore, we prove that for any linear function $g:\mathbb R^n\to\mathbb R$, the order type of the resulting $\mathcal F$ is at most $φ_{n-1}(0)$. Finally, we show that there do exist continuous functions $g:\mathbb R^n\to\mathbb R$ for which the order types of the resulting sets $\mathcal F$ approach the small Veblen ordinal.

math.CO↗
arXiv2023

Provable better quasi orders

It has recently been shown that fairly strong axiom systems such as $\mathsf{ACA}_0$ cannot prove that the antichain with three elements is a better quasi order ($\mathsf{bqo}$). In the present paper, we give a complete characterization of the finite partial orders that are provably $\mathsf{bqo}$ in such axiom systems. The result will also be extended to infinite orders. As an application, we derive that a version of the minimal bad array lemma is weak over $\mathsf{ACA_0}$. In sharp contrast, a recent result shows that the same version is equivalent to $Π^1_2$-comprehension over the stronger base theory $\mathsf{ATR}_0$.

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arXiv2023

The logical strength of minimal bad arrays

This paper studies logical aspects of the notion of better quasi order, which has been introduced by C. Nash-Williams (Mathematical Proceedings of the Cambridge Philosophical Society 1965 & 1968). A central tool in the theory of better quasi orders is the minimal bad array lemma. We show that this lemma is exceptionally strong from the viewpoint of reverse mathematics, a framework from mathematical logic. Specifically, it is equivalent to the set existence principle of $Π^1_2$-comprehension, over the base theory $\mathsf{ATR_0}$.

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arXiv2022

How to escape Tennenbaum's theorem

We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model.

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arXiv2022

Arithmetical and Hyperarithmetical Worm Battles

Japaridze's provability logic $GLP$ has one modality $[n]$ for each natural number and has been used by Beklemishev for a proof theoretic analysis of Peano aritmetic $(PA)$ and related theories. Among other benefits, this analysis yields the so-called Every Worm Dies $(EWD)$ principle, a natural combinatorial statement independent of $PA$. Recently, Beklemishev and Pakhomov have studied notions of provability corresponding to transfinite modalities in $GLP$. We show that indeed the natural transfinite extension of $GLP$ is sound for this interpretation, and yields independent combinatorial principles for the second order theory $ACA$ of arithmetical comprehension with full induction. We also provide restricted versions of $EWD$ related to the fragments $IΣ_n$ of Peano arithmetic. In order to prove the latter, we show that standard Hardy functions majorize their variants based on tree ordinals.

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arXiv2022

Reducing $ω$-model reflection to iterated syntactic reflection

In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper we study connections between these two kinds of reflection principles in the setting of second-order arithmetic. We prove that, for a large swathe of theories, $ω$-model reflection is equivalent to the claim that arbitrary iterations of uniform $Π^1_1$ reflection along countable well-orderings are $Π^1_1$-sound. This result yields uniform ordinal analyses of theories with strength between $\mathsf{ACA}_0$ and $\mathsf{ATR}$. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory $T$, which is the operator on countable ordinals that maps the order-type of $\prec$ to the proof-theoretic ordinal of $T+\mathsf{WO}(\prec)$. We obtain precise results about the growth of proof-theoretic dilators as a function of provable $ω$-model reflection. This approach enables us to simultaneously obtain not only $Π^0_1$, $Π^0_2$, and $Π^1_1$ ordinals but also reverse-mathematical theorems for well-ordering principles.

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arXiv2021

The $Π^1_2$ Consequences of a Theory

We develop the abstract framework for a proof-theoretic analysis of theories with scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis aimed at the study of theorems of complexity $Π^1_2$. This is done by replacing the use of ordinal numbers by particularly uniform, wellfoundedness preserving functors in the category of linear orders. Generalizing the notion of a proof-theoretic ordinal, we define the functorial $Π^1_2$ norm of a theory and prove its existence and uniqueness for $Π^1_2$-sound theories. From this, we further abstract a definition of the $Σ^1_2$- and $Π^1_2$-soundness ordinals of a theory; these quantify, respectively, the maximum strength of true $Σ^1_2$ theorems and minimum strength of false $Π^1_2$ theorems of a given theory. We study these ordinals, developing a proof-theoretic classification theory for recursively enumerable extensions of $\mathsf{ACA}_0$ Using techniques from infinitary and categorical proof theory, generalized recursion theory, constructibility, and forcing, we prove that an admissible ordinal is the $Π^1_2$-soundness ordinal of some recursively enumerable extension of $\mathsf{ACA}_0$ if and only if it is not parameter-free $Σ^1_1$-reflecting. We show that the $Σ^1_2$-soundness ordinal of $\mathsf{ACA}_0$ is $ω_1^{ck}$ and characterize the $Σ^1_2$-soundness ordinals of recursively enumerable, $Σ^1_2$-sound extensions of $Π^1_1{-}\mathsf{CA}_0$.

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arXiv2021

Finitely Axiomatized Theories Lack Self-Comprehension

In this paper we prove that no consistent finitely axiomatized theory one-dimensionally interprets its own extension with predicative comprehension. This constitutes a result with the flavor of the Second Incompleteness Theorem whose formulation is completely arithmetic-free. Probably the most important novel feature that distinguishes our result from the previous results of this kind is that it is applicable to arbitrary weak theories, rather than to extensions of some base theory. The methods used in the proof of the main result yield a new perspective on the notion of sequential theory, in the setting of forcing-interpretations.

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