arXiv subjects

Oriola Gjetaj

Publications and source records attributed to Oriola Gjetaj.

4 recordsLinked to original sources

arXiv2026

Free sets, thin sets and rainbows for barriers

We formulate and prove the generalizations of Friedman's free set and thin set theorems and of the rainbow Ramsey theorem to colorings of barriers. We analyze the strength of these theorems from the point of view of computability theory proving some upper and lower bounds on the complexity of solutions for computable instances and some uniform computable reductions. We obtain as corollaries some proof-theoretical results on the logical strength of the theorems, in the spirit of reverse mathematics.

math.LO↗

arXiv2026

Ramsey-like theorems for the Schreier barrier

The family of finite subsets $s$ of the natural numbers such that $|s|=1+\min s$ is known as the Schreier barrier in combinatorics and Banach Space theory, and as the family of exactly $ω$-large sets in Logic. We formulate and prove the generalizations of Friedman's Free Set and Thin Set theorems and of Rainbow Ramsey's theorem to colorings of the Schreier barrier. We analyze the strength of these theorems from the point of view of Computability Theory and Reverse Mathematics. Surprisingly, the exactly $ω$-large counterparts of the Thin Set and Free Set theorems can code $\emptyset^{(ω)}$, while the exactly $ω$-large Rainbow Ramsey theorem does not code the halting set.

math.LO↗

arXiv2026

Goodstein at the Second Threshold: An Independence Result for $ID_2$

The classical Goodstein process, defined via hereditary base-$k$ exponential normal form, is a well-known example of a principle unprovable in Peano Arithmetic. In this paper, we generalize this framework by constructing a new Goodstein process based on the Hardy hierarchy. We develop an ordinal notation system utilizing a two-step collapsing procedure, which yields a proof-theoretic ordinal of $ψ_0ψ_1(\varepsilon_{Ω_2+1})$. By defining $k$-normal forms for natural numbers within this system, we introduce a Goodstein-type process and demonstrate that the theory of non-iterated positive inductive definitions for two operators ($ID_2$) cannot prove its termination. This result establishes a new independence result at the second proof-theoretic threshold, further extending the reach of Goodstein-type principles beyond the Bachmann-Howard level.

math.LO↗

arXiv2022

Intermediate Goodstein principles

The original Goodstein process proceeds by writing natural numbers in nested exponential $k$-normal form, then successively raising the base to $k+1$ and subtracting one from the end result. Such sequences always reach zero, but this fact is unprovable in Peano arithmetic. In this paper we instead consider notations for natural numbers based on the Ackermann function. We define three new Goodstein processes, obtaining new independence results for $ {\sf ACA}_0$, ${\sf ACA}_0'$ and ${\sf ACA}_0^+$, theories of second order arithmetic related to the existence of Turing jumps.

math.LO↗