## Notre Dame Journal of Formal Logic

### Combinatorial Unprovability Proofs and Their Model-Theoretic Counterparts

#### Abstract

For a function $f$ with domain $[X]^{n}$, where $X\subseteq\mathbb{N}$, we say that $H\subseteq X$ is canonical for $f$ if there is a $\upsilon\subseteq n$ such that for any $x_{0},\ldots,x_{n-1}$ and $y_{0},\ldots,y_{n-1}$ in $H$, $f(x_{0},\ldots,x_{n-1})=f(y_{0},\ldots,y_{n-1})$ iff $x_{i}=y_{i}$ for all $i\in\upsilon$. The canonical Ramsey theorem is the statement that for any $n\in\mathbb{N}$, if $f:[\mathbb{N}]^{n}\rightarrow\mathbb{N}$, then there is an infinite $H\subseteq\mathbb{N}$ canonical for $f$. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey theorem with a largeness condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying $\operatorname{PA}$.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 2 (2014), 231-244.

Dates
First available in Project Euclid: 24 April 2014

https://projecteuclid.org/euclid.ndjfl/1398345782

Digital Object Identifier
doi:10.1215/00294527-2420654

Mathematical Reviews number (MathSciNet)
MR3201834

Zentralblatt MATH identifier
1301.03062

#### Citation

Aghaei, Mojtaba; Khamseh, Amir. Combinatorial Unprovability Proofs and Their Model-Theoretic Counterparts. Notre Dame J. Formal Logic 55 (2014), no. 2, 231--244. doi:10.1215/00294527-2420654. https://projecteuclid.org/euclid.ndjfl/1398345782

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