Refine search
Actions for selected content:
1 results
THE KAUFMANN–CLOTE QUESTION ON END EXTENSIONS OF MODELS OF ARITHMETIC AND THE WEAK REGULARITY PRINCIPLE
- Part of
-
- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 04 March 2025, pp. 1-19
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each
$n\in \mathbb {N}$ and any countable model of 
$\mathrm {B}\Sigma _{n+2}$, we construct a proper 
$\Sigma _{n+2}$-elementary end extension satisfying 
$\mathrm {B}\Sigma _{n+1}$, which answers a question by Clote positively. We also give a characterization of the countable models of 
$\mathrm {I}\Sigma _{n+2}$ in terms of their end extendibility, similar to the case of 
$\mathrm {B}\Sigma _{n+2}$. Along the proof, we introduce a new type of regularity principle in arithmetic called the weak regularity principle, which serves as a bridge between the model’s end extendibility and the amount of induction or collection it satisfies.
