$\Pi^1_1$-Comprehension as a Well-Ordering Principle

Anton Freund

doi:10.1016/j.aim.2019.106767

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arXiv · 1809.06759

$\Pi^1_1$-Comprehension as a Well-Ordering Principle

Anton Freund

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Abstract

A dilator is a particularly uniform transformation $X\mapsto T_X$ of linear orders that preserves well-foundedness. We say that $X$ is a Bachmann-Howard fixed point of $T$ if there is an almost order preserving collapsing function $\vartheta:T_X\rightarrow X$ (precise definition to follow). In the present paper we show that $\Pi^1_1$-comprehension is equivalent to the assertion that every dilator has a well-founded Bachmann-Howard fixed point. This proves a conjecture of M. Rathjen and A. Montalb\'an.

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Anton Freund. 2018-09-18. $\Pi^1_1$-Comprehension as a Well-Ordering Principle. https://doi.org/10.1016/j.aim.2019.106767

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