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PREDICATIVE COLLAPSING PRINCIPLES
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 09 December 2019, pp. 511-530
- Print publication:
- March 2020
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- Article
- Export citation
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We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal α there exists an ordinal β such that
$1 + \beta \cdot \left( {\beta + \alpha } \right)$ (ordinal arithmetic) admits an almost order preserving collapse into β. Arithmetical comprehension is equivalent to a statement of the same form, with 
$\beta \cdot \alpha$ at the place of 
$\beta \cdot \left( {\beta + \alpha } \right)$. We will also characterize the principles that any set is contained in a countable coded ω-model of arithmetical transfinite recursion and arithmetical comprehension, respectively.
