### Abstract

In this paper, we discuss uniform versions of some axioms of second order arithmetic in the context of higher order arithmetic. We prove that uniform versions of weak weak König's lemma WWKL and Σ_{1} ^{0} separation are equivalent to (∃^{2}) over a suitable base theory of higher order arithmetic, where (∃^{2}) is the assertion that there exists Φ^{2} such that Φ f^{1} = 0 if and only if ∃x^{0} (fx = 0) for all f. We also prove that uniform versions of some well-known theorems are equivalent to (∃^{2}) or the axiom (Suslin) of the existence of the Suslin operator.

Original language | English |
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Pages (from-to) | 587-593 |

Number of pages | 7 |

Journal | Mathematical Logic Quarterly |

Volume | 50 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2004 Jan 1 |

Externally published | Yes |

### Keywords

- Higher order arithmetic
- Reverse mathematics
- Second order arithmetic
- Weak König's lemma

### ASJC Scopus subject areas

- Logic

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## Cite this

Sakamoto, N., & Yamazaki, T. (2004). Uniform versions of some axioms of second order arithmetic.

*Mathematical Logic Quarterly*,*50*(6), 587-593. https://doi.org/10.1002/malq.200310122