## On the strength of Ramsey’s theorem.(English)Zbl 0843.03034

Summary: We show that, for every partition $$F$$ of the pairs of natural numbers and for every set $$C$$, if $$C$$ is not recursive in $$F$$ then there is an infinite set, $$H$$, such that $$H$$ is homogeneous for $$F$$ and $$C$$ is not recursive in $$H$$. We conclude that the formal statement of Ramsey’s Theorem for Pairs is not strong enough to prove $$ACA_0$$, the comprehension scheme for arithmetical formulas, within the base theory $$RCA_0$$, the comprehension scheme for recursive formulas. We also show that Ramsey’s Theorem for Pairs is strong enough to prove some sentences in first order arithmetic which are not provable within $$RCA_0$$. In particular, Ramsey’s Theorem for Pairs is not conservative over $$RCA_0$$ for $$\Pi^0_4$$-sentences.

### MSC:

 03F30 First-order arithmetic and fragments
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### References:

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