Journal of Symbolic Logic

Random reals, the rainbow Ramsey theorem, and arithmetic conservation

Chris J. Conidis and Theodore A. Slaman

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We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2-$RAN$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $RAN_0$ and so $RAN_0 +$ 2-$RAN$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over $RAN_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-$RAN$ has non-trivial arithmetic consequences. In Section 4, we show that 2-$RAN$ is conservative over $RCA_0 + B\Sigma_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of 2-$RAN$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^- + B\Sigma_2$.

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J. Symbolic Logic, Volume 78, Issue 1 (2013), 195-206.

First available in Project Euclid: 23 January 2013

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Conidis, Chris J.; Slaman, Theodore A. Random reals, the rainbow Ramsey theorem, and arithmetic conservation. J. Symbolic Logic 78 (2013), no. 1, 195--206. doi:10.2178/jsl.7801130.

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