Tameness for set theory $I$

20 Mar 2020 Viale Matteo

The paper is a first of two and aims to show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a $\Pi_2$-property formalized in an appropriate language for second or third order number theory is forcible from some $T\supseteq\mathsf{ZFC}+$large cardinals if and only if it is consistent with the universal fragment of $T$ if and only if it is realized in the model companion of $T$... (read more)

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