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Tarski's theorem about choice and the alternative axiomatic extension of NFU.

The main advantage of NFU over plain NF is that it does not disprove the axiom of choice. Both the axiom of choice and the axiom of infinity are independent, but relatively consistent with NFU. Therefore, NFU + Inf + AC is a theory that is rich enough to encompass all the existent mathematics— but with some technical difficulties. Namely, it is hard to work with Kuratowski’s ordered pairs because they are not typeleveled, meaning that the ordered pair does not have the same type as its projections. The fortunate circumstance is that everything can be developed irrespective of how we define ordered pairs, but making them type-leveled yields a significant simplification of exposition. A prevalent solution to that problem in contemporary literature is to postulate a new axiom, so-called axiom of ordered pairs. From our point of view, the introduction of that axiom lacks the proper motivation and justification for inclusion, and it also creates new problems ; it can be expressed only by introducing new primitive notions. Such evasions are a common occurrence in contemporary NFU. In theory NFU + Inf + AC we can define type-leveled ordered pairs using Tarski’s theorem about choice which is equivalent to the AC. The main drawback for using NFU+Inf+ACis that we first need to develop all the necessary theory with Kuratowski’s ordered pairs, prove the equivalence of the AC to Tarski’s theorem, and only then define type-leveled ordered pairs. This is apparently unavoidable. However, we propose an approach which does that hard work only once: to start with NFU+Inf+Tarski, then define type- leveled ordered pairs, and then easily prove the equivalence of Tarski’s theorem to the AC. In order to justify that shift of axioms, we must show that in NFU+Inf +AC we can prove the equivalence of the AC to the Tarski’s theorem, but then it becomes a self-sufficient result one can just cite afterwards. It is also worth saying that Tarski seems much more justified as an axiom than the “axiom of ordered pairs”. In order to complete our presentation, we also need to show that the same thing can be done in NFU+Inf+Tarski, but the equivalence proof can be mirrored by the former, and that proof will be in fact much simpler. In effect, those two theories are equiconsistent.

Axiomatization ; Tarski's theorem about choice ; New Foundations with Urelements

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