TL;DR
This paper introduces librationism, a novel logical framework that addresses paradoxes and provides a foundation for mathematics, including a method to interpret ZFC within a weaker system.
Contribution
It develops librationism ({ ext{ extpounds}}), isolates a non-paradoxical domination operation, and extends interpretations of ZF and ZFC using new notions like librationist capture.
Findings
Librationism clarifies mathematical and philosophical paradoxes.
A domination operation is identified that avoids paradoxes.
ZFC can be interpreted within librationism plus additional postulates.
Abstract
We develop librationism, {\pounds}, and clarify some mathematical and philosophical matters which relate to the particular manner in which it deals with the paradoxes and to its usefulness as a foundation for mathematics and type free reasoning. We isolate a domination operation which unlike the power set operation is not paradoxical and which helps us isolate the definable real numbers. We show that {\pounds} plus a postulate and a postulation interprets ZFC; our strategy for achieving this involves extending an interpretation by Harvey Friedman of ZF in a system weaker than ZF with collection minus extensionality and a novel notion of which entails collection, specification and choice in desired contexts.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Philosophy and Theoretical Science