In this thesis, we give a substantial formalisation of classical set theory in the proof system Coq. We assume an axiomatisation of ZF and present a development of the theory containing relations, functions and ordinals. The implementation follows the structure of standard text books. In the context of this theory, we prove Zermelo’s Well-Ordering Theorem and the Axiom of Choice equivalent. In ...

We introduce a new simple first-order framework for theories whose objects are wellorderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gödel’s axiom of constructibility. In list theory there are strong arguments favoring Gödel’...

In our previous paper (J. Combin. Theory Ser. A 103 (2) (2003) 387) we formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed an example of a distance graph on the real line R whose chromatic number depends upon the system of axioms we choo...

In the early years of set theory, Du Bois Reymond introduced a vague notion of infinitary pantachie meant to symbolize an infinity bigger than the infinity of real numbers. Hausdorff reformulated this concept rigorously as a maximal chain (a linearly ordered subset) in a partially ordered set of certain type, for instance, the set N N under eventual domination. Hausdorff proved the existence of...

In a social choice model with an infinite number of agents, there may occur “equal size” coalitions that a preference aggregation rule should treat in the samemanner.We introduce an axiom of equal treatment with respect to ameasure of coalition size and explore its interaction with common axioms of social choice. We show that, provided the measure space is sufficiently rich in coalitions of the...

Here we pursue Cantor’s theory of cardinalities of infinite sets a bit more deeply. We also begin to take a more sophisticated approach in that we identify which results depend upon the Axiom of Choice and strive to give proofs which avoid it when possible. However, we defer a formal discussion of the Axiom of Choice and its equivalents to a later installment, so the reader who has not encounte...

Proof assistants based on higher-order logic frequently use first-order automated theorem provers as proof search mechanisms. The reconstruction of the proofs generated by common tools, such as MESON and Metis, typically involves the use of the axiom of choice to simulate the Skolemisation steps. In this paper we present a method to reconstruct the proofs without introducing Skolem functions. T...

We give a rigorous construction of the Dirac Sea for fermionic quantization in non-separable Hilbert spaces. These CAR-representations depend on Axiom Choice, hence are not unique, nevertheless they unitarily equivalent to classic Fock representation.