Topos as A Model for Set Theory and Independence Proof

Introduction

Part 1:  For any elementary topos, its internal logic, considered as propositional calculus, is a Heyting algebra. Some constructions are required to make a topos a suitable model of set theory, where the propositional calculus needs to be Boolean and two-valued.  We discuss some properties of topos desired for a suitable model: the existence of a natural number object (NNO), being Boolean, satisfying the axiom of choice (AC) and well-pointedness. I gave some proofs for some exercises in [1] and organise them into a thread.

Part 2: We sketch the construction of a Cohen topos, which is Boolean and satisfies the axiom of choice, where the continuum hypothesis (CH) fails [1, pp. 277-290]. After that, we improve the result by a filter-quotient construction for obtaining a well-pointed topos satisfying AC with an NNO, where CH fails. Such a filter-quotient construction indeed provides a stronger model than the Cohen topos, with the four properties mentioned above satisfied.

Full Article

Here is the article Topos as A Model for Set Theory and Independence Proof which is an excerpt from my previous work (2018) [2]. It discusses some desired properties of topos as a suitable model of set theory and ideas on the independence proof of the axiom of choice. I gave detailed proofs for some exercises in [1] and organise them into a thread. I have also cited necessary definitions and propositions for completeness (I omitted the proofs for propositions that can be directly found in [1]). Assuming basic familiarity with category theory,  this article should be self-contained.  Here is an appendix for quick reference: Appendix.

Note: there are some missing cross-references which should not affect reading.

Summary

To show the foundation aspects of topos, we have clarified the differences and connections between external and internal concepts and discussed the properties required for a topos as a suitable model of set theory, that is, the existence of a natural number object (NNO), being Boolean, satisfying the axiom of choice (AC) and well-pointedness.

The universality of the NNO was discussed and we showed how the recursion, addition, multiplication and the partial order can be defined from the NNO and how they correspond to that for sets. The propositional calculus, the Heyting algebra and some equivalent conditions for a topos being Boolean were discussed. The presheaf topos and sheaf topos, as special examples, were illustrated for the Heyting algebras and the equivalent conditions for being Boolean. Some propositions about external and internal projective objects were proved and related to the external and internal axiom of choice (AC and IAC). We proved the equivalent conditions for a presheaf topos satisfying AC or IAC, thus had an example of a presheaf topos satisfying IAC but not AC.
We also proved the equivalence of AC and IAC for a well-pointed topos and some equivalent conditions for well-pointedness.

To show an application of the foundation aspect of topos, we sketched the process of construction of the Cohen topos, which is Boolean and satisfies AC, where the continuum hypothesis (CH) fails, with some details examined. We then proved that the filter-quotient construction on the Cohen topos preserves the cardinal inequality and the NNO. We also proved that the filter-quotient topos constructed from a Boolean topos satisfying AC is well-pointed and again satisfies AC. Together with the fact that a well-pointed topos is both Boolean and two-valued, we concluded that the filter-quotient topos constructed from the Cohen topos satisfies all the properties for sets and violates CH, thus is an improved model in the independence proof.

Further Remark

Later in the book [1], the Mitchell-Benabou language, as a first-order language for topos, is constructed, and then the relation to set theory will become clear. The internal properties we have discussed can be expressed by some formulas that correspond to the formulas expressing related properties in set theory. For example, a topos is Boolean iff the formula  \forall p( p\vee \neg p)  holds; a topos satisfies IAC iff the formula for AC holds. Also, the geometric construction for the independence proof can be translated to the language of forcing. On the other hand, we can say that the symbols in logic can be interpreted diagrammatically through categories. In this way, we see the symbolic and diagrammatic aspects of logic. Topos theory provides a diagrammatic view of logic, which is particularly good when dealing with structures. Though the idea in the independence proof is actually equivalent to that in set theory, the diagrammatic view and by topos theory gives us a different viewpoint.

References

[1] MacLane, S., & Moerdijk, I. (2012). Sheaves in geometry and logic: A first introduction to topos theory. Springer Science & Business Media.

[2] Likun Xie. (2018). Topos Foundation for Set Theory. Unpublished Bachelor Thesis. The University of Manchester, UK.