In the appendix of Herrlich's book ``Axiom of Choice'', there are diagrams illustrating properties of several models for ZF^{-}. In the introduction to the appendix of this book, it is written that they are ZF-models. Among them, there is the second Fraenkel model. Prof. Kunen informed me that Fraenkel's models did not satisfy the axiom of foundation, so the second Fraenkel model is not a model for ZF. I wonder where and how I can check easily whether this particular model of Diagram A.5 from Herrlich's book is a ZF-model or only ZF^{-}-model where ZF^{-} is ZF with foundation deleted. I would be grateful for hints.
Dear Mohamed El Naschie,
As you well know, the axiom of choice is independent of ZF. This axiom is highly controversial and many interesting theorems have been obtained in some models for ZF in which the axiom of choice fails but some weaker assumptions than the axiom of choice are satisfied. Now, I am more interested in the models for ZF +the negation of AC than in the ones in which AC holds. Of course, it is usually more comfortable to use ZFC than ZF+something weaker than AC, however a lot of important knowledge would be unavailable if we proved mathematical theorems carelessly only in ZFC without deeper insight into them.
I am also looking for a system of axioms good for both mathematics and physics, and each time, when I meet AC in physics, something goes wrong with the ``degree of truth '' in science. Simply, sometimes the ``degree of truth'' seems too small for me when a careless usage of AC appears. I am closer to truth when I work with models for ZF which are not necessarily models for ZFC. This is partly why I appreciate the excellent book ``Axiom of Choice'' by Horst Herrlich so much and I want to study it in deep even if there are some minor mistakes in it. I am so sad that Horst Herrlich passed away.
Regards, Eliza
Dear Eliza,
Many of the results obtained for ZF^(-) models can be transferred to ZF models. See p.175 in the book ``Consequences of the Axiom of Choice'' by P. Howard and J. E. Rubin.
All the best,
Kyriakos
Dear Prof. Mohamed El Naschie,
Thank you for your long letter. I realize that AC can be comfortable for physicists and I am afraid that also for most mathematicians ; however, this kind of comfort may lead to statements taken for true statements, while their " truth" does not have satisfactory proof, so it is frequently still unknown whether they are true or false,or just unprovable. For example, for statements p and q, mathematicians check whether the implication "if p, then q" holds. If it is known that the implication "if p, then q" holds and it is unknown whether p holds, one should not deduce that the truth of the implication implies that q holds Nobody has ever proved AC because it is impossible to prove AC. If one assumes ZF and proves correctly that ZF+AC implies q, this means that, in every model M for ZF, if AC were true in M, then q would be true in M. This does not mean at all that q is certainly true. Banach-Tarski theorem can fail in a model for ZF, so physicists should be very careful while using it. There are a lot of descriptions in physics that are inadequate with the reality.
Regards, Eliza Wajch
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