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I am working on a translation from French about, ultimately, the axiom of constructibility. In the opening paragraphs, the author describes how we are going to assume the ZF axioms are consistent to create a stronger theory with this axiom. Then, he goes on to say that it is harmless to also assume that there is a transitive ZF model (a set model). Then, he continues by making a precise statement about this which I translated as "More precisely, the theory of Zermelo-Fraenkel, including the single binary predicate $\in$ admits an expansion of the theory with $\in$ to include a unary predicate $\tau$ with an axiom saying a there is a unique set which is transitive and a ZF model by checking with $\tau$." I'm not sure if I got it right, so my question is:

How do you expand a model of ZF + the axiom of consistency to include that the model is transitive by adding a unary predicate?

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Erin: What do you mean by "the axiom of consistency"? What text is this from (it may be useful to provide a precise reference). –  Andres Caicedo Dec 20 '12 at 19:49
Perhaps it makes sense to give the article you're translating? –  Todd Trimble Dec 20 '12 at 19:50
I mean, he means, the axiom of consistency is the statement: there is a set model of the ZF axioms. Its a book Cours de Logique Mathematique, but I'd rather not have someone just translate the french :) –  Erin Carmody Dec 20 '12 at 19:55

1 Answer 1

up vote 5 down vote accepted

Most likely the author is referring to the following fact:

The Reflection Scheme proves each instance of the following scheme: Let $\Delta$ be a finite subset of the axioms of ZF. ZF proves that there is a transitive set which is a model of $\Delta$.

Let $\tau$ be a constant symbol, and assume ZF is consistent. By the compactness theorem, there is a model of the following set of sentences:

ZF $\cup$ $\bigcup$ { $\tau$ is a transitive model of $\Delta$ : $\Delta$ is a finite subset of ZF }.

Note that $\tau$ is asserted to be a transitive model of ZF via a scheme. The single statement that there exists a transitive model of ZF has (much) greater consistency strength than CON(ZF), by considerations having to do with Gödel's second incompleteness theorem, plus absoluteness of arithmetical statements for transitive models.

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Thanks. This is just what i needed. I think you got what the author was saying. –  Erin Carmody Dec 20 '12 at 21:13
A slightly stronger scheme is "$V_\delta\prec V$", which asserts, in the language with constant symbol $\delta$, that $\delta$ is an ordinal and that $\forall a\in V_\delta, (\varphi(a)\iff V_\delta\models\varphi[a])$, for each formula $\varphi$. The compactness argument shows that ZFC+$V_\delta\prec V$ is equiconsistent with just ZFC, and indeed conservative over ZFC, since by reflection any given ZFC model can be expanded to a model of any finite fragment of ZFC+$V_\delta\prec V$. –  Joel David Hamkins Dec 20 '12 at 22:07
This is a plausible guess as to what the author might have meant, but in addition to the fact that the quotation mentioned an axiom rather than a scheme, there's the fact that the theory in question (ZF plus a scheme saying the transitive set $\tau$ satisfies each axiom of ZF individually) is a conservative extension of just ZF. One doesn't use the assumption that ZF is consistent. This theory (or the analogous one using ZFC) is what Shoenfield uses for the development of forcing in his book "Mathematical Logic". –  Andreas Blass Dec 20 '12 at 22:08

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