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So some first-order sentences are absolute in all transitive models of ZFC - by absolute, it is meant that either they are false in all transitive models, or true in all transitive models.

What would be examples of such absoluteness? And how does one usually prove these?

(For example, a statement like "There is a set that is a $\omega$-model of ZFC" is absolute, and I want to know how one proves this type of statement.)

Edit: I will read Kunen along with Jech, but I would still like to know a reason/proof that shows why $\Delta_1$ sentences are absolute.

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(Provably) $\Delta_1$ statements. This is best possible. Kunen's book on Set theory explains this carefully (Chapter 2 of the new version). –  Andres Caicedo Jun 28 '13 at 6:44

2 Answers 2

Proofs of these statements you can find in the usual books, so let me discuss the reason why do we have the absoluteness to begin with.

First of all, the trait of transitive models is that they are substructures of the universe. If $M$ is a transitive model, and $M\models x\in y$ then $x\in y$. This, and the transitivity of $M$ of course, is the key reason why $\Delta_0$ formula are absolute.

Recall that a $\Delta_0$ formula is a formula is equivalent to a formula written using bounded quantifiers, e.g. $\varphi(x)$ being $(\exists y\in x)(\forall z\in y)(z\notin x)$. But bounded quantification pass between the transitive models exactly because they are transitive, if $M\models(\exists y\in x)(\forall z\in y)(z\notin x)$ then this statement holds for the real $\in$ relation, and it holds for all the members of $x,y,z$ and so on.

On the other hand, if $V$ satisfies $\varphi(x)$, and $x\in M$ then by transitivity it is not hard to see that $y$ which exists also lies in $M$, and that every $z\in y$ also lies in $M$, and therefore $M$ satisfies this formulas as well.

Formally we prove this by induction on the number of quantifiers, or length, or complexity of the formula. Whatever you find the easiest, neither is very difficult.

The reason for $\Delta_1$ absoluteness is slightly trickier, but it's a very nice trick indeed. Let $\varphi(x,y)$ be a $\Delta_0$ formula, if for some $x\in M$ we have that $M\models\exists y\varphi(x,y)$ then $M$ knows about a witness for $\varphi(x,y)$. But now we have this witness in the universe, and $\varphi(x,y)$ is a $\Delta_0$ formula so $\varphi(x,y)$ holds in the universe, so $\exists y\varphi(x,y)$ is true in the universe.

Similarly, if $\forall y\varphi(x,y)$ holds in the universe, and $x\in M$ then it holds for all $y\in M$ as well, and $\varphi(x,y)$ is again a $\Delta_0$ formula so it is absolute and $M\models\varphi(x,y)$ for every $y\in M$, and therefore $M\models\forall y\varphi(x,y)$.

So existential quantification goes up and universal quantification goes down. But what is a $\Delta_1$ formula? It is a formula which is equivalent to an existential quantification of a $\Delta_0$ formula, and equivalent to a universal quantification of a[nother] $\Delta_0$ formula. One is absolute upwards and the other downwards making our $\Delta_1$ formula absolute in both directions.

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One technique to show that a sentence is absolute for transitive models of $ZFC$ is to show that is equivalent to a $\Delta_0$ formula (a formula where every variable is bound). Some examples of this are:

  • $x$ is an ordinal
  • $x = 0$
  • $x$ is a limit/successor ordinal
  • $f$ is a function/injective/surjective
  • etc

For example $x$ is an ordrinal $iff$ $x$ is transitive and well-ordered by $\in$ i.e. $iff$ $\forall y \in x (\forall z \in y (z \in x)) \wedge$ $\forall y \in x (y \notin y) \wedge \forall a,b,c \in x (a \in b \wedge b \in c \implies a \in c) \wedge \forall y,z \in x (y \in z \vee z \in y \vee z = y) \wedge$ $\forall y \subseteq x (y \neq \emptyset \implies \exists z \in y \forall z' \in y (z' \notin z)$

The first line says $x$ is transitive, the second says $\in$ totally orders $x$ and the last says $\in$ is well-founded on $x$ (of course, previously I would have needed to show that $\subseteq$ is also absolute (it is, even more, its $\Delta_0$).

This is the simplest route to proving something is absolute. As Andres points out even more can be shown i.e. if you have a $\Delta_1$ statement then it is absolute as well. In general the technique is just to relativize the sentence to whatever model you are considering, but of course that becomes very tedious very fast. I would say the presentation in Kunen is very good and it will clear a lot of things up for you. Hopefully this helps a little!

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Also, many things are not absolute for transitive models. For example, $\mathcal P(x)$ is not absolute and neither is being a cardinal. –  Ryan Sullivant Jun 28 '13 at 7:08

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