Is the Axiom of Union independent of the rest of ZF? - MathOverflow

Is the Axiom of Union independent of the rest of ZF?

2011-11-24T15:52:50Z

Short version: Is the axiom of union independent of the rest of axioms of ZF?

NO) Tourlakis (2003) says in p. 177 that the axiom of union can be derived from the rest of ZF if an appropriate version of collection axiom[*] is chosen. The quote is:

«Bourbaki (1966b) adopts the axiom of pairing, but adopts collection version (2), and proves both separation and union»

YES) In the other hand, I have read here and here something like "$H_{\kappa}$ is a model for ZF-Union+¬Union", where $\kappa$ was $\beth_\omega$ or a singular cardinal.

Any reference on the subject would be highly appreciated. I apologize in advance if the question is too basic (not a mathematician!). Also, I have googled it and followed some false trails before asking here. Thanks.

[*] The appropriate version of collection is apparently weaker (or equivalent at most) than the collection axiom that he is adopting in his text. I think the statement is:

$(∀x)(∃z)(∀y)({\mathcal P} [x, y] → y ∈ z) → (\forall A)(\exists B)(\forall y)(y\in B\leftrightarrow (\exists x\in A){\mathcal P}[x,y])$

I have translated the notation from III.8.12 and III.2 (obviating any reference to ur-elements).

EDIT: Thank you very much for the answers, they were really helpful.

Answer by Kaveh for Is the Axiom of Union independent of the rest of ZF?

2011-11-24T18:26:00Z

The axiom is

$$\forall x \ \exists y \ \forall z \ (\varphi(z,x) \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ \varphi(y,x)) $$

Consider

$$\forall x \ \exists y \ \forall z \ (z\in x \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ y\in x ) $$

This is an instance of the axiom. Now the left side is true (take $y=x$). The right side expresses the existence of the union of $X$.

The other version of the axiom:

$$\forall x\in X \ \exists y \ \varphi(x,y) \to \ \exists Y \ \forall x\in X \ \exists y\in Y \ \varphi(x,y)$$

Answer by Andreas Blass for Is the Axiom of Union independent of the rest of ZF?

2011-11-24T19:33:28Z

The usual version of collection is the second one in Kaveh's answer. It, with the remaining axioms, won't give the axiom of union. A counterexample is given in the question, except for a slightly unusual definition of $H_\kappa$ for singular $\kappa$; it should be the collection of those sets $x$ such that each member of the transitive closure $TC({x})$ has cardinality $<\kappa$. (This does not imply that the whole $TC({x})$ has cardinality $<\kappa$, which is the usual meaning of $H_\kappa$ .)

The "appropriate version" of collection used by Bourbaki seems appropriate mainly in the sense that the axiom of union has been built in.