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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.

edited Nov 24 2011 at 23:08

asked Nov 24 2011 at 15:52

Tct
53●3

	Can you write down what the "appropiate version of collection" is? – Ramiro de la Vega Nov 24 2011 at 17:47
	If the answers ware helpful, pick one that helped the most and 'accept' it by clicking on the tick. This is better than editing your question. – David Roberts Nov 24 2011 at 23:10

David Feldman · Accepted Answer · 2011-11-24 19:46:53Z UTC

8

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.

edited Nov 24 2011 at 19:46

David Feldman

answered Nov 24 2011 at 19:33

Andreas Blass
33.8k●4●59●111

	The claim about the "weakness" of the "appropriate" version of the collection axiom still puzzled me; but I realize now that it is probably irrelevant, since the implication is proved using the rest of axioms of ZF (including union). – Tct Nov 24 2011 at 23:05
	Shoenfield, Mathematical Logic also uses the "appropriate" version. Do I understand correctly that you are saying there are reasons to prefer the "usual" version? – rgrig Nov 25 2011 at 13:08
	As far as I can see, I think there is no special reason to adopt the "appropriate" version of Bourbarki or Shoenfield (but remember: I am not an specialist, not even a mathematician!). I simply got confused about of the independence of Unions by the statements in Tourlakis (2003). – Tct Nov 25 2011 at 14:14
	@Tct: Well, I'm asking if there is a reason to choose the usual one, which is not the same as asking if there is a reason not to choose the appropriate one. :P – rgrig Nov 25 2011 at 17:06

Kaveh · Answer 2 · 2011-11-24 19:37:40Z UTC

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:

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

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

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