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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.

4 typo

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:

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

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

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. See III.8.12 and p. 165.

PS: I apologize in advance if think the question statement istoo basic :

$(∀x)(∃z)(∀y)({\mathcal P} [x, y] y z) (not a mathematician!). Also, \forall A)(\exists B)(\forall z)(z\in B\leftrightarrow (\exists w\in A){\mathcal P}[x,y])$

I have googled it translated the notation from III.8.12 and followed some false trails before asking hereIII.2 (obviating any reference to ur-elements).

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