Is the Axiom of Union independent of the rest of ZF? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:54:18Z http://mathoverflow.net/feeds/question/81815 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81815/is-the-axiom-of-union-independent-of-the-rest-of-zf Is the Axiom of Union independent of the rest of ZF? Tct 2011-11-24T15:52:50Z 2011-11-24T23:08:11Z <p>Short version: Is the axiom of union independent of the rest of axioms of ZF?</p> <p>NO) <a href="http://books.google.es/books?id=zwv0RgAACAAJ&amp;dq=tourlakis+set+vol+2&amp;hl=es&amp;ei=vWbOTu6YHYmb8QP8-MzODw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CDEQ6AEwAA" rel="nofollow">Tourlakis (2003)</a> 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:</p> <p>«Bourbaki (1966b) adopts the axiom of pairing, but adopts collection version (2), and proves both separation and union»</p> <p>YES) In the other hand, I have read <a href="http://mathoverflow.net/questions/48365/minimal-subset-of-axioms-for-zfc/54328#54328" rel="nofollow">here</a> and <a href="http://tiddlyspace.com/bags/oxkunengroup_public/tiddlers/4.7%3A%2520Models%2520for%2520ZFC-Replacement%2520and%2520ZFC-Unions" rel="nofollow">here</a> something like "$H_{\kappa}$ is a model for ZF-Union+¬Union", where $\kappa$ was $\beth_\omega$ or a singular cardinal.</p> <p>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.</p> <p>[*] 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:</p> <p>$(∀x)(∃z)(∀y)({\mathcal P} [x, y] → y ∈ z) → (\forall A)(\exists B)(\forall y)(y\in B\leftrightarrow (\exists x\in A){\mathcal P}[x,y])$</p> <p>I have translated the notation from III.8.12 and III.2 (obviating any reference to ur-elements).</p> <p>EDIT: Thank you very much for the answers, they were really helpful.</p> http://mathoverflow.net/questions/81815/is-the-axiom-of-union-independent-of-the-rest-of-zf/81828#81828 Answer by Kaveh for Is the Axiom of Union independent of the rest of ZF? Kaveh 2011-11-24T18:26:00Z 2011-11-24T19:37:40Z <p>The axiom is</p> <p>$$\forall x \ \exists y \ \forall z \ (\varphi(z,x) \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ \varphi(y,x))$$</p> <p>Consider</p> <p>$$\forall x \ \exists y \ \forall z \ (z\in x \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow \exists x\in X \ y\in x )$$</p> <p>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 $X$.</p> <hr> <p>The other version of the axiom:</p> <p>$$\forall x\in X \ \exists y \ \varphi(x,y) \to \ \exists Y \ \forall x\in X \ \exists y\in Y \ \varphi(x,y)$$</p> http://mathoverflow.net/questions/81815/is-the-axiom-of-union-independent-of-the-rest-of-zf/81831#81831 Answer by Andreas Blass for Is the Axiom of Union independent of the rest of ZF? Andreas Blass 2011-11-24T19:33:28Z 2011-11-24T19:46:53Z <p>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 <code>$H_\kappa$</code> for singular $\kappa$; it should be the collection of those sets $x$ such that each member of the transitive closure $TC({x})$ has cardinality $&lt;\kappa$. (This does not imply that the whole $TC({x})$ has cardinality $&lt;\kappa$, which is the usual meaning of <code>$H_\kappa$</code>.)</p> <p>The "appropriate version" of collection used by Bourbaki seems appropriate mainly in the sense that the axiom of union has been built in.</p>