User tct - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:41:40Z http://mathoverflow.net/feeds/user/19498 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/81831#81831 Comment by Tct Tct 2011-11-25T14:14:57Z 2011-11-25T14:14:57Z As far as I can see, I think there is no special reason to adopt the &quot;appropriate&quot; 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). http://mathoverflow.net/questions/81815/is-the-axiom-of-union-independent-of-the-rest-of-zf/81831#81831 Comment by Tct Tct 2011-11-24T23:05:21Z 2011-11-24T23:05:21Z The claim about the &quot;weakness&quot; of the &quot;appropriate&quot; 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).