Timeline for About a weakening of "union axiom" on ZF set theory
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8 events
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| Nov 28, 2011 at 14:46 | comment | added | Emil Jeřábek | @Sridhar: The axiom of pairing shows that each element of $C=\mathcal P(A)\times\mathcal P(B)$ exists (so to speak), but I do not see how to prove that the class $C$ is a set. | |
| Nov 23, 2011 at 11:28 | comment | added | Sridhar Ramesh | [Note that this is essentially just Pare's argument for the construction of coproducts from products, with the axiom of replacement slapped on at the end. The key is that ZF' and its extensions still form topoi; they are just topoi which lack the particular large cardinal consequences of the synergy of the axioms of union and replacement] | |
| Nov 23, 2011 at 11:17 | comment | added | Sridhar Ramesh | I believe ZF' proves that the union of two sets always exists (at least, on what I think is a natural interpretation of what ZF' is to be): Given sets $A$ and $B$, ZF' can establish the existence of $P(A) \times P(B)$ [powersets being given directly by an axiom and cartesian products requiring only the axiom of pairing (on top of extensionality)]. From this, we can use Separation to extract the subset of such pairs with one component an empty set and the other a singleton (in either A or B); finally, we can apply replacement to this to obtain the union of A and B. | |
| Nov 21, 2011 at 18:59 | comment | added | François G. Dorais | When interpreted in category-theoretic terms, the Union Axiom is mostly there to ensure that Set (viewed as a topos) is closed under internal coproducts. This is true in any elementary topos: if $f:A\to B$ is any morphism, then $f^*$ has both a left adjoint (internal coproduct) and a right adjoint (internal product). | |
| Nov 21, 2011 at 18:37 | comment | added | Buschi Sergio | I dont know if exist a reference for ZF', i think to the WU because the general set union is too bad for have a categorical traduction, in the weak form WU is easly definible in terms (internal logic) of topos theory. Anyway I seem that Comprehension axiom allow the union of "definible" (by some formulas) subset. Thank for your interest anyway. | |
| Nov 21, 2011 at 18:20 | comment | added | Emil Jeřábek | ZF' + “unions of two sets exist” is still much weaker than ZF (in the sense that ZF proves its consistency: see mathoverflow.net/questions/48365/…). | |
| Nov 21, 2011 at 18:09 | comment | added | François G. Dorais | The axiom WU is an instance of the Comprehension Axiom, so ZF' is just ZF without the Union Axiom. I believe that ZF' cannot prove that the union of two sets always exists, but that might depend on how the other axioms are formulated. Could you give a reference version for the axioms of ZF' (especially the Replacement Axiom)? | |
| Nov 21, 2011 at 17:59 | history | asked | Buschi Sergio | CC BY-SA 3.0 |