User kate.r - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:24:20Z http://mathoverflow.net/feeds/user/17106 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72736/zfc-set-membership-and-fol ZFC, set membership and FOL kate.r 2011-08-12T03:11:44Z 2011-08-13T06:28:35Z <p>Hi,</p> <p>Is set membership <em>defined</em> in the signature of ZFC, or is it *specified" in the signature of ZFC? The wikipedia article <a href="http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory" rel="nofollow">http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory</a> says that the signature <em>has</em> set membership, but what does it mean by <em>has</em>?</p> <p>If I understand correctly, the axioms specify the properties of set membership. Given set membership, how come ZFC can be foramlised in FOL when some axioms, e.g., axiom of infinity, require quantification of sets? Aren't sets unary predicates since S(a) = True iff a is a member of S? Quantification over unary predicates is a feature of second-order though.</p> <p>I must be missing something...</p> <p>Thanks</p> http://mathoverflow.net/questions/72736/zfc-set-membership-and-fol/72746#72746 Comment by kate.r kate.r 2011-08-12T12:21:15Z 2011-08-12T12:21:15Z Just to get my terminology right: If I understand correctly, a theory specifies a symbol in its signature and defines the symbol's properties in its axioms -- is that right?