▁notation . ▁For ▁example , ▁the ▁even ▁integers ▁can ▁be ▁constructed ▁as ▁the ▁subset ▁of ▁the ▁integers ▁ ▁satisfying ▁the ▁con gru ence ▁mod ulo ▁predicate ▁: ▁▁▁▁ ▁In ▁general , ▁the ▁subset ▁of ▁a ▁set ▁ ▁obey ing ▁a ▁formula ▁ ▁with ▁one ▁free ▁variable ▁ ▁may ▁be ▁written ▁as : ▁▁▁▁ ▁The ▁ax iom ▁schema ▁of ▁specification ▁states ▁that ▁this ▁subset ▁always ▁exists ▁( it ▁is ▁an ▁ax iom ▁schema ▁because ▁there ▁is ▁one ▁ax iom ▁for ▁each ▁). ▁Form ally , ▁let ▁ ▁be ▁any ▁formula ▁in ▁the ▁language ▁of ▁Z FC ▁with ▁all ▁free ▁variables ▁among ▁ ▁( ▁is ▁not ▁free ▁in ▁). ▁Then : ▁▁▁▁ ▁Note ▁that ▁the ▁ax iom ▁schema ▁of ▁specification ▁can ▁only ▁construct ▁subsets , ▁and ▁does ▁not ▁allow ▁the ▁construction ▁of ▁entities ▁of ▁the ▁more ▁general ▁form : ▁▁▁▁ ▁This ▁restriction ▁is ▁necessary ▁to ▁avoid ▁Russell ' s ▁par ado x ▁and ▁its ▁variants ▁that ▁accomp any ▁na ive ▁set ▁theory ▁with ▁un rest rict ed ▁compreh ension . ▁ ▁In ▁some ▁other ▁ax iom at izations ▁of ▁Z F , ▁this ▁ax iom ▁is ▁redundant ▁in ▁that ▁it ▁follows ▁from ▁the ▁ax iom ▁schema ▁of ▁replacement ▁and ▁the ▁ax iom ▁of ▁the ▁empty ▁set . ▁ ▁On ▁the ▁other ▁hand , ▁the ▁ax iom ▁of ▁specification ▁can ▁be ▁used ▁to ▁prove ▁the ▁existence ▁of ▁the ▁empty ▁set , ▁denoted ▁, ▁once ▁at ▁least ▁one ▁set ▁is ▁known ▁to ▁exist ▁( see ▁above ). ▁One ▁way ▁to ▁do ▁this ▁is ▁to ▁use ▁a ▁property