▁strictly ▁increasing ▁bij ection . ▁This ▁result ▁implies , ▁for ▁example , ▁that ▁there ▁exists ▁a ▁strictly ▁increasing ▁bij ection ▁between ▁the ▁set ▁of ▁all ▁rational ▁numbers ▁and ▁the ▁set ▁of ▁all ▁real ▁algebraic ▁numbers . ▁ ▁any ▁two ▁count ably ▁infinite ▁atom less ▁Boolean ▁al gebras ▁are ▁is omorphic ▁to ▁each ▁other . ▁ ▁any ▁two ▁equivalent ▁count able ▁atomic ▁models ▁of ▁a ▁theory ▁are ▁is omorphic . ▁ ▁the ▁Erd ős – R ény i ▁model ▁of ▁random ▁graphs , ▁when ▁applied ▁to ▁count ably ▁infinite ▁graphs , ▁always ▁produces ▁a ▁unique ▁graph , ▁the ▁R ado ▁graph . ▁ ▁any ▁two ▁many - complete ▁recurs ively ▁enumer able ▁sets ▁are ▁recurs ively ▁is omorphic . ▁ ▁Application ▁to ▁dens ely ▁ordered ▁sets ▁ ▁Suppose ▁that ▁▁ ▁( A , ▁ ≤ A ) ▁and ▁( B , ▁ ≤ B ) ▁are ▁linear ly ▁ordered ▁sets ; ▁ ▁They ▁are ▁both ▁un bounded , ▁in ▁other ▁words ▁neither ▁A ▁nor ▁B ▁has ▁either ▁a ▁maximum ▁or ▁a ▁minimum ; ▁ ▁They ▁are ▁dens ely ▁ordered , ▁i . e . ▁between ▁any ▁two ▁members ▁there ▁is ▁another ; ▁ ▁They ▁are ▁count ably ▁infinite . ▁ ▁Fix ▁enumer ations ▁( without ▁repet ition ) ▁of ▁the ▁underlying ▁sets : ▁ ▁A ▁= ▁{ ▁a 1 , ▁a 2 , ▁a 3 , ▁… ▁}, ▁B ▁= ▁{ ▁b 1 , ▁b 2 , ▁b 3 , ▁… ▁} . ▁ ▁Now ▁we ▁construct ▁a ▁one - to - one ▁correspond