Looking at consecutive numbers and their English names:

• zero (z,e,r,o) and one (o,n,e) share: e, o
• one (o,n,e) and two (t,w,o) share: o
• two (t,w,o) and three (t,h,r,e,e) share: t
• three (t,h,r,e,e) and four (f,o,u,r) share: r
• four (f,o,u,r) and five (f,i,v,e) share: f, r
• five (f,i,v,e) and six (s,i,x) share: i
• six (s,i,x) and seven (s,e,v,e,n) share: s
• seven (s,e,v,e,n) and eight (e,i,g,h,t) share: e
• eight (e,i,g,h,t) and nine (n,i,n,e) share: e, i, n, t
• nine (n,i,n,e) and ten (t,e,n) share: n, e
• ten (t,e,n) and eleven (e,l,e,v,e,n) share: e, n
• eleven (e,l,e,v,e,n) and twelve (t,w,e,l,v,e) share: e, l
• twelve (t,w,e,l,v,e) and thirteen (t,h,i,r,t,e,e,n) share: t, e
• thirteen and fourteen share: t, e, r
• fourteen and fifteen share: f, e
• fifteen and sixteen share: i, e
• sixteen and seventeen share: s, e
• seventeen and eighteen share: e, t
• eighteen and nineteen share: e, i, n
• nineteen and twenty share: n, e, t

After checking many consecutive pairs, we need to find two that share no letters. This requires careful examination of each pair.