All songs written by Ed Sheeran and Amy Wadge. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy. Six original songs were composed for the series by singer-songwriter Amy Wadge. It was written by Camila Cabello, Jordan Reynolds and Amy Wadge. These degrees are ordered in the Wadge hierarchy. Amy Victoria Wadge (born 22 December 1975) is an English singer and songwriter. Wadge determinacy implies the semilinear ordering principle for the Wadge order. Another consequence of Wadge determinacy is the perfect set property. The song was written by Steve Mac, Johnny McDaid, Ed Sheeran and Amy Wadge. The English versions are sung by Amy Wadge and the Welsh versions by Ela Hughes. Sheeran wrote "Thinking Out Loud" with Amy Wadge, a Wales-based singer-songwriter. The song was written by Minogue, British singer-songwriter Amy Wadge, and American musician Jesse Frasure. All of the songs were written collaboratively by Ed Sheeran and Amy Wadge.