Why do the names of emperors of the Ming Dynasty contain the five elements radicals "gold, wood, water, fire, and earth"?

They belong to different lineages. During the Hongwu period, Zhu Yuanzhang drew up twenty-cross characters for each of the East Palace and the kings' lineage. The Yanwang line (the later emperor line) was "Gao, Zhan, Qi, Jian, You, Hou, Zai, Yi, Chang, You, Ci, He, Yi, Bo, Zhong, Jian, Jing, Di, Xian and You". Wuzong and Sejong are cousins, so the second character in their names is the same. It also stipulates that the third character is preceded by the radicals of the five elements, namely metal, wood, water, fire, and earth. The father of Wuzong Zhu Houzhao, Xiaozong Zhu Youtang, was the third son of Xianzong, and the father of Sejong Zhu Houcong, King Xingxian Zhu Youqi, was the fourth son of Xianzong. The difference in the radical of the third character is to distinguish them from different lineages. Wuzong belongs to the Xiaozong line, while Shizong belongs to the Xingwang line. Xiaozong also had a son named Zhu Houwei, who was also Huobian. This is a naming habit among ancient Chinese families, not just the royal family.