The statement that Achilles believes that he will survive the Trojan war and return home to Phthia before the death of his aged father Peleus is false.
Achilles or Achilleus, in Greek mythology, was a hero of the Trojan War and the greatest of all the Greek warriors. He is the central character of Homer's Iliad. He was the son of the mortal Peleus, king of the Myrmidons, and the Nereid, or sea nymph, Thetis. According to Homer, Achilles was brought up by his mother in Phthia. According to the consensus of Historians and scholars Achilles was a legend as Homer's skill developed a character that encompassed both the heroism and the failings of the warriors who held Troy's walls against a siege. There are no records that Achilles believes that he will survive the Trojan war and return home to Phthia before the death of his aged father Peleus.
The nuclear arms race was an arms race competition for supremacy in nuclear warfare between the United States, the Soviet Union, and their respective allies during the Cold War.