I have athlete's foot. Is this incurable?

Very curable. Many over thr counter antifungals or if not effective prescription topical r oral meds.
No. Athletes foot (or tinea pedis) is almost always curabale. The problem is recurrence. Fungi like dark, moist areas especially where they can imbed and adhere i.e. Sports shoes. Besides using the otc sprays and creams, try to rotate athletic shoes so they dry out fully between use and spray the inside of the shoes if you have athlete's foot. Always wear socks of some sort. Clean and dry is the key.