I would have to agree technically K. Nelfson, but "lasting forever is a relative thing."
If a well designed cobb oven is protected from the elements with a pavillion or other structure that effectively mitigates
water, and the design is good, the cobb is a proper blend, and the foundation is well made. One built of just wood timber on stone plinth, and cobb could last a minimum of 2000 years, so I really don't see the need for a heavy stone foundation for an oven.
Many forms of architecture are a tad out of balance, especially today's modern architecture, which has virtually no balance between it and the environment at all. The oldest undocumented timber frame (once a bread bakery) now destroyed by war in Syria, was 7000 years old. It was built of cobb, timber, and stone plinth foundation. The oldest wood frame structure, sitting on stone plinth, is a Japanese timber frame, which is 2000 years old and still stand strong. There really is no need to go to the expense of building a masonry foundation for a cobb oven, other than personal aesthetics and motif, if your only goal is longevity.