Reflecting on countable nonstandard models of arithmetic, we first get two results that seem affirmative for the indeterminacy-of-reference skeptic: 

• Nonstandard models of PA exist, and
• There are continuum-many distinct (non-isomorphic) ones.

But then we get two results that seem to turn the tables on the skeptic:

• They all have the same goofy order-type (N+QZ), and
• Addition and multiplication are uncomputable in all of them. (Tennenbaum)

last modified September 24, 2022