[Reminder: JB=John Baez, MW=Michael Weiss.]
What you and Enayat are calling the “standard” model of arithmetic is what I used to call “an omega”, i.e., the omega of a model of ZF. Is that the new standard terminology for it? I don’t like it, for philosophical reasons I won’t get into. (Reminds me of the whole “interpretations of QM” that books have to skirt around, when they just want to shut up and calculate.)
Leaving ZF out of it, a friend in grad school used to go around arguing that 7 is non-standard. Try and give a proof that 7 is standard using fewer than seven symbols. And of course for any element of a non-standard model, there is a “proof” of non-standard length that the element is standard. I think he did this just to be provocative. Amusingly, he parlayed this line of thought into some real results and ultimately a thesis.
JB: Fun! It reminds me a bit of this:
I have seen some ultrafinitists go so far as to challenge the existence of 2100 as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 21, 22, 23,…, 2100, do we stop having “Platonic reality”? Here this “…” is totally innocent, in that it can easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 21 and asked him if this was “real” or something to that effect. He virtually immediately said yes. Then I asked about 22 and he again said yes, but with a perceptible delay. Then 23 and yes, but with more delay. This continued for a couple more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2100 times as long to answer yes to 2100 as he would to answering 21. There is no way that I could get very far with this.
Harvey M. Friedman, “Philosophical Problems in Logic”
Both of them were real characters. Yessenin-Volpin was a dissident in the Soviet Union, locked in a psychiatric hospital for what Vladimir Bukovsky jokingly called “pathological honesty”. Edward Nelson was another student of my Ph.D. advisor, Irving Segal. He did ground-breaking work on mathematically rigorous quantum field theory. Much later he wrote a fascinating paper on internal set theory, which I’d like to talk about sometime. In his final years he thought he had proved the inconsistency of Peano arithmetic! Terry Tao caught the mistake in the proof, and Nelson quickly admitted his error. That was an embarrassing incident, but he came out of it fine. Crackpots never admit they’re wrong; he was not like that!