Sean Walsh organized a one-day workshop on the philosophical significance of Tennenbaum’s Theorem on Saturday. It kicked off with me presenting a short piece that Tim Button and I have forthcoming in Philosophia Mathematica: here’s a preprint of our paper.
But for a quicker read, my overheads give the headline idea — that’s there no implication about how we grasp the standard model to be got out of the elegant but non-trivial Tennenbaum’s Theorem that you can’t get out of the very easy theorem that every model of PA where every element has a finite number of predecessors is isomorphic to the standard model. Tennenbaum’s Theorem has no extra oomph against the Skolemite sceptic. Indeed, appealing to either model theoretic result just doesn’t touch the sceptic’s worries. (The talk timed nicely, and having Tim there to help fend questions made giving it a lot more fun!)
The current temporal parts of Walter Dean and Leon Horsten were agreed, contra earlier parts, that Tennenbaum’s Theorem cuts no ice against the model-theoretic sceptic (I wasn’t so clear where Paula Quinlon now stands). But I think all three other speakers in different ways wanted to squeeze something philosophical out of Tennenbaum’s Theorem. If/when published pieces emerge, I’ll say why I wasn’t so convinced. But a fun occasion (as such closely-focused workshops tend to be).