Abstract: 

We will go over my pre-AI-era result that ZF does not prove there is a model of PA with full standard system. Then we will discuss my AI-assisted result that ZFC+GCH is consistent with there not being any ordinal-definable \aleph_1-sized model of PA with full standard system, which implies that the unique-up-to-isomorphism \aleph_1-sized saturated elementary extension of \omega has no ordinal-definable presentation.

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