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I have long held similar views as put forth here. As professor Hamkins points out though, $(M,\subseteq)$ is insufficient to model most of mathematics. However, this is unsatisfactory to me, as this seems to conflate the inclusion of mereology with that of set theory. We have no reason to assume that these two inclusion relations are related. So how do actual systems of mereology hold up? If $\mathcal{M}$ is a model of GEM, then does there exist an interpretation function from $\mathcal{M}$ to some model of ZFC? If there is, then it is not obvious to me what it should be.

tox123
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    Am I understanding correctly that GEM has finite models? If so, it by itself is certainly not going to be strong enough to interpret ZFC. – James Hanson Sep 24 '21 at 00:12
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    It seems furthermore that Tsai showed that GEM is decidable, which also precludes the possibility of interpreting ZFC. – James Hanson Sep 24 '21 at 00:16
  • @JamesHanson interesting result (although deeply unfortunate) thanks for sharing that. I also wasn't aware that GEM has finite models, but that does make sense I suppose. – tox123 Sep 24 '21 at 16:51

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