$\mathsf{D}_{\preceq}$: $\hspace{5mm} x \preceq y \leftrightarrow y = x \oplus y$.Crucial for that result is the restriction of any axiom schemes of $T$ to the original language. At the end of the paper, "What Difference Does it Make?", I wondered if extending Peano arithmetic, $\mathsf{PA}$, with fusion theory and full induction yields a non-conservative extension.

$\mathsf{D}_{O}$: $\hspace{5mm} xOy \leftrightarrow \exists w(w \preceq x \wedge w \preceq y)$.

$\mathsf{D}_{At}$: $\hspace{3mm} At(x) \leftrightarrow \forall z(z \preceq x \rightarrow z = x)$.

$\mathsf{F}_1$: $\hspace{6mm} x \oplus x = x$.

$\mathsf{F}_2$: $\hspace{6mm} x \oplus y = y \oplus x$.

$\mathsf{F}_3$: $\hspace{6mm} x \oplus (y \oplus z) = (x \oplus y) \oplus z$.

$\mathsf{UF}$: $\hspace{5mm} \exists x \phi(x) \rightarrow \exists! z [\forall x(\phi(x) \rightarrow x \preceq z) \wedge \forall y \preceq z \exists x(\phi(x) \wedge xOy)]$.

This second paper, "Arithmetic with Fusions", is very much a preliminary draft, and it would be nice if any reader could find a snag. (Some earlier attempts did run into snags, but eventually, it now seems that the two main interpretability results are correct.) In the paper, we argue that the fusion extension of Peano arithmetic, denoted $\mathsf{PAF}$, interprets full second-order arithmetic, $Z_2$. If this is right, then $\mathsf{PAF}$ is a very strong theory indeed. One might also conclude that this sort of non-conservation

*undermines*a fictionalist view of mereological fusions. (This is joint work with Thomas Schindler.)

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