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There are no ZFC or arithmetic axioms, for example second-order arithmetic, in the physical world. Why can I solve differential equations in the electricity subject?

We may also assume the world has discrete time and space, or just we don't care about the physical rule in the small scale. Then the differential equation is something approximate to a difference equation, and we don't need real numbers of infinite digits, and its existence in real world.

Assuming the consistency of ZFC, we could prove there are provable error bound between the solution of difference equation and the solution of differential equation.

Geoffrey Thomas
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user779130
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    What does it mean "There are (there are no) ZFC or arithmetic axioms in the physical world" ? – Mauro ALLEGRANZA Sep 21 '21 at 05:51
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    But yes, (maybe) we can do only with approximation. See [Finitism](https://en.wikipedia.org/wiki/Finitism) as well as [Hartry Field](https://en.wikipedia.org/wiki/Hartry_Field)'s *Science Without Numbers*, 1980 – Mauro ALLEGRANZA Sep 21 '21 at 05:53
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    I think you are mixing "exists in the physical world" with "needed in models of the physical world". Not even positive integers "exist in the physical world", they are abstractions used to *represent* it, and there are physical models where the world has no time or space at the fundamental level, and is represented by algebraic structures other than numbers, see e.g. [Evolution in Quantum Causal Histories](https://arxiv.org/abs/hep-th/0302111). For a different sense of "existence" see [Do numbers exist independently from observers?](https://philosophy.stackexchange.com/q/451/9148) – Conifold Sep 21 '21 at 09:34

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