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hyperlinked fragment and first order
The original and the current variant sound strange, since it makes it sound like a mere fragment of the known axioms. The alteration is probably also not the full correction, since only two of the axioms involve that predicate symbol.
I presume the predicate $\mathcal{C}$ is meant to capture the “feasible” numbers. If so, this should be mentioned, and also some mention of ultrafinitism.
The reference defines the arithmetic to be an extension of Peano arithmetic, and includes induction for all formulae not containing the predicate $\mathcal{C}$. The definition has been changed to correspond to the definition used in the paper (which extends Peano arithmetic, not just a fragment thereof), and the content that relies on not having induction now refers to the theory “Nelson arithmetic without induction”.
Simplified the presentation of some models, and fixed some typos in the definitions of additionable and multiplicable numbers. A couple of “models” didn’t satisfy all of the axioms, so were modified or removed.
A bunch of other miscellaneous edits too.
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