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Arturo, one does not need the axiom of infinity to define the natural numbers, or their operations (of course, one cannot then ensure the existence of $ {\mathbb n}$, but it and addition can be. However, here some users stated that addition was not completely defined, or that the above argument i presented was fallible, in a post i made about whether these. The slickest definition is perhaps the definition of addition for surreal numbers

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The real numbers can be seen as a subset of the surreals, so this also defines addition for real. The theory of the natural numbers with multiplication and addition is undecidable, but the restriction to just addition is decidable, and the restriction with just multiplication is decidable. It's mathematically fine but conceptually.

I'm reading russell's introduction to mathematical philosophy russell defines the sum of two numbers in terms of successors

It is possible to define the natural numbers, and the addition and multiplication, in terms of set theory, starting with the binary predicate symbol $\in$ (is a member of), and. I am trying to solve this following exercise in axler Define an addition and scalar. The natural numbers with addition are the free.

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