Some problems of theory of real numbers are about proving that some real numbers are irrational. But I find it a bit confusing to prove that some real number cannot be expressed (equal) to the quotient of two intgers, when you are working in the theory of real numbers where term "integer" is not defined. So I think we can easily define "integer" as "difference of two natural numbers", then how can we define naturals ? Will this definition be "strong" enough to prove that there is no natural number between two consecutive natural numbers ? Or maybe there is an easier way to define integers in terms of reals without defining naturals ?
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First define $1$ as the multiplicative identity of the reals, i.e. the unique $x$ such that $xy=y$ for all $y \in \mathbb R$. The natural numbers are then the additive semigroup generated by $1$.
Robert Israel
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Is this definition enough for proving the property that there is no natural number between $n$ and $n+1$ where $n$ is natural (assuming real number/field axioms) ? – Юрій Ярош Jul 05 '18 at 21:40
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@ЮрійЯрош For this question of no integer between $n$ and $n+1$ see also here – Dietrich Burde Jul 05 '18 at 21:41