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Since $\mathbb{Z} \subset \mathbb{R},$ all integers $n$ are also a real number. However, $\mathbb{Z}$ and $\mathbb{R}$ is defined by Peano's axiom and Dedekind cut, two very different methods. Does this imply that $n$ as an integer is different as $n$ as a real number? If so, how? Is $n$ as a rational number also different?

Kenta S
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    There are canonical embeddings from $\mathbf N$ into $\mathbf Z$ , from $\mathbf Z$ into $\mathbf Q$, from $\mathbf Q$ into $\mathbf R$, &c. Each time, we identify a number set with its image by the embedding. – Bernard Jun 30 '18 at 12:41
  • See also https://math.stackexchange.com/questions/1553537/why-mathbbr-is-a-subset-of-mathbbc – lhf Jun 30 '18 at 13:19

2 Answers2

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I read your question as one about intuition on numbers, and I will answer from that point of view.

There are many equivalent ways to construct differ kinds of numbers. For example, you can build the reals with Dedekind cuts or with Cauchy sequences. If you construct the natural numbers first and then expand to integers, rationals, and reals, you have a large number of options.

As others have pointed out, at each step the previous kind of numbers can be seen as a subset of the new ones via an embedding. The exact embedding depends on your choice of constructions.

I see all of this as formalization or rigorous definition of the various kinds of numbers, not the essence of numbers. The way I see numbers is different. I might think of numbers as quantities, points on the geometrical real line, or something else. I think of them flexibly in different ways. To me Dedekind cuts and the like come from the intuitive meaning of a number and are secondary. While rigorous definitions are crucial for building a sound theory, on an intuitive level I see definitions as descriptions.

The number two can be thought of as a Dedekind cut, a supremum of a set of rationals, the formal limit of a Cauchy sequence, the number of elements in some finite sets, the set consisting of the empty set and the set containing the empty set, the successor of one, the ratio or difference of two naturals or integers, or something yet different. It is useful to see the same thing in many ways.

Joonas Ilmavirta
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What comes at play here is the notion of embedding, which is an injective map $f : X \to Y$ preserving some structure.

For example, the integers $\mathbb Z$ are defined using equivalence classes on $\mathbb N \times N$ by the equivalence relation $(a,b) \sim (c,d)$ if and only if $a+b=c=d$.

You can embed $\mathbb N$ into $\mathbb Z$ by associating $n \in \mathbb N$ to the class $f(n) = \widetilde{(n,0)}$. This is injective, and you have interesting properties preserved like

$$f(n+m)=f(n)+f(m)$$

At the end, $\mathbb N$ is not a subset of $\mathbb Z$ but the image of $\mathbb N$ by the embedding is.

You can proceed with similar embedding to embed $\mathbb Z$ into $\mathbb Q$ and $\mathbb Q$ into $\mathbb R$.

mathcounterexamples.net
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