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At school we are often taught that natural are the same positive integers, integers are subset of the rational, these in turn are subset of real, etc.

But then in set theory, building natural according to the definition of Von Neumann is done, and there the naturals are not the same positive integers. Integers are equivalence classes of ordered pairs of natural numbers. Similarly, integers are not subset of rationals.

So if someone asks me if the naturals are subset of integers, should I answer no, because there is no construction in which they are?

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  • $\begingroup$ Constructions aside, ultimately number systems are defined by their properties. And the set of positive integers has all the properties we use to define the natural numbers. $\endgroup$ – user137731 Sep 28 '16 at 22:16
  • $\begingroup$ Just work from your definition of subset. Every element of N is a member of the rationals so N is a subset of the rationals. The construction of the numbers doesn't change the actual properties of them. $\endgroup$ – lordoftheshadows Sep 28 '16 at 22:18
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    $\begingroup$ @lordoftheshadows That's not really correct - the point of the OP is that in the usual formal definition of $\mathbb{Q}$ and $\mathbb{N}$, $\mathbb{N}$ is not a subset of $\mathbb{Q}$. Remember that a rational number is (usually) defined as an ordered pair of natural numbers - even the ones which are natural numbers! Basically, there's a type error: the natural number $0$ is different set-theoretically from the rational number $0$. (This has lead a number of people to argue that the usual set-theoretic framework is silly - not unreasonably . . .) $\endgroup$ – Noah Schweber Sep 28 '16 at 22:21
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    $\begingroup$ The integers might not be a subset of the rationals in a particular construction, but in any construction there is a subset of the rationals which has all of the properties of the integers. The constructions are there to formalize things, we do them once and then forget the dirty details, nobody think about ordered pairs as $\{\{a\},\{a,b\}\}$ and nobody writes $8\in 75$, even though that's true in some constructions of the naturals $\endgroup$ – Alessandro Codenotti Sep 28 '16 at 22:22
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    $\begingroup$ @Alessandro: I'm not the OP, but I think your comment, or something like it, should be written up as an answer. I would upvote it. $\endgroup$ – Brian Tung Sep 28 '16 at 22:27
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One must make the distinction between constructing a model versus constructing the model. The latter is merely a hallucination. In more detail: when we state the axioms of, say, the naturals, we are merely listing some properties. One may have a strong feeling that these properties uniquely describe just one and only system, namley that of the natural numbers. In other words, one believes there is just one model for the axioms. But that is not true. Any model of the axioms will have to employ completely arbitrary names for the elements. Any formalism which produces a model must include an ad-hoc naming convention. Given a model you can simply change the names of the elements and obtain another model of the same axioms. Then, of course, one replaces the initial feeling of "there is a unique model" by "there is, up to isomorphism, a unique model". Now that is a totally different assertion.

So, if someone asks you if the naturals are a subset of the integers you should ask to first phrase the question correctly, since 'the naturals' and 'the integers' are illusions. That someone may then ask you: is there a model of the naturals in any model of the integeres, to which the answer is yes.

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