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More formally (though not too formal, as I am pretty under-educated in mathematical logic) what does it mean when one "constructs" another mathematical object from another within the context of introducing a new theory or new definitions.

For example I've found when one writes of lets say "constructions of the real numbers" what they mean is they are going to use some deductive system that has no notion of a real number to define an algebraic structure that is isomorphic to the field of real numbers.

Likewise I've seen this done in "constructions of the integers" where they might take lets say ordered pairs of natural numbers (the point being as before never to reference or use anything about the objects we are going to "construct") to create the integers

Again I've seen this also done in "constructions of the rationals" where they might take lets say ordered pairs of integers (again never referencing/using anything involving rational numbers) to create the rationals.

So to re-iterate what does "construction" mean formally in this context of introducing new objects?

Asaf Karagila
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user3865123
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    See Construction of the real numbers: compared to the axiomatic definition of the real number system as a complete ordered field, the construction of the reals assume a pre-existing structure (basically: the naturals) and define progressively new objects (the rational, the reals) in terms of the previous ones. Having defined how to "contsruct" them, the procedure prove that the new defined objects satisfy the corersponding axioms. – Mauro ALLEGRANZA Apr 05 '18 at 06:01
  • @Mauro ALLEGRANZA So in general a "construction of x" in these sorts of contexts is just a proof of the existence of a mathematical structure isomorphic to x? – user3865123 Apr 05 '18 at 06:03
  • Not exactly, IMO: is the proof that there is a strucure satisfying a specified set of axioms (the structure $\mathbb R$ satisfying the axioms for complete ordered fileds, in my example) ... provided that the "initial" structure ($\mathbb N$ in my exanple) exists. – Mauro ALLEGRANZA Apr 05 '18 at 06:16
  • You can refer to the "construction of straightedge and compass" of Euclidean geometry: it is a way to prove that, provided that straight lines and circles exist, also other geometrical figures: angles, triangles, etc. exist because we can define procedures (algorithms) to build them up from pre-existing entities. – Mauro ALLEGRANZA Apr 05 '18 at 06:35

2 Answers2

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This is a bit ambiguous, as all natural language is ambiguous (especially when used in proximity to mathematical language which is not, or at least should not be, ambiguous)

Generally speaking, in mathematics you work in a context where you have a universe of objects. These could be sets, or functions, or numbers, or all of them.

When we construct something, we show that there is a way to define an object (or sometimes a collection of objects) which satisfies the properties making it "worth the name" of its construction. This formaly validates our claim as to the existence of something.

  1. When we say that we "construct a sequence", then we mean to say that we define the sequence.

  2. When we say that we "construct the real numbers", then we argue that given just the rational numbers and some background universe with "enough sets and functions", then we can define an object which has the same behavior we expect from the real numbers. We can then show that this structure is indeed unique up to isomorphism, so the method of construction (Dedekind completion, Cauchy completion, etc.) is in fact irrelevant.

The idea is that a construction usually involves some objects to start with. It could be the rationals, or a specific number used to bootstrap a sequence, or just the empty set. And from that object we define another object, in a reasonably explicit way.

For example, you cannot define a function $f\colon\Bbb R\to\Bbb R$ such that $f(x+y)=f(x)+f(y)$ and $f$ is not continuous, we know that because there are universes of mathematics where no such function exists. However, if you are given a Hamel basis for $\Bbb R$ over $\Bbb Q$, then from that basis you can construct such function (and it follows that in the aforementioned universes, no Hamel bases exist either).

Asaf Karagila
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  • Thanks I appreciate your answer. Also from looking at your profile I get the feeling you are very well trained in the realm of mathematical logic, now I have almost no experience in the subject outside of maybe naive set theory and propositional logic. Thus I wanted to know if you could recommend a book to me on the subject, if possible something that goes over set theory in detail and explores foundations other then and including $\text{ZFC}$. – user3865123 Apr 05 '18 at 19:50
  • Hmm. Well, I don't know all that much about this actually. Homotopy Type Theory should have a free online book dealing with this. Not entirely sure about the other stuff. Let me also remark that I don't like my answer all that much. Its essence is fine, but the presentation is shit. I'll probably sit to rewrite it tomorrow or the day after that. – Asaf Karagila Apr 05 '18 at 19:52
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What does "construction" mean formally in this context of introducing new objects?

If you were to set out to formally "construct," say, the set of real numbers, from scratch, you would have to begin with a list of axioms and rules of inference that would have to include at least one axiom that postulates the existence of some object, usually some set X with certain properties.

Your axioms and rules of inference should allow you to infer the existence of (i.e. construct) new sets given the existence of others. So, starting with set X, you would construct one set after another in this way until you have the set of real numbers.

Dan Christensen
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