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?