TITLE: What is exactly the meaning of being isomorphic?
QUESTION [11 upvotes]: I know that the concept of being isomorphic depends on the category we are working in. So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or geometry, I often hear that people say that such a structure is complete in the sense that any other set that satisfy their properties is isomorphic. What is the real meaning and implications of being isomorphic? At least to me it is not very clear.
Let's see for example the axioms of geometry. To begin with, there are different systems of axioms, all of them trying to define that object called geometry. So if we are talking about the same object, I suppose the theories generated should be isomorphic each other. Does this make sense? Again what is the real implication of being isomorphic? According to the definition there should be a bijection between the sets, which means that it cannot be larger sets (with a higher cardinality, or even different cardinality) that hold the propierties of the given structure and also the operations defined on each structure should be mantained in the bijection. Is this all? Please, if I'm saying bullshit forgive me I just want to really understand this concept. I will really appreciate any comment.
Edit: In my example I meant Euclidian Geometry.

REPLY [3 votes]: Each category defines its own definition of isomorphic. It may be helpful to consider similar categories with the same objects but different definitions of isomorphic.
Example with shapes
Triangles in the Euclidean plane are a nice collection of objects. We'll define three categories, each with their own idea of what isomorphic means, and give three different definitions of a triangle. The completeness of the definition will depend on which category we work in.
(Arbitrary) definitions of isomorphic
The first category contains all Euclidean triangles, and calls two triangles isomorphic if one can be taken to the other using reflections.
The second category contains all Euclidean triangles, and calls two triangles isomorphic if one can be taken to the other using reflections and dilations.
The third category contains all Euclidean triangles, and calls two triangles isomorphic if one can be taken to the other using “shadows” (as in, draw one triangle on the plane z=1, the other on the plane z=0; can you shine a light from somewhere so that the shadow of the z=1 triangle is exactly the z=0 triangle?)
Summaries of isomorphic
In the first category, two triangles are isomorphic iff they are congruent.
In the second category, two triangles are isomorphic iff they are similar.
In the third category, all triangles are isomorphic.
Complete definitions
A definition of a triangle is “complete” means that if two triangles satisfy it, then they are isomorphic.
In the first category, the "SSS congruence theorem" says that giving the three side lengths is a complete definition of a triangle.
In the second category, the "AAA similarity theorem" says that giving the three angle measures is a complete definition of a triangle. This is not complete in the first category.
In the third category, the "projective equivalence theorem" says that "it has three corners" is a complete definition of a triangle. This is not complete in the first or second category.