These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case).
These are two different but isomorphic implementations of natural numbers in set theory.
They are isomorphic as models of Peano axioms, that is, triples (N,0,S) where N is a set, 0 an element of N, and S (called the successor function) a map of N to itself (satisfying appropriate conditions).
As emphasized in Benacerraf's identification problem, the two implementations differ in their answer to the question whether 0 ∈ 2; however, this is not a legitimate question about natural numbers (since the relation ∈ is not stipulated by the relevant signature(s), see the next section).
[details 1] Similarly, different but isomorphic implementations are used for complex numbers.
The successor function S on natural numbers leads to arithmetic operations, addition and multiplication, and the total order, thus endowing N with an ordered semiring structure.
The two isomorphic implementations of natural numbers mentioned in the previous section are isomorphic as triples (N,0,S), that is, structures of the same signature (0,S) consisting of a constant symbol 0 and a unary function S. An ordered semiring structure (N, +, ·, ≤) has another signature (+, ·, ≤) consisting of two binary functions and one binary relation.
Such relation between structures of different signatures is sometimes called a cryptomorphism.
However, not all fixed points of this action correspond to species of structures.
[9] A pair of mutually inverse procedures of deduction leads to the notion "equivalent species".
The two corresponding procedures of deduction coincide; each one replaces all given subsets of X with their complements.
In the general definition of Bourbaki, deduction procedure may include a change of the principal base set(s), but this case is not treated here.
[10] Equivalence between two species of structures leads to a natural isomorphism between the corresponding functors.
However, in general, not all natural isomorphisms between these functors correspond to equivalences between the species.
[details 6] In practice, one makes no distinction between equivalent species of structures.
Nevertheless, the communication is successful, which means that such different definitions may be thought of as equivalent.
A person acquainted with topological spaces knows basic relations between neighborhoods, convergence, continuity, boundary, closure, interior, open sets, closed sets, and does not need to know that some of these notions are "primary", stipulated in the definition of a topological space, while others are "secondary", characterized in terms of "primary" notions.
Thus, in practice a topology on a set is treated like an abstract data type that provides all needed notions (and constructors) but hides the distinction between "primary" and "secondary" notions.
"[14] As was mentioned, equivalence between two species of structures leads to a natural isomorphism between the corresponding functors.
Moreover, every permutation of the index set { 0, 1, 2, ... } leads to a natural isomorphism; they are uncountably many.
A structure of a (simple) graph on a set V = { 1, 2, ..., n } of vertices may be described by means of its adjacency matrix, a (0,1)-matrix of size n×n (with zeros on the diagonal).
"Natural" is a well-defined mathematical notion, but it does not ensure uniqueness.