1. Richard Dedekind ( )
The grandfather of mathematical structuralism
Structuralism: a./ Paul Benacerraf: „What numbers cannot be” (1965)
b./ William Lawvere’s works on category theory (from the 60’s)
1872: Continuity and irrational numbers
Dedekind cut: Divide the rational numbers into two classes so that every member of the first (lower) class is less than any member of the second (upper) class. Such a classification is called cut. There are three cases:
The upper class has a minimal member.
The lower class has a maximal member.
Neither of 1. or 2.
Let us postulate that in the case 3. there is an (irrational) number defined by the cut. It is less than the members of the upper class and greater than the members of the lower class of the cut. Rational numbers can be identified with Dedekind cuts of the 1. és the 2. cases (as you like it).
But what are the natural numbers?

2. 1887: What numbers are and what they ought to be?
„In science nothing capable of proof ought to be accapted without proof.”
System (= set), subset, union, intersection.
Transformation (= function) of a system (= on a set), composition.
Similar transformation (= injective function)
S’=(S) is the system consisting of the -pictures of S. If  is a similarity transformation, then it has a converse that is a similarity transformation again and  is bijective between S and S’.
Two systems are similar iff there is a similarity transformation between them.
We can divide all systems into (equivalence) classes by similarity. A class can be defined as the class of the systems similar to a system R. R is the representative of the class and it may be chosen as any member of the class.

3. Transformation of a system into itself.
A transformation  for which (S)  S
K  S is a chain (for ) iff (K)  K.
S is a chain, (K) is a chain.
Union, intersection of chains is a chain.
If A  S, then the intersection of all chains containing A is a chain containing A and contained by S. It is the A0 (or 0(A)) chain of A.
0((A)) =  (0(A)). This system is called the chain-transform or transform-chain of A, shortly A’0.
Theorem of complete induction: For any system , if
A  ,
for any x  A0  , (x)  A0  ,
then A0  .

4. V. The finite and the infinite
A system is (Dedekind-)infinite iff it is similar to a proper part of itself. Finite in the other case.

5. VI. Simply infinite systems – natural numbers
N is simply infinite iff there is a similarity  and an element 1 of N s.t. N =0({1}) and 1   (N)
Every infinite system contains as a part a simply infinite system.
Natural numbers: the elements of any simply infinite system N if
Every natural number m generates a chain m0 and m  m0.
Every natural number different from 1 is a follower of some natural number.
Complete induction: If
A(m) holds;
for any n  m0, if A(n), then A((n)),
then A(x) holds for any member of m0.
The axioms of second-order PA hold for simply infinite systems.

6. X. The class of simply infinite systems
Theorem 132. All the simply infinite systems are similar.
In other words: the theory of simply infinite systems is categorical.
On each model, the same propositions are true.
Every proposition of the language of this theory is either true in every simply infinite system and therefore a semantical consequence of the second-order Peano-axioms, or the same holds for its negation
Therefore, the theory is negation complete.
It can’be formalized within first-order logic.
It contains second-order Peano arithmetics (in fact, it is identical with it).
Therefore, second-order logic cannot be made deductively complete.
Semantical consequence in second-order logic is not compact: there are valid inferences with infinitely many premises where the conclusion does not follow from any finite subset of the premises.
Let us identify arithmetical truth with being a consequence of second-order Peano arithmetics (or being true in every simply infinite system).

7. Is second-order logic logic yet?
Quine: No, it is set theory in disguise
Neo-Fregeans and structuralists: yes, it is logic on the same right as first-order logic is logic. It has the disadvantage that it has no complete deductive theory but it has the advantages that it makes possible a complete arithmetics.
Boolos: second-order logic does have existential consequences but weaker ones than set theory.