1. 1884 – Grundlagen der Arithmetik (Foundations of Arithmetics)
Philosophical program:
There are absolute and eternal truths
Anti-empiricism, anti-historicism
„Anti-psychologism”
Basic principles (Introduction):
Subjective and objective, psychologiacal and logical should be distinguished
Always ask for the meaning of the words in the context of sentences
Distinction between concept and object
[Concept is the semantical value of an unary predicate]

2. Critical analysis: what numbers are not – they are neither physical nor mental.
Most important target of the criticism: the Euclidian definition of unit and number
Elements VII.:
Frege’s question: Are the units distinguishable or not?
Two basic results of the critical analysis:
Cardinality propositions (like ‘I have two hands’ , ‘There were twelve apostles’ are about predicates („concepts”). The expressions ‘there are two’, ‘there are twelve’ and the like express concepts of second grade (they are second order predicates) – as well as the expression ‘there are’ or ‘there exists’.
These second-grade concepts [numerical quantifiers] can be defined in a simple way [within first-order logic].
But from this sequence of definitions, we cannot get an answer the question „Is Julius Caesar a number?”
[We didn’t define numbers as objects. Julius Caesar problem.]

3. (Hume principle:) Two concepts have the same cardinality iff there is a one-to-one mapping between the objects falling under them.
(H) (Nx:F(x) = Nx:G(x))  b(1-1(b)  x(F(x)  G(b(x)) y(G(y) x(F(x)  b(x)=y))))

4. Digression about abstraction
Abstraction: not a psychological process where we don’t consider the differences between some objects and on that way we get their common property.
We have an equivalence relation between the objects and we can say that the equivalent objects have the common property.
Against a set-theoretic background it can be said that the common property is nothing but belonging to the same equivalence class.
We can render the same object to the members of the same equivalence class and different objects to the members of different classes. They are the abstract objects defined by the method.
Example: direction of straight lines on a plane.
Equivalence relation: paralellism.
The direction of a straight line should be the same as the direction of any straight line parallel with it, and different from the direction of any straight line not parallel with it.
You can even identify directions with the equivalence classes of straight lines for parallellism.

5. Frege’s proposal:
„Having the same cardinality” (equinumerosity, Equinum) is an equivalence relation between concepts,
and it can be defined (by (H)) without reference to any previously defined concept of cardinality.
Equinum(F, G) def
b(1-1(b)  x(F(x)  G(b(x)) y(G(y) x(F(x)  b(x)=y))))
So let’s do the same with concepts to gain the cardinalities of concepts.
[But unfortunately, the equivalence classes for „having the same cardinality” are proper classes, not sets.
Therefore, numbers gained on this way are not objects and the class of natural numbers doesn’t exist.]
(D) Nx:Fx =def ˇG(Equinum(F, G))
[If H() is a function, ˇH() is its value range.
Especially, if H(x) is a concept, then ˇxH(x) is its extension.
Equinum(F, G) is a concept of second grade (with fixed F and variable G).]
Hume principle follows from (D).
Number(n) def F(Nx:Fx =n)

6. 0 = def Nx:(xx)
ISucc(m, n)  def Fy(Nx:F(x) = n  F(y)  Nx:(F(x)  x  y) = m)
1 = def Nx:(x = 0)
ISucc(0, 1)
m < n  def Isucc*(m, n)
m  n  def m = n  m < n
NNum(n)  def (0  n)
Nnum(n)   ISucc(n, n)
n = Nx:(x < n)
For the natural numbers defined on this way, axioms of primitive Peano arithmetics are true [Frege’s theorem].

7. (D) Nx:Fx =def ˇG(Equinum(F, G))
(D) is a natural definition of (cardinal) number.
But according to it, even the natural numbers are proper classes (except of 0). E.g.,1 is the set of all unit sets.
(D) is used only to deduce (H).
Boolos (1987): let us substitute (D) with (H). We get a theory that is equiconsistent with PA.
Basic Laws: Value ranges are introduced by the axiom (V):
(ˇxf(x) = ˇyg(y))  x(f(x) = g(x))
(V) gives an identification criterion for value ranges. But unfortunately, it is inconsistent.
The definition of direction gives an identification criterion for directions and introduces directions on this way. It is consistent if elementary geometry is consistent.
(H) gives an identification criterion for numbers and it is consistent.
Neo-fregeanism:
(Let us call abstraction principles propositions that give identification criteria for some sort of new entities via an equivalence relation.)
An abstraction principle makes possible to introduce a new sort of abstract objects, provided it is consistent.