# The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Elementary Ontology

**OE**) is:

A0 | $x\epsilon y\leftrightarrow \mathsf{\Sigma}z\left(z\epsilon x\right)\wedge \mathsf{\Pi}zu\left(z\epsilon x\wedge u\epsilon x\to z\epsilon u\right)\wedge \mathsf{\Pi}z\left(z\epsilon x\to z\epsilon y\right)$ |

R1 | $x\epsilon y\u2215x\epsilon x$ |

R2 | $x\epsilon y\wedge y\epsilon z\u2215x\epsilon z$ |

R3 | $x\epsilon y\wedge y\epsilon z\u2215y\epsilon x$ |

D$\mathrm{V}$ | $x\epsilon \mathrm{V}\leftrightarrow x\epsilon x$ | x is an object; |

D$\mathsf{\Lambda}$ | $x\epsilon \mathsf{\Lambda}\leftrightarrow x\epsilon x\wedge ~x\epsilon x$ | x is a contradictory object. |

Dex | $ex\left(x\right)\leftrightarrow \mathsf{\Sigma}z\left(z\epsilon x\right)$ | x exists; |

Dsol | $sol\left(x\right)\leftrightarrow \mathsf{\Pi}zu\left(z\epsilon x\wedge u\epsilon x\to z\epsilon u\right)$ | at most one object is x; |

Dob | $ob\left(x\right)\leftrightarrow x\epsilon x$ | is object x; |

D⊂ | $x\subset y\leftrightarrow \mathsf{\Pi}z\left(z\epsilon x\to z\epsilon y\right)$ | x is included in y (weak inclusion); |

D⊏ | $x\u228fy\leftrightarrow \mathsf{\Sigma}z\left(z\epsilon x\right)\wedge \mathsf{\Pi}z\left(z\epsilon x\to z\epsilon y\right)$ | x is included in y (strong inclusion); |

D○ | $x\u25cby\leftrightarrow \mathsf{\Pi}z\left(z\epsilon x\leftrightarrow z\epsilon y\right)$ | x is extension identical with y; |

D= | $x=y\leftrightarrow x\epsilon y\wedge y\epsilon x$ | x is identical with y; |

Dn | $x\epsilon ny\leftrightarrow x\epsilon x\wedge $$\text{}~x\epsilon y$ | x is non y. |

#### 2.2. Peano’s Axiomatics

LA1 | $1\epsilon N$ |

LA2 | $a\epsilon N\to Sa\epsilon N$ |

LA3 | $a\epsilon N\to ~Sa\epsilon 1$ |

LA4 | $a\epsilon N\wedge b\epsilon N\wedge Sa=Sb\to a=b$ |

LA5 | $1\epsilon x\wedge \mathsf{\Pi}b(b\epsilon N\wedge b\epsilon x\to Sb\epsilon x$$)\text{}\wedge a\epsilon N\to a\epsilon x$ |

#### 2.3. Elementary Ontology with Frege’s Predication Scheme

**OE**

^{sub}) have the following shapes (see [4]):

SA1 | $x\epsilon sub\left(y\right)\to sub\left(x\right)\subset sub\left(y\right)$ |

SA2 | $x\epsilon sub\left(y\right)\to ~y\subset sub\left(x\right)$ |

SA3 | $x\epsilon sub\left(y\right)\to y\epsilon y$ |

SA4 | $sub\left(x\right)\u25cb\mathrm{s}ub\left(y\right)\to y\u25cby$ |

## 3. Idea

#### 3.1. An Indefinite Numerical Phrase with the Term Anzahl

#### 3.2. A Definite Numerical Phrase with the Term Anzahl

## 4. Arithmetic of Natural Numbers

A1 | $0\epsilon N$ |

A2 | $a\epsilon N\to Sa\epsilon N$ |

A3 | $a\epsilon N\to ~Sa\epsilon 0$ |

A4 | $a\epsilon N\wedge b\epsilon N\wedge Sa=Sb\to a=b$ |

A5 | $0\epsilon x\wedge \mathsf{\Pi}b(b\epsilon N\wedge b\epsilon x\to Sb\epsilon x$)$\text{}\wedge a\epsilon N\to a\epsilon x$ |

**OE**

^{sub}enriched with these axiomatics. The language of this system (

**OE**

^{sub}[A1,A2,A3,A4,A5]) is extended with number nominal variables (a,b,c), referring to natural numbers. Apart from the standard rule of detachment (MP):

B1 | $0\epsilon A\mathsf{\Lambda}$ |

B2 | $a\epsilon Ax\to \mathsf{\Sigma}z\left(Sa\epsilon Az\right)$ |

B3 | $a\epsilon Ax\to ~Sa\epsilon A\mathsf{\Lambda}$ |

B4 | $a\epsilon Ax\wedge b\epsilon Ay\wedge Sa\epsilon Sb\to a\epsilon b$ |

B5 | $0\epsilon x\wedge \mathsf{\Pi}by(b\epsilon x\wedge b\epsilon Ay\to Sb\epsilon x$)$\text{}\wedge a\epsilon Az\to a\epsilon x$ |

(A1) | $0\epsilon N$ | [B1,DN] |

(A2) | $a\epsilon N\to Sa\epsilon N$ | [DN,B2] |

(A3) | $a\epsilon N\to ~Sa\epsilon 0$ | [DN,B1,R2,B3] |

(A4) | $a\epsilon N\wedge b\epsilon N\wedge Sa=Sb\to a=b$ | [DN,D=,B4] |

(A5) | $0\epsilon x\wedge \mathsf{\Pi}b(b\epsilon N\wedge b\epsilon x\to Sb\epsilon x$)$\text{}\wedge a\epsilon N\to a\epsilon x$ | [DN,B5] |

C1 | $a\epsilon a$ |

C2 | $a\epsilon Ax\to ex\left(nx\right)$ |

C3 | $~ex\left(x\right)\to 0\epsilon Ax$ |

C4 | $a\epsilon Ax\wedge b\epsilon Ax\to a\epsilon b$ |

C5 | $0\epsilon x\wedge \mathsf{\Pi}by(b\epsilon x\wedge b\epsilon Ay\to Sb\epsilon x$)$\text{}\wedge a\epsilon Az\to a\epsilon x$ (=B5) |

T1.1 | $sol\left(Ax\right)$ | [C4,Dsol] | |

T1.2 | $A\mathsf{\Lambda}=0$ | [OE,C3,T1.1,R3,D=] | |

T1.3 | $a\epsilon Ax\wedge b\epsilon Ax\to a=b$ | [C4,D=] | |

T1.4 | $\begin{array}{ll}\hfill Sa\epsilon Sa\leftrightarrow \text{}Sa\epsilon na& \wedge \mathsf{\Pi}xz(a\epsilon Ax\wedge z\epsilon nx\\ & \leftrightarrow Sa\epsilon A\left(x\cup z\right)\wedge z\epsilon nx)\\ & \wedge \mathsf{\Pi}xz\left(0\mathsf{\epsilon}Ax\wedge z\mathsf{\epsilon}nx\to Sa\mathsf{\epsilon}Az\right)\\ & \wedge \mathsf{\Pi}yz(\mathrm{S}a\mathsf{\epsilon}Az\\ & \wedge ~ex\left(z\right)\to ~a\mathsf{\epsilon}Ay)\wedge \mathsf{\Pi}z(Sa\mathsf{\epsilon}Az\wedge z\mathsf{\epsilon}z\to a=0)\end{array}$ | [DS] | |

T1.5 | $Sa\epsilon A\left(x\cup z\right)\wedge z\epsilon nx\to a\epsilon Ax$ | [R1,T1.4] | |

T1.6 | $Sa\mathsf{\epsilon}Ax\wedge x\mathsf{\epsilon}x\to a=0$ | [R1,T1.4] | |

T1.7 | $~Sa\mathsf{\epsilon}a$ | [R1,T1.4.Dn] | |

T2.1 | $0\epsilon A\mathsf{\Lambda}$ | (=B1) | [OE,C3] |

T2.2 | $a\epsilon Ax\to \mathsf{\Sigma}z(Sa\epsilon A)$ | (=B2) | [C2,C1,Dex,T1.4] |

T2.3 | $a\epsilon Ax\to ~Sa\epsilon A\mathsf{\Lambda}$ | (=B3) | [R1,T1.4,OE] |

T2.4 | $a\epsilon Ax\wedge b\epsilon Ay\wedge Sa\epsilon Sb\to a\epsilon b$ | (=B4) | [OE,T1.4,C2,Dex,T1.5,T1.3] |

T2.5 | $\varphi \left(0\right)\wedge \mathsf{\Pi}b(b\epsilon N\wedge \varphi \left(b\right)\to \varphi \left(Sb\right)$)$\text{}\wedge a\epsilon N\to \varphi \left(a\right)$ | [C1,Dstsf,DN,C5] |

## 5. Addition and Multiplication: Logical and Philosophical Analysis

#### 5.1. Addition

#### 5.2. Multiplication

## 6. The New Formulation of the Arithmetic of Natural Numbers

**OE**

^{sub}[C1,C2,C3*,C4,C5]. In the formulation of axiom C5, the functor of successor occurs, which we are introducing similarly, by means of the already given definition DS.

T2.6 | $0\u25cbA\mathsf{\Lambda}$ | [D0,D○] |

T2.7 | $~ex\left(x\right)\to 0\epsilon Ax$ (=C3) | [Dex,OE,Dd,D○,C3*,R1,T2.6,R2] |

## 7. The Term Anzahl in the Indefinite Sense

## 8. Conclusions

**OE**

^{sub}). The notion of a pair (list) was introduced by means of rules OL, IL, and RL, the last of them playing a significant role. I have recently noticed that it is possible to define an ordered pair in the framework of the

**OE**

^{sub}system.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Słupecki, J.S. “Leśniewski Calculus of names”. Studia Log. Int. J. Symb. Log.
**1955**, 3, 7–70. [Google Scholar] - Peano, G. Arithmetices Principia Nova Methodo Exposito; Bocca: Turin, Italy, 1889. [Google Scholar]
- Leśniewski, S. Lecture Notes in Logic; Srzednicki, J.T.J., Stachniak, Z., Eds.; Kluwer Academic: Dordrecht, The Netherlands, 1988. [Google Scholar]
- Wojciechowski, E. “Rachunek nazw i schemat predykacji z Begriffschrift Gottloba Fregego” (Calculus of Names and Predication Scheme from Gottlob Frege’s Begriffsschrift). In Predykacja, negacja i kwantyfikacja (Predication, Negation and Quantification); Wojciechowski, E., Ed.; Aureus: Kraków, Poland, 2019. [Google Scholar]
- Frege, G. Die Grundlagen der Arithmetik: Eine Logisch Mathematische Untersuchung über den Begriff der Zahl; Verlag von Wilhelm Koebner: Breslau, Germany, 1884. [Google Scholar]
- Patzig, G. Gottlob Frege und die logische Analyse der Sprache. In Sprache Und Logik, 2nd ed.; Patzig, G., Ed.; Vandenhoeck & Ruprecht: Göttingen, Germany, 1981; pp. 77–100. [Google Scholar]
- Wojciechowski, E. “Klasy dystrybutywne i klasy kolektywne” (Distributive Classes and Collective Classes). In Predykacja, negacja i kwantyfikacja (Predication, Negation and Quantification); Wojciechowski, E., Ed.; Aureus: Kraków, Poland, 2019. [Google Scholar]
- Wojciechowski, E. “Rachunek nazw z listami” (The Calculus of Names with Lists). Rocz. Filoz.
**2011**, 59, 35–50. [Google Scholar] - Borkowski, L. Logika Formalna (Formal Logic), 2nd ed.; PWN: Warszawa, Poland, 1977. [Google Scholar]
- Grote, A. Anzahl, Zahl und Menge. Die Phänomenologischen Grundlagen der Arithemitk; Felix Meiner Verlag: Hamburg, Germany, 1983. [Google Scholar]
- Von Helmholz, H. Zählen und Messen, erkenntnisstheoretisch betrachtet. In Philosophische Aufsätze, Eduard Zeller Ze Seinem Fünfzigjährigen Doctorjubiläum Gewidmet; Fues Verlag: Leipzig, Germany, 1887; pp. 17–52. [Google Scholar]

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**MDPI and ACS Style**

Wojciechowski, E. The *Zahl-Anzahl* Distinction in Gottlob Frege: Arithmetic of Natural Numbers with *Anzahl* as a Primitive Term. *Axioms* **2020**, *9*, 6.
https://doi.org/10.3390/axioms9010006

**AMA Style**

Wojciechowski E. The *Zahl-Anzahl* Distinction in Gottlob Frege: Arithmetic of Natural Numbers with *Anzahl* as a Primitive Term. *Axioms*. 2020; 9(1):6.
https://doi.org/10.3390/axioms9010006

**Chicago/Turabian Style**

Wojciechowski, Eugeniusz. 2020. "The *Zahl-Anzahl* Distinction in Gottlob Frege: Arithmetic of Natural Numbers with *Anzahl* as a Primitive Term" *Axioms* 9, no. 1: 6.
https://doi.org/10.3390/axioms9010006