Ackermann set theory

The Ackermann set theory is an axiomatic set theory that was given in 1955 by Wilhelm Ackermann . In it he tried to translate Cantor's definition of sets into a precise system of axioms.

The Ackermann set theory extends the Zermelo-Fraenkel set theory ZFC by classes (there: totalities), but differs from the more well-known Neumann-Bernays-Gödel set theory in that real classes can also be elements of other classes and therefore also small real ones Classes there. The ZFC axioms are only valid there in a real sub-area that fulfills the foundation axiom (it can be sorted out with Neumann's cumulative hierarchy ). The Ackermann set theory therefore contains an extended range of sets with unfounded sets and can be viewed as a generalization of the usual ZFC set theory and Zermelo set theory .

The Ackermann axioms

Ackermann's remarkably simple axiom system is based on the first-order predicate logic with identity, the two-place element relation and the one-place predicate and has an axiom scheme and an axiom for classes and for sets: ${\displaystyle \in }$ ${\displaystyle \,{\text{ist Menge}}}$

• Class Comprehension : Classes of sets exist:

The following applies to single-digit predicates : ${\displaystyle \varphi }$

${\displaystyle \forall A\colon (\varphi (A)\to A{\text{ ist Menge}})\to \exists B\colon \forall C\colon (C\in B\leftrightarrow \varphi (C))}$

The class is denoted by. ${\displaystyle \,B}$ ${\displaystyle \{C\mid \varphi (C)\}}$

• Class extensionality : Classes with the same elements are the same:

${\displaystyle \forall C\colon (C\in A\leftrightarrow C\in B)\to A=B.}$

• Set Comprehension : Classes of sets that are exclusively assigned to sets are sets:

For formulas in which exactly the variables occur freely and in which the predicate does not occur, the following applies: ${\displaystyle \varphi }$ ${\displaystyle A,T_{1},\ldots ,T_{n}}$ ${\displaystyle \,{\text{ist Menge}}}$

${\displaystyle T_{1}{\text{ ist Menge }}\land \dots \land \;T_{n}{\text{ ist Menge }}\land \;\forall A\colon (\varphi (A)\to A{\text{ ist Menge}})\to }$ ${\displaystyle \exists B\colon (B{\text{ ist Menge }}\land \;\forall A\colon (A\in B\leftrightarrow \varphi (A)))}$

• Elements and subclasses of sets are sets:

${\displaystyle A{\text{ ist Menge }}\land (B\in A\lor \forall C\colon (C\in B\to C\in A))\to B{\text{ ist Menge}}}$

Nota bene: This axiom excludes that real classes are set members, but not that real classes are members of real classes.

The selection Axiom replaced Ackermann through the ε-axiom of Hilbert , an axiom schema in a by the predicate extended language: ${\displaystyle \varepsilon _{x}\varphi (x)}$

• Each non-empty class contains a selected element:

The following applies to single-digit predicates : ${\displaystyle \varphi }$

${\displaystyle \exists X\colon \varphi (X)\leftrightarrow \varphi (\varepsilon _{X}\varphi (X))}$

The foundation axiom was not aware of Ackermann.

variants

Ackermann also formulated axioms that take into account Cantor's objects of perception from his definition of sets and, in addition to sets, also provide non-sets as set elements. Objects are quantity elements and are recorded using a definable predicate:

${\displaystyle A{\text{ ist Objekt}}=\exists M\colon (A\in M\land M{\text{ ist Menge}})}$.

• Class Comprehension : Classes of objects exist:

The following applies to single-digit predicates : ${\displaystyle \varphi }$

${\displaystyle \forall A\colon (\varphi (A)\to A{\text{ ist Objekt}})\to \exists B\colon \forall C\colon (C\in B\leftrightarrow \varphi (C))}$

• Class extensionality as above.

Nota bene: Objects that are not sets are not primary elements in the Zermelo sense. Because here is the strongest form of the axiom of extensionality , which only allows a single empty class and no further empty primitive elements. So additional objects are real classes .

• Set Comprehension : Only classes of objects assigned to objects are sets:

For formulas in which exactly the variables occur freely and in which the predicates and do not occur, the following applies: ${\displaystyle \varphi }$ ${\displaystyle A,T_{1},\ldots ,T_{n}}$ ${\displaystyle \,{\text{ist Menge}}}$ ${\displaystyle \,{\text{ist Objekt}}}$

${\displaystyle T_{1}{\text{ ist Objekt}}\land \dots \land \;T_{n}{\text{ ist Objekt}}\land \;\forall A\colon (\varphi (A)\to A{\text{ ist Objekt}})\to }$ ${\displaystyle \exists B\colon (B{\text{ ist Menge }}\land \;\forall A\colon (A\in B\leftrightarrow \varphi (A)))}$

• Elements and subclasses of sets are objects:

${\displaystyle A{\text{ ist Menge }}\land (B\in A\lor \forall C\colon (C\in B\to C\in A))\to B{\text{ ist Objekt}}}$

As a third variant, Ackermann gave a version based on type theory .

literature

• Wilhelm Ackermann: On the axiomatics of set theory , 1955. In: Mathematische Annalen . Vol. 131, 1956, pp. 336-345.

Web links

• Klaus Gloede: Set theory script WS 2000/01, chapter on Ackermann set theory (PDF file; 23 kB)

Individual evidence

1. ↑ David Hilbert: Problems of the foundation of mathematics , 1929, in: Mathematische Annalen 102 (1930), 1–9, there p. 3.