Scott's system of axioms

The Scott's axiom system , named after the mathematician Dana Scott , is a system of axioms of set theory , which as an alternative access to the system of axioms of Zermelo-Fraenkel set theory , short-IF can be viewed. It uses the principle of reflection , which can be proven in ZF, as an axiom and in this way can dispense with some ZF axioms.

motivation

The Von Neumann hierarchy divides the entire universe of quantities into levels , with the ordinal numbers running through. We describe here three consequences which then, conversely, become axioms of Scott's system of axioms. ${\displaystyle V_{\alpha }}$ ${\displaystyle \alpha }$

• ${\displaystyle \forall x\,\exists \alpha \,(x\in V_{\alpha })}$

That is, every amount is in one level. This is exactly the equivalent of the foundation axiom of the Von Neumann hierarchy, according to which every set is already in a level or is already limited by a level with regard to the relation; one therefore speaks of the limitation lemma . ${\displaystyle x}$ ${\displaystyle \in }$

• ${\displaystyle \forall \alpha \,\forall x\,(x\in V_{\alpha }\,\leftrightarrow \,\exists \beta \,(\beta <\alpha \land (x\in V_{\beta }\lor x\subset V_{\beta })))}$

So if in a stage is, it is already at a lower level with or is included as a subset in such lower level. This so-called cumulation lemma follows directly from the recursive definition of the levels as a union of all predecessors or as a power set of the predecessor, depending on whether a Limes ordinal number is or not. ${\displaystyle x}$ ${\displaystyle V_{\alpha }}$ ${\displaystyle V_{\beta }}$ ${\displaystyle \beta <\alpha }$ ${\displaystyle V_{\alpha }}$ ${\displaystyle \alpha }$

The reflection principle says that every statement that can be formulated in ZF is already reflected by a step , more precisely: ${\displaystyle \varphi =\varphi (x)}$ ${\displaystyle V_{\alpha }}$

• ${\displaystyle \forall x\,\exists \alpha \,(x\in V_{\alpha }\land V_{\alpha }{\text{ spiegelt }}\varphi (x))}$

The reflection by the level means the reflection by the predicate defined by " "; Details on the concept of mirroring can be found in the article Relativization (set theory) . ${\displaystyle V_{\alpha }}$ ${\displaystyle x\in V_{\alpha }}$

These three properties - boundedness lemma, cumulation lemma and reflection principle - should now be raised to axioms without using the levels defined in ZF. For this we need a new predicate “ is level”, which we call. The spelling is therefore to be read as “ is level”, and one can imagine something similar to the levels of the Von Neumann hierarchy. The exact properties of these levels are, however, determined by the axioms of Scott's system of axioms, which is now presented. ${\displaystyle x}$ ${\displaystyle \Sigma }$ ${\displaystyle \Sigma x}$ ${\displaystyle x}$

The system of axioms

We use small Latin letters as variables for quantities and the symbols , where = stands for equality and for the element relationship, is a one-digit predicate and the remaining symbols are the usual logical symbols. In the following axioms denote a set-theoretical formula with the variable and possibly further variables (parameters) . ${\displaystyle =,\in ,\Sigma ,\neg ,\land ,\lor ,\rightarrow ,\leftrightarrow ,\exists ,\forall }$ ${\displaystyle \in }$ ${\displaystyle \Sigma }$ ${\displaystyle \varphi =\varphi (x,x_{1},\ldots ,x_{n})}$ ${\displaystyle x}$ ${\displaystyle x_{1},\ldots ,x_{n}}$

• Existence : ${\displaystyle \exists x(x=x)}$

The axiom of existence requires that there is at least a set in the set universe .

• Extensionality : ${\displaystyle \forall x\,y\,(\forall z(z\in x\leftrightarrow z\in y)\rightarrow x=y)}$

The axiom of extensionality describes the quantitative aspect of the concept of sets, if two sets contain the same elements, then they are the same.

• Disposal : ${\displaystyle \forall x_{1},\ldots ,x_{n}\forall x\exists y\forall z\,(z\in y\leftrightarrow z\in x\land \varphi (z,x_{1},\ldots ,x_{n}))}$

At any amount and at any property can discard the set of elements that fulfill this property, more precisely: For a given formula and given parameters, there are at any amount , the amount which exactly those elements of is that the property suffice. This is not a single axiom, but a so-called scheme of axioms, since one gets an axiom for every formula . ${\displaystyle \varphi }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle x}$ ${\displaystyle \varphi (z,x_{1},\ldots ,x_{n})}$ ${\displaystyle \varphi }$

• Limitations : . ${\displaystyle \forall x\exists v\,(\Sigma v\land x\in v)}$

Any amount is in one level.

• Cumulation : ${\displaystyle \forall v\,(\Sigma v\rightarrow \forall x\,(x\in v\leftrightarrow \exists w(\Sigma w\land w\in v\land (x\in w\lor x\subset w))))}$

The focus is , as usual for . In words, the axiom of accumulation says: If there is a level, then for each of this level there is a level contained in which lies as an element or as a subset. ${\displaystyle x\subset w}$ ${\displaystyle \forall z(z\in x\rightarrow z\in w)}$ ${\displaystyle v}$ ${\displaystyle x}$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle x}$

• Reflection principle : ${\displaystyle \forall x\,\exists v\,(\Sigma v\land x\in v\land v{\text{ spiegelt }}\varphi (x,x_{1},\ldots x_{n}))}$

Here all formulas should run through without the symbol , so it is again a scheme of axioms. The expression means ${\displaystyle \varphi }$ ${\displaystyle \Sigma }$ ${\displaystyle v{\text{ spiegelt }}\varphi (x,x_{1},\ldots x_{n})}$

${\displaystyle \forall x_{1},\ldots x_{n}\,(x_{1}\in v\land \ldots \land x_{n}\in v\rightarrow (\varphi (x,x_{1},\ldots x_{n})\leftrightarrow [\varphi (x,x_{1},\ldots x_{n})]^{v}))}$

where is the formula resulting from the relativization according to . ${\displaystyle [\varphi (x,x_{1},\ldots x_{n})]^{v}}$ ${\displaystyle v}$ ${\displaystyle \varphi (x,x_{1},\ldots x_{n})}$

The entirety of these axioms will be denoted in the following . ${\displaystyle \Sigma }$

Equivalence to ZF

The first three axioms from are also ZF axioms, and the introductory remarks show that the definition defines a predicate that also satisfies the other three axioms. Conversely, it can be made to derive all ZF axioms, that is, the union axiom , power set axiom , axiom of infinity , the foundation axiom and the scheme of substitution axioms . ${\displaystyle \Sigma }$ ${\displaystyle \Sigma x:\Leftrightarrow \exists \alpha \,(x=V_{\alpha })}$ ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$

As in ZF, you can therefore introduce ordinal numbers and the Von Neumann hierarchy of . In then the sentence applies: ${\displaystyle \Sigma }$ ${\displaystyle V_{\alpha }}$ ${\displaystyle \Sigma }$

${\displaystyle \forall x\,(\Sigma x\leftrightarrow \exists \alpha \,(x=V_{\alpha }))}$.

Thus the axiom systems are ZF and equivalent. In both axiomatizations, the same sets can prove the lack in ZF through is to be replaced. ${\displaystyle \Sigma }$ ${\displaystyle \Sigma x}$ ${\displaystyle \exists \alpha \,(x=V_{\alpha })}$

literature

• Heinz-Dieter Ebbinghaus : Introduction to set theory , Spektrum Verlag 2003, ISBN 3-8274-1411-3 , especially Chapter X, §3
• Dana Scott : Axiomatizing set theory in Axiomatic Set Theory II , Proceedings of Symposia in Pure Mathematics 13 (1974), American Mathematical Society , pp. 207-214