|Axiom: principle or rule for linking elements of a theory that is not proven within the theory. It is assumed that axioms are true and evident. Adding or eliminating axioms turns a system into another system. Accordingly, more or less statements can be constructed or derived in the new system. > System.|
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|Thiel I 208
Axioms/Dedekind/Thiel: From axioms, evidence, i.e. a brief insight into their truth, is required. Euclid's axioms are manageable, today's axiom systems can grow rapidly and can become unclear. From the axioms, every theorem should be derivable.
This derivability, however, exists separately for each sentence. The plural of "geometries" shows a change in the concept of geometry itself.
Dedekind was the first to try to axiomatize the calculating discipline of arithmetic (not Peano).
Definition "basic properties"/Dedekind: are those which cannot be derived from each other. > Property.
Dedekind Peano Axioms:
(1) 1 ε Z
(2) (m)((m ε Z) > (m' ε Z))
(3) (m ε Z)(n ε Z)((m' = n') > (m = n))
(4) (m ε Z) ~(m' = 1)
(5) (m ε Z)((E(m) > E(m')) >(E(1) > (n ε Z)((E)(n))
Dedekind and Peano use in the 5th axiom instead of "ε" "m in the set M".
Thiel: that is not necessary.
We convince ourselfs that the natural numbers satisfy the axiom system by inserting. The five axioms are then transformed into true sentences, for which we also say that the natural numbers with the properties and relations mentioned form a > model of the axiom system.
The constructive arithmetic with the calculus N and the construction equality of counting signs provides an operative model of the axioms. Mathematicians do not work like this either in practice or in books. The practice is not complete.
Insisting on "clean" solutions only arises with metamathematical needs.
Philosophie und Mathematik Darmstadt 1995