Lexicon of Arguments

 

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.
>Derivation, >Derivability, >Axiom systems.
The plural of "geometries" shows a change in the concept of geometry itself.
>Geometry.
I 209
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.
Cf. >Properties.
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))

I 210
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.
>Models.
I 211
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.
I 213
Insisting on "clean" solutions only arises with metamathematical needs.