Philosophy Lexicon of Arguments

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.

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.
Author Item Excerpt Meta data

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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.
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. > 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))
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.
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.

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
Dedekind, R.
Chr. Thiel
Philosophie und Mathematik Darmstadt 1995

> Counter arguments against Dedekind
> Counter arguments in relation to Axioms

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Ed. Martin Schulz, access date 2017-06-24