|Waismann I 75
Loewenheim-Skolem/Waismann: proved that it is impossible to characterize the series of numbers with finitely many axioms.
Every statement which applies in the arithmetic of the natural numbers is also valid for other types of structure, so that it is impossible to distinguish the series of numbers by any inner properties from series of a different kind.
Example: Frappantes and non-trivial example is that the re-interpretation of the concept of the integer is possible, although so much is required that one might think that these sentences could apply only to the integers.
The integers are:
1. linearly ordered
2. they reproduce by addition, multiplication, subtraction. comm. asso., distr. etc. zero element, one element,
3. concepts of divisibility, of units, relative prime, etc., if a and b are relatively prime, then there are two numbers x and y such that ax by = 1.
Now one can also form polynomials of the form
antn + an 1tn 1 + ....a1t + ao
according to the 5 axioms, which were originally intended only for natural numbers.
Theory of Loewenheim Skolem: such a re-interpretation will always be possible, by how many properties the concept of natural numbers is to be grasped._____________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. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976