Berka I 294
Definition/Axiom/Hilbert: the established axioms are at the same time the definitions of the elementary concepts whose relations they regulate. ((s) Hilbert speaks of relationships, not of the use of concepts).
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Definitions, >
Definability, >
Basic concepts.
Independence/Axiom/Hilbert: the question is whether certain statements of individual axioms are mutually dependent, and whether the axioms do not contain common components which must be removed so that the axioms are independent of each other
(1).
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Independence.
1. D. Hilbert: Mathematische Probleme, in: Ders. Gesammelte Abhandlungen (1935), Vol. III, pp. 290-329 (gekürzter Nachdruck v. S 299-301).
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Thiel I 262
We consider the first three axioms of Hilbert:
1. There are exactly two straight lines at each of two distinct points P, Q, which indicate
(2) with P and Q.
2. For every line g and to any point P, which does not indicate with it, there is exactly one line that is indicated with P, but with no point of g.
3. There are three points which do not indicate with one and the same straight line.
In Hilbert's original text, instead of points one speaks of "objects of the first kind" instead of straight lines of "objects of the second kind" and instead of the incidence of "basic relation". Thus, the first axiom is now:
For each of two different objects of the first kind, there is precisely one object of the second kind, which is in a basic relation with the first two.
Thiel I 263
If the axioms are transformed quantifier-logically, then only the schematic sign "π" (for the basic relation) is free for substitutions, the others are bound by quantifiers, and can no longer be replaced by individual names of points or lines.
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Quantification, >
Quantifiers.
They are thus "forms of statements" with "π" as an empty space.
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Propositional functions.
They are not statements like those before Hilbert's axioms, whose truth or falsehood is fixed by the meanings of their constituents.
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Truth values.
In the Hilbert axiom concept (usually used today), axioms are forms of statements or propositional schemata, the components of which must be given a meaning only by interpretation by specifying the variability domains and the basic relation. The fact that this can happen in various ways, shows that the axioms cannot determine the meaning of their components (not their characteristics, as Hilbert sometimes says) themselves by their co-operation in an axiom system.
Thiel I 264
Multiple interpretations are possible: e.g. points lying on a straight line, e.g. the occurrence of characters in character strings, e.g. numbers.
Thiel I 265
All three interpretations are true statements. The formed triples of education regulations are models of our axiom system. The first is an infinite, the two other finite models.
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Models, >
Infinity.
Thiel I 266
The axioms can be combined by conjunction to form an axiom system.
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Conjunction.
Through the relationships, the objects lying in the subject areas are interwoven with each other in the manner determined by the combined axioms. The regions V .. are thereby "structured" (concrete and abstract structures).
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Domains, >
Structures (Mathematics).
One and the same structure can be described by different axiom systems. Not only are logically equivalent axiom systems used, but also those whose basic concepts and relations differ, but which can be defined on the basis of two systems of explicit definitions.
Thiel I 267
Already the two original axiom systems are equivalent without the assumption of reciprocal definitions, i.e. they are logically equivalent.
This equivalence relation allows an abstraction step to the fine structures. In the previous sense the same structures, are now differentiated: the axiom systems describing them are not immediately logically equivalent, but their concepts prove to be mutually definable.
For example, "vector space" "group" and "body" are designations not for fine structures, but for general abstract structures. However, we cannot say now that an axiom system makes a structure unambiguous. A structure has several structures, not anymore "the" structure.
Thiel I 268
E.g. body: the structure Q has a body structure described by axioms in terms of addition and multiplication.
E.g. group: the previous statement also implies that Q is also e.g. a group with respect to the addition. Because the group axioms for addition form part of the body axioms.
Modern mathematics is more interested in the statements about structures than in their carriers. From this point of view, structures which are of the same structure are completely equivalent.
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Indistinguishability.
Thiel: in algebra it is probably the most common to talk of structures. Here, there is often a single set of carriers with several links, which can be regarded as a relation.
Thiel I 269
E.g. relation: sum formation: x + y = z relation: s (x, y, z).
In addition to link structures, the subject areas often still carry order structures or topological structures.
Thiel I 270
Bourbaki speaks of a reordering of the total area of mathematics according to "mother structures". In modern mathematics, abstractions, especially structures, are understood as equivalence classes and thus as sets.
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N. Bourbaki, >
Equivalence classes.
2. Indicate = belong together, i.e. intersect, pass through the point, lie on it.
