User:Boris Tsirelson/Sandbox1: Difference between revisions

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==Univalent or multivalent==
==Decidable or undecidable==


Plane geometry (called also "planar geometry") is a part of solid geometry that restricts itself to a single plane ("the plane") treated as a geometric universe. The question "which plane?" is inappropriate, since planes do not differ in their geometric properties. Every two planes α, β are isomorphic, that is, there exists an isomorphism ''f'' between α and β. Treating α and β as sets of points one defines isomorphism as an invertible (one-to-one and onto) map f : α to β preserving all primitive relations. Namely: f maps lines into lines; the distance between f(A) and f(B) on β is equal to the distance between A and B on α; etc. The same is required of the inverse map <math>f^{-1}:\be\to\al</math>.
Theorems of a theory are, by definition, statements that follow from
the given axioms according to the given rules (called by different
authors inference rules, derivation rules, deduction rules,
transformation rules). Note that "a theorem" does not mean "a motivated
theorem", "an important theorem" etc., not even "an already discovered
theorem". All theorems are just an indiscriminate stream of logical
consequences of the axioms. It is impossible to list all theorems,
since they are infinitely many. However, an endless algorithmic
process can generate theorems, only theorems, and all theorems in the
sense that every theorem will be generated, sooner or later. (In more
technical words: the set of all theorems is recursively enumerable.)


Axioms of the plane Euclidean geometry leave no freedom, they determine uniquely all geometric properties of the plane. More exactly: all Euclidean planes are mutually isomorphic. In this sense we have "the" Euclidean plane. In terms of Bourbaki, the plane
An open question (in a mathematical theory) is a statement neither
Euclidean geometry is an univalent theory. In contrast, axioms of a linear space (called also vector space) leave a freedom: a linear space may be one-dimensional, two-dimensional, three-dimensional, four-dimensional and so on (infinite dimension is also possible). The corresponding theory is multivalent.
proved nor disproved. It is possible (in principle, not necessarily in
practice) to run the theorem-generating algorithm waiting for one of
two events: either the given statement appears to be a theorem, or its
negation does; in both cases the (formerly) open question is
decided. However,
(a) there is no guarantee that it will be decided in the first
10<sup>6</sup> steps of the algorithm, nor in the first
10<sup>1000</sup> steps, nor in any time estimated beforehand;
(b) worse, there is no guarantee that it will be decided at all.


According to Bourbaki, the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.
A statement is called independent (in other words, undecidable) in the
 
given theory, if it is not a theorem, but also its negation is not a
A similar idea occurs in mathematical logic: a theory is called categorical if all its models are mutually isomorphic. However, for Bourbaki a theory is embedded into the set theory, while in logic a theory is standalone (embedded into the first-order logic).
theorem.
 
==Motivated or indiscriminate==
 
Monkeys could type into a computer a sequence of hardware instructions; the computer could execute them; but the result of such "programming" has almost no chance to be fascinating or useful. Fascinating computer games reflect human predilections. Useful programs reflect human needs. A computer is dull for humans unless its software reflects human life in one way or another.
 
Likewise, a theorem is of no interest for humans unless it is motivated in one way or another by human life. The motivation may be quite indirect; many theorems "only" help to prove other theorems, many are appreciated "only" for their aesthetic value, etc. But some kind of motivation is necessary. Indiscriminate stream of logical consequences of the axioms is not publishable in the mathematical literature.

Revision as of 12:48, 5 June 2010

Decidable or undecidable

Theorems of a theory are, by definition, statements that follow from the given axioms according to the given rules (called by different authors inference rules, derivation rules, deduction rules, transformation rules). Note that "a theorem" does not mean "a motivated theorem", "an important theorem" etc., not even "an already discovered theorem". All theorems are just an indiscriminate stream of logical consequences of the axioms. It is impossible to list all theorems, since they are infinitely many. However, an endless algorithmic process can generate theorems, only theorems, and all theorems in the sense that every theorem will be generated, sooner or later. (In more technical words: the set of all theorems is recursively enumerable.)

An open question (in a mathematical theory) is a statement neither proved nor disproved. It is possible (in principle, not necessarily in practice) to run the theorem-generating algorithm waiting for one of two events: either the given statement appears to be a theorem, or its negation does; in both cases the (formerly) open question is decided. However, (a) there is no guarantee that it will be decided in the first 106 steps of the algorithm, nor in the first 101000 steps, nor in any time estimated beforehand; (b) worse, there is no guarantee that it will be decided at all.

A statement is called independent (in other words, undecidable) in the given theory, if it is not a theorem, but also its negation is not a theorem.

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