﻿ Christian Thiel on Arithmetics - Philosophy Dictionary of Arguments

# Philosophy Dictionary of Arguments

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Thiel I 225
Arithmetics/Lorenzen/Thiel: Arithmetics is the theory in which the infinite occurs in its simplest form, it is essentially nothing more than the theory of the infinite itself.
Arithmetics as the theory of the set of signs (e.g. tally-list) is universal in the sense that the properties and relations of any other infinite set of signs can always be "mapped" in some way.
The complexity of matter has led to the fact that a large part of the secondary literature on Gödel has put a lot of nonsense into the world on metaphors such as "reflection", "self-reference", etc.
I 224
The logical arithmetic full formalism is denoted with F. It contains, among other things, inductive definitions of the counting signs, the variables for them, the rules of quantifier logic and the Dedekind-Peanosian axioms written as rules.
I 226
The derivability or non-derivability of a formula means nothing other than the existence or non-existence of a proof figure or a family tree with A as the final formula.
Therefore also the metamathematical statements "derivable", respectively "un-derivable" each reversibly correspond unambiguously to a basic number characterizing them. > Theorem of Incompleteness > Gödel.
Terminology/Writing: S derivable, \$ not derivable.
"\$ Ax(x)" is now undoubtedly a correctly defined form of statement, since the count for An(n) is uniquely determined. Either \$An(n) is valid or not.
Thiel I 304
The centuries-old dominance of geometry has aftereffects in the use of language. For example "square", "cubic" equations etc.
Arithmetics/Thiel: has today become a number theory, its practical part degraded to "calculating", a probability calculus has been added.
I 305
In the vector and tensor calculus, geometry and algebra appear reunited.
A new discipline called "invariant theory" emerges, flourishes and disappears completely, only to rise again later.
I 306
Functional analysis: is certainly not a fundamental discipline because of the very high level of conceptual abstraction.
I 307
Bourbaki contrasts the classical "disciplines" with the "modern structures". The theory of prime numbers is closely related to the theory of algebraic curves. Euclidean geometry borders on the theory of integral equations. The ordering principle will be one of the hierarchies of structures, from simple to complicated and from general to particular.

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Chr. Thiel