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Proof Theory: Proof Theory in mathematics and logic is about the existence or nonexistence of finite strings of symbols allowing to derive a statement. Therefore, proof theory is a part of the syntax, as opposed to the model theory, which belongs to the semantics. See also model theory, syntax, semantics.
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 Concept Summary/Quotes Sources

David Hilbert on Proof Theory - Dictionary of Arguments

Berka I 384
Proof Theory/Hilbert: first, the concepts and propositions of the theory to be examined are represented by a formal system, and treated without reference to their meaning only formally.
I 385
Proof Theory: this (subsequent) investigation is dependent on the logical meaning of its concepts and conclusions. Thus formal theory is compared with a meaningful meta theory (proof theory)(1).
Berka I 395
Proof Theory/Hilbert: basic thought, thesis: everything that makes up existing mathematics is strictly formalized, so that the actual mathematics becomes a set of formulas. New: the logical signs "follow" (>) and "not".
Final scheme:

S › T

Where each time the premises, i.e. (S and S > T) are either an axiom, or are created by inserting an axiom or coincide with the final formula.
Definition provable/Hilbert: a formula is provable if it is either an axiom or an axiom by insertion from it, or if it is the final formula of a proof.
, >Provability.
Meta-Mathematics/proof theory/Hilbert: meta mathematics is now added to the actual mathematics: in contrast to the purely formal conclusions of the actual mathematics, the substantive conclusion is applied here. However, only to prove the consistency of axioms.
>Axioms, >Axiom systems, >Axioms/Hilbert.
In this meta-mathematics, the proofs of the actual mathematics are operated upon, and these themselves form the subject of the substantive investigation.
Thus the development of the mathematical totality of knowledge takes place in two ways:
A) by obtaining new provable formulas from the axioms by formal concluding and
B) by adding new axioms together with proof of the consistency by substantive concluding.
>Consistency, >Material implication.
Berka I 395
Truth/absolute truth/Hilbert: axioms and provable propositions are images of the thoughts which make up the method of the previous mathematics, but they are not themselves the absolute truths.
Def absolute truth/Hilbert: absolute truths are the insights provided by my proof theory with regard to the provability and consistency of the formula systems.
Through this program, the truth of the axioms is already shown for our theory of proof(2).

1. K. Schütte: Beweistheorie, Berlin/Göttingen/Heidelberg 1960, p. 2f.
2. D. Hilbert: Die logischen Grundalgen der Mathematik, in: Mathematische Annalen 88 (1923), p. 151-165.

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. Translations: Dictionary of Arguments
The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

Berka I
Karel Berka
Lothar Kreiser
Logik Texte Berlin 1983

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