Axiomatic set theory as a basis for the construction of mathematics

It is widely known that one of the major tasks of 'Foundations' is to construct a formal system which can he said to contain the whole of mathematics. For various reasons axiomatic set theory is a very suitable choice for such a system and it is one which has proved acceptable to both logicians and mathematicians. The particular demands of mathematicians and logicians, however, are not the same. As a result there exist at the moment two different formulations of set theory which can be roughly said to cater for logicians and mathematicians respectively. It is these systems which are the subject of this dissertation. The system of set theory constructed for logicians is by P. Bernays. This will be discussed in chapter II. For mathematicians No Bourbaki has constructed a system of set theory within which he has already embedded a large part of mathematics. This system will be discussed in chapter III. Chapter I is historical and contains some of Cantor's original ideas. The relationship between Bernays' system and (essentially) Bourbaki's system is commented upon in chapter IV. <p

Graduate Catalogue, 1966-1967

https://encompass.eku.edu/g_cat/1010/thumbnail.jp

Syntactical consistency proofs for term induction revisited

GIn [6] wird von Gerhard Gentzen die Widerspruchsfreiheit der Peano Arithmetik erster Stufe PA bewiesen. Die Methode geht dabei folgendermaßen vor: Man deniert einen simplen Teil SPA der Peano Arithmetik (SPA enthält im speziellen keine Anwendung des Induktionsschemas) und zeigt zuerst die von SPA. Der Rest des Arguments verläuft indirekt. Man nimmt an, dass PA einen Widerspruch ableitet und zeigt das dessen Deduktion zu einer Deduktion in SPA transformiert werden kann, was der Widerspruchsfreiheit von SPA widerspricht. Diese Transformation verläuft wie folgt: Jeder Deduktion in PA wird eine Ordinalzahl (oder genauer, ein Ordinalzahlterm eines Ordinalzahlnotations Systems) zugeordnet, diese wird der Rang der Deduktion genannt. Dann wird gezeigt, dass es zu jeder Deduktion eines Widerspruches (die nicht in SPA verläuft) eine Deduktion (ebenfalls eines Widerspruches) gibt die einen kleineren Rang hat. Diese Methode benötigt daher die Wohlfundiertheit des verwendeten Ordinalzahlnotations Systems (in diesem Fall bis "0). Bei näherer Betrachtung von Gentzens Methode fällt auf, dass sie lediglich folgende Eigenschaften von PA verwendet: 1. Alle Axiome von PA sind Allsätze oder Instanzen des Induktionsschemas φ(0) ^ ∀ x[φ(x) -> φ(x + 1)] -> ∀ xφ(x): 2. Alle geschlossenen Terme sind beweisbar (in SPA) gleich zu einem Term der lediglich aus 0 und dem Symbol der Nachfolgerfunktion aufgebaut ist. Dies erlaubt eine Verallgemeinerung von Gentzens Methode. In dieser Diplomarbeit werden wir daher Theorien Tf = (Tf )0 [ (Tf )Ind betrachten die folgende Eigenschaften erfüllen: 1. (Tf )0 besteht lediglich aus Allsätzen. 2. (Tf )Ind beinhaltet alle Instanzen des Induktionsschemas φ(c1)^:::^φ(cm)^∀ ~x[φ(x1)^:::^φ(xn) -> φ(f(x1; :::; xn))] -> ∀ xφ(x): 3. Der simple Teil von Tf beweist für jeden geschlossenen Term t, dass t gleich einem Term _t ist der lediglich aus den symbolen c1; :::; cm und f aufgebaut ist. Die Widerspruchsfreiheit von Tf kann nun, wie in [6] für PA, relativ zu ihrem simplen Teil (wo Induktion wie zuvor bei Gentzen nicht möglich ist) gezeigt werden. Eine Konsequenz dieses Resultates ist das folgende Korollar. Korollar. Sei φ(a) quantorenfrei und Tf widerspruchsfrei. Wenn Tf j= 9xφ(x), dann (Tf )0 j= 9xφ(x). Insbesondere ist Tf _0 1 -konservativ über (Tf )0. Es scheint mir als wäre die Methode, die von Kurt Schütte in seinem Widerspruchsfreiheitsbeweis von PA verwendet wird, eine gänzlich andere. Schütte, Tait und Andere verwenden Kalküle mit unendlichen Deduktionsregeln um, in einem gewissen Sinne, die Beweistheoretische Ordinalzahl einer Theorie zu berechnen. Dies erfolgt über eine Transformation der endlichenDeduktionen der Theorie (in der Logik erster Stufe) in Deduktionen ineinem unendlichen Kalkül, das Schnittelimination erlaubt. Im Gegensatz zu Gentzens Methode hat die von Schütte eine enge Beziehung zu den beweistheoretischenOrdinalzahlen. Auf die Unterschiede der beiden Methoden wird nicht weiter eingegangen werden. Anstatt dieses Vergleiches wird lediglich eine Variante von Taits Methode dazu verwendet die _1 1-Ordinalzahl, wie von Wolfram Pohlers in [∀ ] beschrieben, von Theorien TAf (aufgefasst als Taitkalkühl) zu messen. Es wird angenommen das TAf folgende Eigenschaften erfüllt: 1. TAf enthält für jede primitiv rekursive Funktion die denierenden Formeln als Axiome. 2. Weiters enthält TAf alle Instanzen des Schemas φ(c1)^:::^φ(cm)^∀ ~x[φ(x1)^:::^φ(xn) -> φ(f(x1; :::; xn))] -> ∀ xφ(x): Hierbei ist f ein m-stelliges Symbol einer gleichstelligen primitiv rekursiven Funktion und c1; :::; cl Individuenkonstanten. 3. Es wird außerdem angenommen das jedes n 2 N gleich einer Komposition aus fN und den natürlichen Zahlen cN1;:::; cN l ist.Gerhard Gentzen proves the consistency of first-order Peano arithmetic \textbf{PA}. His method works as follows: Define a simple part \textbf{SPA} of peano arithmetic (\textbf{SPA} does in particular not contain induction) and first show the consistency of \textbf{SPA}. Now assume towards a contradiction that \textbf{PA} deducts an contradiction. Show that this deduction can be transformed into a deduction in \textbf{SPA}, this contradicts the consistency of \textbf{SPA}. How to get a deduction in \textbf{SPA}: We assign an ordinal (more exact an ordinal term of an ordinal notation system) to each deduction in \textbf{PA}, called the rank of the deduction. Next show that for each deduction which deducts a contradiction (and is not in \textbf{SPA}) there is a deduction (also deducting and contradiction) with smaller rank. This method requires that the ordinal notation system (which goes up to $\varepsilon_0$) is well-founded. It turns out that Gentzen's method requires only to the following properties of \textbf{PA}: \begin{enumerate} \item All axioms of \textbf{PA} are universal sentences or instances of the induction schema $$\varphi(0) \wedge \forall x [\varphi(x) \rightarrow \varphi(x+1)] \rightarrow \forall x \varphi(x).$$ \item All closed terms are provable equal to a term build up just from $0$ and the symbol of the successor function. \end{enumerate} This allows a slight generalisation of Gentzen's method. In this Diploma Thesis we consider theories $T_f = (T_f)_0 \cup (T_f)_{\text{Ind}}$ with the following properties: \begin{enumerate} \item $(T_f)_0$ contains only universal sentences. \item $(T_f)_{\text{Ind}}$ contains all instances of the general induction schema $$\varphi(c_1)\wedge ... \wedge \varphi(c_m) \wedge \forall \vec{x} [\varphi(x_1)\wedge...\wedge \varphi(x_n) \rightarrow \varphi(f(x_1,...,x_n))] \rightarrow \forall x \varphi(x).$$ \item The simple part of $T_f$ proves for every closed term $t$ the equality of $t$ to a term $\bar{t}$ build up just from $c_1,...,c_m$ and $f$. \end{enumerate} As Gentzen did for \textbf{PA}, the consistency of $T_f$ can be shown with respect to their simple part which corresponds to the simple part of Gentzen (also without induction). As a consequence, one gets the following result for all such theories. \begin{cor2} Assume $\varphi(a)$ is quantifier free and $T_f$ consistent.\\ If $T_f \models \exists x \varphi(x)$, then $(T_f)_0 \models \exists x \varphi(x)$.\\ I.e., $T_f$ is $\Sigma_1^0$-conservative over $(T_f)_0$. \end{cor2} It seems that this method is different in an essential way to the method Kurt Sch\"{u}tte uses in his consistency proof of \textbf{PA}. Sch\"{u}tte, Tait and others uses calculi with infinite deduction rules. These methods compute, in some sense, the proof theoretical ordinal of the considered theory by embedding the deductions of the theory (in ordinary first-order logic) in an infinite system which allows cut-elimination. In contrast to Gentzen's method Sch\"{u}tte's and Tait's methods are closely related to the proof theoretical ordinals.\\ We do not provide an analysis of the disparities of both methods. Instead we present the point of view Wolfram Pohlers propose, to measure the $\Pi_1^1$-ordinal of theories $\mathsf{TA}_f$ (presented as a Tait-calculus) satisfying the following conditions: \begin{enumerate} \item $\mathsf{TA}_f$ includes all defining axioms for primitive recursive functions. \item All instances of the schema $$\varphi(c_1)\wedge ... \wedge \varphi(c_m) \wedge \forall \vec{x} [\varphi(x_1)\wedge...\wedge \varphi(x_n) \rightarrow \varphi(f(x_1,...,x_n))] \rightarrow \forall x \varphi(x)$$ are included. Here $f$ is an $m$-array primitive recursive function constant and $c_1,...,c_l$ are individual constants. \item Every $n \in \mathbb{N}$ is equal to a composition of $f^\mathbb{N}$ and the elements $c_1^\mathbb{N},...,c_l^\mathbb{N}$. \end{enumerate

Interactive program verification using virtual programs

This thesis is concerned with ways of proving the correctness of computer programs. The first part of the thesis presents a new method for doing this. The method, called continuation induction, is based on the ideas of symbolic execution, the description of a given program by a virtual program, and the demonstration that these two programs are equivalent whenever the given program terminates. The main advantage of continuation induction over other methods is that it enables programs using a wide variety of programming constructs such as recursion, iteration, non-determinism, procedures with side-effects and jumps out of blocks to be handled in a natural and uniform way. In the second part of the thesis a program verifier which uses both this method and Floyd's inductive assertion method is described. The significance of this verifier is that it is designed to be extensible, and to this end the user can declare new functions and predicates to be used in giving a natural description of the program's intention. Rules describing these new functions can then be used when verifying the program. To actually prove the verification conditions, the system employs automatic simplification, a relatively clever matcher, a simple natural deduction system and, most importantly, the user's advice. A large number of commands are provided for the user in guiding the system to a proof of the program's correctness. The system has been used to verify various programs including two sorting programs and a program to invert a permutation 'in place' the proofs of the sorting programs included a proof of the fact that the final array was a permutation of the original one. Finally, some observations and suggestions are made concerning the continued development of such interactive verification systems

Bowdoin College Catalogue (1982-1983)

https://digitalcommons.bowdoin.edu/course-catalogues/1263/thumbnail.jp