Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics

By David B. Malament | Go to book overview

APPENDIX

This appendix consists of a note by Bernays that is found in the Hilbert Nachlaß, cod. 685:9, 2 and was written, presumably, between 1925 and 1928. It is entitled Existenz and Widerspruchsfreiheit. The note is preceded by a brief note on the finitist standpoint (containing only well-known observations) and followed by a note analyzing the criticism of axiomatic set theory by Skolem and von Neumann. For our purposes the latter note is of interest only by the way in which Bernays explicates “consistency of the countable infinite.” The consistency proof of arithmetic (including the transfinite axioms for the epsilon operator) establishes the consistency of the countable infinite in the following sense: “An axiom system has been recognized as consistent that cannot be satisfied by a finite system of objects.” Clearly, this topic is taken up in a very illuminating way in Bernays 1950.


Existence and Consistency

The claim: “Existence = consistency” can only refer to a system as a whole. Within an axiomatic system the axioms decide about the existence of objects.

If, for a system as a whole, consistency is to he synonymous with existence, then the proof of consistency must consist in an exhibition [of a model].

(All consistency proofs up to now have been either direct exhibitions or indirect ones by reduction; in the latter case a certain other system is already taken as existent. Frege has defended with particular emphasis the view that any proof of consistency has to be given by the actual presentation of a system of objects.)

In proof theory, laying a new foundation of arithmetic, consistency proofs are not given by exhibition. From this foundational standpoint it does not hold any longer that existence equals consistency. Indeed, it is not the opinion that the possibility of an infinite system is to be proved, rather it is only to be shown that operating with such a system does not lead to contradictions in mathematical reasoning [beim Schliessen].


Existenz und Widerspruchsfreiheit

Die Behauptung: “Existenz = Widerspruchsfreiheit” kann sich immer nur auf ein System als Ganzes beziehen. Innerhalb eines axiomarischen Systems wird über die Existenz von Dinger) durch die Axiome entschieden.

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