[FOM] From theorems of infinity to axioms of infinity

Panu Raatikainen panu.raatikainen at helsinki.fi
Mon Mar 11 01:12:25 EDT 2013

Surely Hilbert's non-traditional view of what axioms are and do was  
behind Zermelo.

Best, Panu

Lainaus Michael Detlefsen <mdetlef1 at nd.edu>:

> I'd like to understand what were the forces underlying the  
> transition from treating existence claims for infinite collections
> as theorems (i.e. propositions that require proof) to propositions  
> that can be admitted as axioms.
> In the latter half of the nineteenth century, both Bolzano  
> (Paradoxes of the Infinite (1851), sections 13, 14) and Dedekind  
> (Theorem 66
> of "Was sind …" (1888)) offered proofs of the existence of infinite  
> collections (using similar arguments).
> By Zermelo's 1908 paper, it had become an axiom (Axiom VII). Zermelo  
> remarked that he found Dedekind's proof unsatisfying because
> it appealed to a "set of everything thinkable", and, in his view,  
> such a collection could not properly form a set.

Panu Raatikainen

Ph.D., University Lecturer
Docent in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki

E-mail: panu.raatikainen at helsinki.fi


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