[FOM] picture of V

José Ferreirós Domínguez josef at us.es
Thu Oct 20 04:08:40 EDT 2011

Dear all:

let me ask a very simple question. Who started the tradition of  
picturing the set-theoretic universe V with the usual "inverted  
cone"? Maybe some of you can make suggestions, perhaps even based on  
personal experience?

The idea behind the picture, of course, can be traced back to von  
Neumann and Zermelo in the late 1920s (see e.g. Zermelo's paper of  
1930, translated in W. Ewald, "From Kant to Hilbert", vol. 2). At the  
basis of the famous iterative conception there were sophisticated  
technical developments showing that any model of ZFC can be developed  
in the form of a cumulative hierarchy. Only afterwards somebody (in  
fact, Goedel) proposed the iterative concepcion, the intuitive view  
of a step-by-step development of the universe of sets with steps  
indexed by the ordinals, the development guided by the powerset  
operation. The idea is spelled out in 1947 (What is Cantor's  
continuum problem?) and seems to be already clear in his mind in a  
lecture of 1933.

But I am curious about the picture itself, which I believe has  
special appeal for the individual mathematician learning set theory  
and trying to understand what it is about.


Jose Ferreiros
Departamento de Filosofia y Logica
Universidad de Sevilla

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