About existence-as-consistency

joseph.vidal.rosset at gmail.com joseph.vidal.rosset at gmail.com
Wed Jun 30 09:11:00 EDT 2021

Le dim. 06/27/21 juin 2021  à 08:57:33 , <sambin at math.unipd.it> a envoyé
ce message:
> Dear Fomers,
> I am deeply interested in the historical origin and explanation of the
> principle by  which consistency  of an  axiomatic theory  T (typically
> ZFC) is sufficient to justify it  and derive that what it speaks about
> exists (in the  case of ZFC, sets satisfying  the properties described
> by  its  axioms).  I call  this  principle:  existence-as-consistency,
> shortly EaC.
> [...]
> A related  question is:  is there  a way to  avoid assuming  EaC while
> keeping classical logic (and hence validity of LEM)?

Dear Giovanni (cc. FOMers),

What you call  EaC is involved by minimal logic  and therefore also by
intuitionistic and classical logic.

Proof by contrapositive of the following axiom: in minimal logic, from
contradiction the void is correctly deduced.
Therefore,  from  the  assumption  "not the  void",  the  negation  of
contradiction  is deducible  (contrapositive  of 1).  In other  words,
because any set  which is not the  empty set is a  set where something
exists,   necessarily  any   set  where   something  exists   is  *not
contradictory* that is to say  *is consistent*. This point proves that
EaC is involved  by minimal logic and  that it has nothing  to do with
the validity of the classical Law of Excluded Middle.

Best wishes,


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