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,
Jo.
More information about the FOM
mailing list