Foundations, consistency, and Ex Falso Quodlibet

[Joseph Vidal-Rosset]

Hello,

In a thread about "Bourbaki and foundations"

Le 14/05/2022, 22:53, Tennant, Neil Tennant wrote:
> Larry,
> Certainly those students shouldn’t mock AC.
> But EFQ is an altogether different kettle of fish!
> It’s needed NOWHERE in foundations.
> Best regards,
> Neil (Tennant)

In my opinion, this claim deserves another thread (but if FOM editor 
disagrees, please keep "Bourbaki and foundations" thread).

I believe that Tennant's claim is false, if the following usual argument 
given about Gödel's incompleteness theorem is valid. See for example 
this webpage:

https://math.stackexchange.com/questions/2889291/is-consistency-an-axiom-of-mathematics 


where there is this usual argument:

> If math is inconsistent then every mathematical sentence is a theorem. (...) Godel: Let S be a recursive system of axioms that is powerful enough to discuss the arithmetic of N. Then S is incomplete or inconsistent.  DanielWainfleet

It is usually said that an inconsistent system S could prove all 
sentences, including the sentence that S is consistent.

> A logical system is Consistent in the sense of Post (with
> respect to a certain category of primitive symbols designated as ‘propositional
> variables’) if a wff consisting of a propositional variable alone is not a theorem.

It is obvious that what I call "the usual argument" as well as what is 
called "Post consistency" is based on Ex Falso Quodlibet. Therefore, 
either Tennant's claim is false, or the usual argument and Post 
consistency are ill founded (that is to say somewhere false).

Best wishes,

Jo.