About existence-as-consistency

[WILLIAM TAIT]

Cantor, in his FOUNDATIONS 1( OF A GENERAL THEORY OF MANIFOLDS (1883), [av expanded version of “On infinite linear sets of points, #5],   wrote  “Mathematics in its development is entirely free a d is only bound in the self-evident respect that it’s concepts must both be consist with each other and also stand in exact relationships, established by definitions, to those concepts which have been previously introduced and are already at hand a d established.” 

So this preceded Hilbert’s statement that, having stated the axioms of Euclidean geometry or the real number, for example, all that was needed for a foundation was a consistency proof. But Hilbert's call for consistency proofs didn’t begin in the 1920’s, Giovanni: it was at the turn of the century.

 The relation to the finitism of the twenties was this: there were two problems with the call for consistency proofs in the beginning of yer 20th  century. One was that there was no precise definition of consistency, because there was   no precise and adequate definition of proof.  The only way to ‘prove consistency’ was to present a model—-which won’t work for the theory of a Dedekind infinite set. The second problem, raised at the time by Poincaré, was that any consistency proof for number theory, for example, would require some form of complete induction, and so would be circular. It took Hilbert’s at least 17 years to solve the first problem: Building on the logical work of Frege and Russell-Whitehead, he-or maybe I should say he and Bernays-developed the concept of nth  order predicate logic as the possible logical frameworks for mathematical theories.  He took longer to respond, in the early 1920’s (unsuccessfully, as it turned out) to the second problem: that the consistency. Proof would be finitist, and finitist complete induction has an intuitive justification not present in the general use of complete induction.

May be more than you needed, Giovanni!

Bill





Sent from my iPad
 
> On Jun 28, 2021, at 5:48 PM, sambin at math.unipd.it wrote:
> 
> 
> 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.
> I am thinking for instance of the appearance of EaC in Hilbert's program in the 1920s. I suspect that the standard model theoretic explanation of EaC (by which T is consistent iff it has a model) came later. 
>  
> A related question is: is there a way to avoid assuming EaC while keeping classical logic (and hence validity of LEM)?
>  
> I thank in advance for any information and comments.
> Giovanni Sambin
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