[FOM] Strong Hypotheses and the Theory of N

[joeshipman@aol.com]

I got 3 interesting and almost incompatible responses to my query about 
natural arithmetical axioms for the arithmetical consequences of ZF.

Richard Heck points out that Craig’s trick could be applied to the 
arithmetical consequences of ZF (for those who don’t know what Craig’s 
trick is, it is a way to turn an enumerable set of statements A1, A2, 
A3, … into a decidable set of statements with the same consequences: 
A1, A1&A2, A1&A2&A3, etc.)  I already knew this trick which is why my 
question should be interpreted to be asking for something “more 
natural” than that particular example.

Aatu Koskensilta says “No” and cites a theorem that the set of 
arithmetical consequences of ZF is not axiomatizable by a finite number 
of schemata in 1st-order (or, indeed, any definably higher-order) 
arithmetic.  But what does “finite number of schemata” mean in this 
case? I presume  a Turing machine that recognizes the decidable set 
Heck indicated, which has a finite number of states, is not 
reinterpretable as recognizing a finite number of schemata. Must a 
“scheme” involve a bounded number of alternations of quantifiers? Then 
the result would make sense but you can certainly have natural sets of 
axioms which follow a clear pattern but don’t have a bounded number of 
quantifiers.

Ali Enayat says “Yes”, citing his own paper, using the doubly indexed 
set of axioms

S --> Con(S* + T)

where S ranges over sentences of arithmetic, S* is the set-theoretical 
sentence saying S is arithmetically true, and T ranges over finite 
fragments of ZF.

This is better than using Craig’s trick, and presumably avoids 
Koskensilta’s result, but it still requires exhaustive enumerations.

Would the problem be easier if I asked for a (presumably consistent) 
“natural” set of arithmetical axioms which implied all the arithmetical 
consequences of ZF but was not necessarily limited to them?

-- JS