[FOM] strong hypotheses and the theory of N

joeshipman@aol.com joeshipman at aol.com
Mon Mar 15 20:00:54 EDT 2010

What more can you say about what "natural" means, and can you explain 
the "minimal requirements" and show that they are necessary?

I have long maintained that if we ever do discover an alien 
civilization with its own mathematics, they are likely to disagree with 
us about all sorts of inifinitary statements, but there will never be 
an arithmetical statement that we regard as proven and they regard as 
refuted (assuming the proofs have been sufficiently scrutinized that we 
are really sure there are no actual mistakes in the proofs).

A related question: is there a natural way to represent the 
"arithmetical content" of ZF by arithmetical axioms; in other words, a 
natural decidable set of arithmetical statements which have the same 
arithmetical consequences as ZF?

-- JS

-----Original Message-----
From: Harvey Friedman <friedman at math.ohio-state.edu>

Here are two relevant observed facts.

1. Any two natural formal systems that interpret EFA = exponential
function arithmetic, are comparable under interpretability.
2. Any two natural formal systems of set theory satisfying minimal
requirements, are comparable under interpretations that preserve the
ordered ring of natural number parts.

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