[FOM] strong hypotheses and the theory of N
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?
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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