[FOM] An objective approach to expanding mathematics
paul at mtnmath.com
Wed Apr 1 12:01:28 EDT 2009
The primary justification for going beyond what I consider to be
objective mathematics raised in the recent discussion is the
implications of parts of ZF for arithmetical and other questions of
obvious practical value.
Following is a brief outline of an approach that I believe could
ultimately lead to systems stronger than ZF in terms of their ability to
solve arithmetical problems and that extend definability in an objective
way that can be further extended indefinitely. I would appreciate
feedback on why this may not be possible or how it relates to existing
work. I am familiar with the survey paper "What's so Special About
Kruskal's Theorem And The Ordinal Γ0?".
The intention is to define sets as properties or recursive processes
that are determined by a recursively enumerable sequence of events. The
starting point is the recursive ordinals. These can be defined by a
property of recursive processes that accept an arbitrarily long integer
sequence of inputs. The property asks if an instance of this TM halts
for every possible input sequence. I call a TM with this property well
founded (WF). An obvious next step is to ask if a TM that accepts a
sequence of integer inputs is WF for all integer sequences restricted to
Godel numbers of TMs that themselves are first order WF. This can be
iterated up to any integer and can be iterated up to any previously
defined ordinal provided the parameters have labels that recursively
encode what parameter types they are WF for. This encoding for anything
at or above the ordinal of the recursive ordinals will be incomplete but
it must be indefinitely expandable.
Instead of building things up we need a property that encompasses
everything that can built in this way not unlike the way the WF property
defines the recursive ordinals. It is this property that I think may not
be provably definable in ZF.
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