[FOM] An objective approach to expanding mathematics

[Paul Budnik]

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.

Paul Budnik
www.mtnmath.com