7 An Objective Interpretation of ZF
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I suspect it is consistent to assume the power set axiom in ZF,
because all subsets of the integers (and larger cardinals) that are
provably definable in ZF form a definite, albeit
countable, collection. These are definite collections only relative
to a specific formal system. Expand ZF with an axiom like "there
exists an inaccessible cardinal" and these collections expand.
Uncountable sets in ZF suggest how the objective parts of ZF can be
expanded. Create an explicitly countable definition of the
countable ordinals defined by the ordinals that are uncountable
within ZF. Expand ZF to ZF+ with axioms that assert the existence
of these structures. This approach to expansion can be repeated
with ZF+. The procedure can be iterated and it must have a fixed
point that is unreachable with these iterations.
Ordinal collapsing functions do something like this.
They use uncountable ordinals as notations for recursive ordinals
to expand the recursive ordinals. Ordinal collapsing can also use
countable ordinals larger than the recursive ordinals. This is
possible at multiple places in the ordinal hierarchy. I suspect
that uncountable ordinals provide a relatively weak way to expand
the recursive and larger countable ordinal hierarchies. The
countable ordinal hierarchy is a bit like the Mandelbrot
set. The hierarchy definable in any
particular formal system can be embedded within itself at many
places. The next version of the ordinal calculator will focus
on general ways to index this embedding to create large recursive
and countable ordinals.
The objective interpretation of ZFC see it as a recursive process
for defining finite sets, properties of finite sets and properties
of properties. These exist either as physical objects that embody
the structure of finite sets or as expressions in a formal language
that can be connected to finite objects and/or expressions that
define properties. Names of all the objects that provably satisfy
the definition of any set in ZF are recursively enumerable because
all proofs in any formal system are. These names and their
relationships form an interpretation of ZF.
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- W. Buchholz. A new system of proof-theoretic ordinal
functions. Ann. Pure Appl. Logic, 32:195-207, 1986.
- Paul Budnik. A Computational Approach
to the Ordinal Numbers. Mountain Math Software, Los Gatos,
- Benoît Mandelbrot. Fractal aspects of the iteration of z
→ λ z (1−z) for complex λ and z.
Annals of the New York Academy of Sciences, 357:49-259,
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