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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[1] 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[3]. The hierarchy definable in any particular formal system can be embedded within itself at many places. The next version of the ordinal calculator[2] 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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References

[1]
W. Buchholz. A new system of proof-theoretic ordinal functions. Ann. Pure Appl. Logic, 32:195-207, 1986.
[2]
Paul Budnik. A Computational Approach to the Ordinal Numbers. Mountain Math Software, Los Gatos, CA, 2010.
[3]
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, 1980.



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