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To Set a Stone with Six Birds: The Free and Purchased Parts of ZFC
Which parts of set theory come for free and which are bought.
In plain words
The axioms of set theory are usually a flat list adopted because it works. This paper shows the list factors through one construction: packaging under a membership lens, audited by rank.
The finite part comes for free. The hereditarily finite sets are exactly the fixed points of the packaging map, and they model ZF minus Infinity, axiom by axiom. Four infinitary axioms enter only as priced hypotheses.
Godel's incompleteness reads as saturation of the construction. Cohen's independence reads as strict extension. The finite forcing lemma is shown to measure exact density defects of the Cohen poset.
What it shows
- A staged construction that recovers ZF minus Infinity for free.
- Infinity, Power set, Replacement and Choice as four priced commitments.
- Cohen genericity as non definability from the ground lens.
What it does not claim
Classical forcing facts are imported, not reproved.
Cite
Tsiokos, I. (2026). To Set a Stone with Six Birds: The Free and Purchased Parts of ZFC. Zenodo. https://doi.org/10.5281/zenodo.20712823