Published October 29, 2025
| Version v1
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Absolute Constructive Limit: An Improved Upper Bound for the Proof-Theoretic Ordinal of CZF
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Description
This paper introduces the Absolute Constructive Limit (ACL) – a universal measure of expressive power for formal systems, defined as the supremum of the Fast-Growing Hierarchy below the proof-theoretic ordinal. We prove ACL is finite, maximal, and base-invariant.
Our main result establishes a new upper bound for Constructive Zermelo-Fraenkel Set Theory (CZF):
PTO(CZF) ≤ ψ₀(ε_{Ω₁ + 1}),
improving upon Rathjen's 1994 bound of ψ₀(ε_{Ω+1}). Consequently:
ACL(CZF) ≤ f_{ψ₀(ε_{Ω₁ + 1})}(3).
This work precisely delineates the boundary of constructive googology and provides the current strongest estimate of CZF's mathematical reach.
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Dates
- Accepted
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2025-10-29
References
- Rathjen, M. (1994). Proof-theoretic analysis of KPM. Archive for Mathematical Logic
- Buchholz, W. (1986). A new system of proof-theoretic ordinal functions. Annals of Pure and Applied Logic
- Aczel, P. (1978). The type theoretic interpretation of constructive set theory. Studies in Logic