Hierarchies of Consistency Strength (From Gödel to Feferman)
[2025-10-22T16:06:00+01:00] User Prompt
"hierarchies of consistency strength."
Hierarchies of consistency strength rank formal theories by what they can prove about the consistency of other theories. If a theory A proves Con(B) (“B is consistent”), then A is stronger than B. Gödel’s Second Incompleteness Theorem implies that no sufficiently strong, consistent theory proves its own consistency; hence, foundations unfold as an unending ladder of stronger systems (Feferman), often measured via proof-theoretic ordinals.
From Gödel’s Limit to a Ladder
| Idea | Formal Content | Consequence |
|---|
| Gödel II | No consistent, arithmetic-capable theory proves Con(T) about itself. | Self-certification is impossible. |
| Relative consistency | A ⊢ Con(B) | A stronger than B. |
| Hierarchy | Iterate: B < A < A' etc. | Infinite ascent of strength. |
Landmarks in the Hierarchy
| Level | Representative Theory | Can Prove Consistency Of | But Not |
|---|
| 1 | Fragments of arithmetic (IΔ0, EFA) | Weaker fragments | Themselves |
| 2 | PA (Peano Arithmetic) | Fragments; elementary arithmetic | Con(PA) |
| 3 | ZFC (Zermelo–Fraenkel + Choice) | PA; many arithmetic theories | Con(ZFC) |
| 4 | ZFC + Inaccessible cardinal | ZFC | Its own consistency |
| 5 | ZFC + Measurable cardinal | ZFC + Inaccessible | Its own consistency |
| … | Stronger large-cardinal axioms | Weaker large-cardinal theories | Self-consistency |
Measuring Strength: Proof-Theoretic Ordinals
Modern proof theory (Gentzen, Feferman, et al.) assigns ordinals as “coordinates of strength.” Examples:
| Theory | Approx. Ordinal | Notes |
|---|
| PA | ε₀ | Gentzen’s 1936 proof used transfinite induction up to ε₀ to show Con(PA)—but in a stronger meta-theory. |
| Predicative Analysis | Γ₀ | Feferman–Schütte analysis of predicative provability. |
| Beyond predicativity | much larger ordinals | Linked to large cardinals; towers of reflection principles. |
Reflection Principles and Iteration
- Reflection: Add axioms stating, roughly, “If the theory proves φ, then φ is true (in some schema).”
- Iterated reflection: Strength increases as you iterate reflection or consistency assertions:
T, T + Con(T), T + Con(T) + Con(T+Con(T)), …
- Outcome: A structured, transfinite climb in strength while Gödel’s barrier persists at each step.
Why It Matters
| Area | Relevance |
|---|
| Foundations | Replaces the search for “the” foundation with a calibrated hierarchy of stronger frameworks. |
| Set Theory | Large cardinals organize high-level strength; consequences cascade down to arithmetic. |
| Philosophy | Epistemic humility: every lens needs a wider lens; no final self-grounding. |
| Computation | Parallels with hierarchies of computability and oracle strength (Turing jumps). |
Summary
Consistency strength forms an ascending ladder: stronger theories verify the safety of weaker ones, yet none verifies itself. Proof-theoretic ordinals and large-cardinal axioms chart this ascent with remarkable precision.
References