Questions: Consistency Strength and the Large-Cardinal Hierarchy

Question 1

ZFC + 'a measurable cardinal exists' (call it T₂) can prove that ZFC + 'an inaccessible cardinal exists' (T₁) is consistent, but T₁ cannot prove T₂ is consistent. What does this tell you about their consistency strengths?

• A. T₁ and T₂ have equal consistency strength because both extend ZFC
• B. T₂ has strictly higher consistency strength than T₁: T₂ proves Con(T₁) but T₁ does not prove Con(T₂)
• C. T₁ has higher consistency strength because inaccessible cardinals are foundational — measurable cardinals build on top of them
• D. The comparison is meaningless because both theories are unprovably consistent by Gödel's theorem

Answer: (B)

Consistency strength is defined precisely by this asymmetric provability relation: T₂ > T₁ in consistency strength if T₂ proves Con(T₁) but T₁ does not prove Con(T₂). The existence of a measurable cardinal implies there are inaccessibly many inaccessibles — T₂ is far stronger. Option A is wrong; extending ZFC by different axioms produces theories of different strength. Option C confuses the informal notion of 'foundational' with consistency strength — inaccessibles are weaker, not stronger, in consistency strength. Option D misreads Gödel: Gödel's theorem says a theory cannot prove its *own* consistency, but stronger theories can prove the consistency of weaker ones.

Question 2

If an inaccessible cardinal κ exists, what does this reveal about ZFC and the structure of the set-theoretic universe?

• A. ZFC is inconsistent, because the existence of such a large cardinal leads to a paradox
• B. V_κ (the universe of sets of rank below κ) is a model of ZFC, implying ZFC is consistent — which by Gödel's theorem cannot be proved within ZFC
• C. ZFC has infinitely many axioms and therefore cannot be complete, regardless of large cardinals
• D. The existence of κ proves that all large-cardinal axioms are consistent, since κ is the smallest large cardinal

Answer: (B)

An inaccessible cardinal κ is a regular strong limit cardinal, meaning V_κ (all sets of rank below κ) satisfies every axiom of ZFC. V_κ is therefore a model of ZFC, which means ZFC is consistent. But Gödel's second incompleteness theorem states that no consistent extension of ZFC can prove its own consistency. Therefore, if ZFC is consistent, it cannot prove 'V_κ exists for some inaccessible κ' — the existence of an inaccessible is a genuinely new assumption that transcends ZFC. Option A is incorrect — large cardinals do not produce inconsistency (so far as we know). Option C is about incompleteness, not large cardinals. Option D is wrong — inaccessibles being the first large cardinal does not imply all stronger axioms are also consistent.

Question 3

A cardinal with higher ordinal value (i.e., larger as an infinite cardinal) usually has higher consistency strength as a large-cardinal axiom.

• T. True
• F. False

Answer: False

Ordinal size and consistency strength are different orderings. A weakly compact cardinal, for example, is defined by a combinatorial property and sits very low on the large-cardinal hierarchy in consistency strength — yet the cardinals it describes are inaccessible (hence very large ordinally). Measurable cardinals are much stronger in consistency strength than weakly compact ones, even though both types of cardinals are huge ordinals. The consistency hierarchy is ordered by provability strength (what each axiom implies about consistency of weaker theories), not by the raw size of the cardinals. This is the core misconception identified in Common Misconceptions.

Question 4

Gödel's incompleteness theorem implies that if ZFC is consistent, then ZFC cannot prove 'there exists an inaccessible cardinal.'

• T. True
• F. False

Answer: True

If an inaccessible cardinal κ exists, V_κ is a model of ZFC, which means ZFC is consistent. So 'inaccessible exists' → 'ZFC is consistent.' Gödel's second incompleteness theorem says ZFC cannot prove its own consistency (assuming ZFC is consistent). Therefore ZFC cannot prove 'inaccessible exists,' because doing so would prove Con(ZFC) — which it cannot do. This argument applies to every large-cardinal axiom: each one implies the consistency of ZFC (and much more), so none can be proved within ZFC. Large-cardinal axioms are genuine new assumptions, not theorems of ZFC.

Question 5

Why does Gödel's incompleteness theorem ensure that no large-cardinal axiom can be proved consistent within ZFC alone, and what does this mean for how set theorists use large-cardinal hypotheses?

Model answer: Gödel's second incompleteness theorem states that any consistent theory T (extending a weak base theory) cannot prove its own consistency. Every large-cardinal axiom A implies Con(ZFC) — because the large cardinal's existence provides a model of ZFC (e.g., V_κ for an inaccessible κ). Therefore ZFC + A proves Con(ZFC), which means ZFC cannot prove ZFC + A is consistent (since that would allow ZFC to prove Con(ZFC), violating Gödel). This makes every rung of the large-cardinal hierarchy a genuine new assumption: no large-cardinal axiom is a theorem of ZFC, and no stronger axiom's consistency can be established within weaker systems. Set theorists use large-cardinal hypotheses as explicit axioms that calibrate how much mathematical strength a theorem requires.

The practical implication is that the large-cardinal hierarchy serves as a measuring rod for mathematical boldness. A theorem that requires measurable cardinals is provably stronger (in the consistency-strength sense) than one requiring only inaccessibles. This gives a rigorous meaning to the intuition that some mathematical claims 'go further' than others, without collapsing into the naive view that all unprovable statements are equally mysterious.