Questions: Measurable Cardinals and Ultrafilters

Question 1

The partitioning argument shows that no nonprincipal ultrafilter on ℕ can be ω₁-complete. Why does this same argument imply that no successor cardinal (like ℵ₁ or ℵ₂) can be measurable?

• A. Successor cardinals are too small to carry any nonprincipal ultrafilter at all
• B. Any successor cardinal can be partitioned into its predecessor-many singletons, and a predecessor-complete nonprincipal ultrafilter cannot decide them consistently
• C. Successor cardinals are not regular, so the κ-completeness condition fails trivially
• D. The ultrapower construction requires inaccessible cardinals to be well-founded, not just successor cardinals

Answer: (B)

The partition argument runs as follows: κ is partitioned into κ-many singletons {α} for α < κ. A κ-complete nonprincipal ultrafilter must contain none of them (nonprincipal) yet must decide every subset (maximality). But the singletons cover κ entirely, so their complements cannot all be in the ultrafilter consistently. This applies to any cardinal reachable from below — including all successor cardinals — forcing measurable cardinals to be inaccessible. Option A is wrong because ultrafilters on ω exist; the issue is completeness. Option C is wrong: successor cardinals are regular.

Question 2

In the ultrapower construction Ult(V, 𝒰), what is the significance of the critical point κ of the elementary embedding j: V → M?

• A. It is the largest cardinal that is moved by j — everything above κ maps to itself
• B. It is the smallest cardinal moved by j — j(α) = α for all α < κ, but j(κ) > κ
• C. It is the cardinal at which V and M first disagree about which sets exist
• D. It marks the boundary of κ-completeness: the ultrafilter is exactly κ-complete but not κ⁺-complete

Answer: (B)

The critical point is the *smallest* ordinal moved by j. All ordinals below κ are fixed — j(α) = α — but j(κ) > κ, witnessing that κ is genuinely 'unreachable from below.' This is the precise sense in which measurable cardinals cannot be assembled from smaller sets: any ordinal-building process internal to the universe fixes everything below κ. Option A reverses the definition. Option C describes a related but different phenomenon (agreement between V and M about sets).

Question 3

Every measurable cardinal is inaccessible.

• T. True
• F. False

Answer: True

This follows from two separate arguments. First, measurability implies regularity: if κ were singular (reachable as a union of fewer-than-κ sets of size less than κ), the κ-completeness condition on the ultrafilter would be violated. Second, measurability implies strong limit (cannot be reached by power sets from below). Both properties together define inaccessibility. Measurable cardinals are not just inaccessible — they are vastly stronger — but inaccessibility is a necessary consequence.

Question 4

Most inaccessible cardinal is measurable.

• T. True
• F. False

Answer: False

This is a tempting but false reversal. Inaccessibility is a *necessary* condition for measurability, not a sufficient one. An inaccessible cardinal merely cannot be reached by successor operations or power sets from below. Measurability additionally requires the existence of a κ-complete nonprincipal ultrafilter on κ — a much stronger large cardinal axiom. The existence of inaccessible cardinals is already unprovable in ZFC; measurable cardinals sit far higher in the large cardinal hierarchy, with much greater consistency strength.

Question 5

Why must a measurable cardinal κ be inaccessible, and why does κ-completeness play the central role in forcing this?

Model answer: κ-completeness means the ultrafilter is closed under intersections of fewer than κ of its members. If κ were a successor cardinal λ⁺, we could partition κ into λ-many pieces of size λ; the ultrafilter's κ-completeness would require it to contain an intersection over λ-many complements of singletons — but that contradicts the nonprincipal requirement (no singleton can be in 𝒰) together with maximality. This forces κ to be a limit cardinal. The regularity argument then shows κ cannot be singular. Together, these properties give inaccessibility as a minimum threshold, below which the κ-complete ultrafilter simply cannot exist.

The key is that κ-completeness is a closure condition on intersections, and this closure condition clashes with the partition structure of any 'reachable' cardinal. Each time you try to construct κ from smaller pieces, those pieces generate a partition that breaks the ultrafilter. Only a cardinal truly unreachable from below can sustain the required completeness. This is why measurable cardinals belong to the large cardinal hierarchy above inaccessibles, Mahlo cardinals, and many other intermediate large cardinals.