Alphabeta MathHow statement and proof provenance work
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The least inaccessible is not Mahlo
Example
In ZFC, if an inaccessible cardinal exists, the least inaccessible is not Mahlo and therefore is not weakly compact.
Facts & Assumptions
Given: ZFC conditional on an inaccessible. The club of infinite strong limits below the least one avoids every uncountable regular, explicitly witnessing non-Mahloness.
Weak compactness implies stationary reflection and Mahloness: Weak compactness implies Mahloness.
Size and rank bounds below an inaccessible: An inaccessible has a club of infinite strong-limit cardinals below it.
The Axiom of Choice: Every family of nonempty sets has a choice function.
Verification
Given an inaccessible, minimize the ordinals at or below it satisfying that property; Separation and ordinal well-ordering give a least one kappa. Let C be the club of infinite strong-limit cardinals below kappa from F2. No uncountable regular cardinal belongs to C, because such a member would be uncountable regular strong limit, an inaccessible below kappa. Thus the set of uncountable regular cardinals used in the page's definition of Mahloness misses the actual club C; omega is not in that set. AC is retained through F2's cardinal-size estimates; the least-ordinal selection itself uses only Separation and ordinal well-ordering.
Missing C means that set is nonstationary, so kappa is not Mahlo. If kappa were weakly compact, F1 would make it Mahlo, contradicting step 1.1. Therefore it is not weakly compact. AC is also retained through F1's cardinal estimates and pressing-down argument. The argument is conditional on the initial existence hypothesis and does not prove that hypothesis consistent.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Monk Theorem 17.27 p.363 (standard reference, not scraped)