Cumulative Hierarchy

Set Theory · Axiom Academy

LESSON The Cumulative Hierarchy Building the universe of sets stage by stage through transfinite iteration 1. The Construction: Base and Successor Cases We build the hierarchy recursively by transfinite induction on ordinals: At each stage, we include all subsets of what we've built so far. At limit stages, we simply collect everything from earlier stages. Let's see what the first stages of the hierarchy look like: Notice how each stage contains exponentially more sets than the previous stage. The finite stages contain only finite sets, but at V_ω we get our first infinite collection. The universe V is defined as the union of all stages: Every set appears at some stage of the hierarchy. The rank of a set x is the smallest ordinal α such that x ∈ V_ α+1 . Equivalently, rank(x) is the supremum of rank(y) + 1 : y ∈ x . The rank measures the "complexity" or "depth" of a set - how many levels of nesting it contains. 4. The Iterative Conception of Set The cumulative hierarchy embodies the iterative conception of sets: sets are formed in stages, and at each stage we can form any collection of sets from previous stages. This conception avoids paradoxes (like Russell's paradox) because we can never form "the set of all sets" - at any stage, we can only form sets of objects from earlier stages. V is a proper class (not a set)

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