properties of ranks of sets
A set A is said to be grounded, if A⊆Vα in the cumulative hierarchy for some ordinal α. The smallest such α such that A⊆Vα is called the rank of A, and is denoted by ρ(A).
In this entry, we list derive some basic properties of groundedness and ranks of sets. Proofs of these properties require an understanding of some of the basic properties of ordinals.
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1.
∅ is grounded, whose rank is itself. This is obvious.
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2.
If A is grounded, so is every x∈A, and ρ(x)<ρ(A).
Proof.
A⊆Vρ(A), so x∈Vρ(A), which means x⊆Vβ for some β<ρ(A). This shows that x is grounded. Then ρ(x)≤β, and hence ρ(x)<ρ(A). ∎
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3.
If every x∈A is grounded, so is A, and .
Proof.
Let . Then is a set of ordinals, so that is an ordinal. Since each , we have . So , showing that is grounded. If , then for some , , which means , and therefore . This shows that . ∎
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4.
If is grounded, so is , and . This is a direct consequence of the previous result.
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5.
If are grounded, so is , and .
Proof.
Since are grounded, every element of is grounded by property 2, so that is also grounded by property 3. Then . ∎
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6.
If is grounded, so is , and .
Proof.
Every element of , as an element of the grounded set , is grounded, and therefore is grounded. So . Since and are both ordinals, . ∎
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7.
If is grounded, so is , and .
Proof.
Every subset of is grounded, since is by property 6. So is grounded. Furthermore, . Since for any , and , we have as a result. ∎
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8.
If is grounded, so is , and .
Proof.
Since is grounded, every is grounded. Let . Then is an ordinal. Since , or . In either case, , since is a transitive set for any ordinal . Since , for every . This means , showing that is grounded. If , then for some , which means , or as a result. Therefore . ∎
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9.
Every ordinal is grounded, whose rank is itself.
Proof.
If , then apply property 1. If is a successor ordinal, apply properties 4 and 5, so that . If is a limit ordinal, then apply property 8 and transfinite induction
, so that . ∎
References
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1
H. Enderton, Elements of Set Theory
, Academic Press, Orlando, FL (1977).
- 2 A. Levy, Basic Set Theory, Dover Publications Inc., (2002).
Title | properties of ranks of sets |
---|---|
Canonical name | PropertiesOfRanksOfSets |
Date of creation | 2013-03-22 18:50:31 |
Last modified on | 2013-03-22 18:50:31 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Derivation |
Classification | msc 03E99 |
Defines | grounded |
Defines | grounded set |