properties of ranks of sets
A set is said to be grounded, if in the cumulative hierarchy for some ordinal . The smallest such such that is called the rank of , and is denoted by .
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 is grounded, so is every , and .
Proof.
, so , which means for some . This shows that is grounded. Then , and hence . ∎
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3.
If every is grounded, so is , 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
- 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 |
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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 |