Hereditary system generated by
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Demote grade once content is put in place, I really want to pull the "smallest" note out and give it a name (generated system of sets?) as this occurs a lot, and can be applied any time the type of system in question is closed under arbitrary intersection, that is the intersection of an arbitrary family of a type of system is a system of that type in and of itself.
- Warning:This page is little more than notes at the moment, however everything stated here is verified and correct
Contents
Definition
The hereditary system generated by a collection of sets, [ilmath]S[/ilmath], which we denote: [ilmath]\mathcal{H}(S)[/ilmath] is the smallest^{[Note 1]} hereditary system containing [ilmath]S[/ilmath]^{[1]}.
- Claim 1: [ilmath]\mathcal{H}(S)=\{V\in\mathcal{P}(T)\ \vert\ T\in S\}[/ilmath]^{[Note 2]} where [ilmath]\mathcal{P}(A)[/ilmath] denotes the power set of [ilmath]A[/ilmath].
Proof of claims
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Pretty routine proof by contradiction. I've done it on some paper somewhere
Notes
- ↑ We do not mean smallest in the sense of cardinality arguments, we also do not mean smallest in the sense of [ilmath]\subset[/ilmath] relation (as given any two hereditary systems containing [ilmath]S[/ilmath] we cannot be sure that either one is a subset (proper or not) of the other! Instead we mean smallest in the following sense:
- [math]\mathcal{H}(S):=\bigcap_{\text{All hereditary systems of sets, }\mathcal{H}\text{, where }S\subseteq\mathcal{H} }\mathcal{H}[/math]
Grade: AThis page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:Prove that the intersection of an arbitrary family of hereditary systems of sets (among other systems of course!) is itself a hereditary system of sets - ↑ There are many ways to write this and this may not be the best. The "defining property" (Note to self: explore notion between FOL sentences and sets) is:
- [ilmath]\left[A\in\mathcal{H}(S)\right]\iff[\exists B\in S(A\in\mathcal{P}(B))][/ilmath]