In set theory you sometimes read statements like "hereditarily finite" or "hereditarily well founded", in the presence of the axiom of foundation Wikipedia says ordinals are sets that are "hereditarily transitive".

I haven't been able to find a good introductory definition of what being hereditory means. It seems to revolve around a property being inherited by the subsets of a set?

Can someone give some basic definitions and direction please, hopefully by explaining what "$A$ is the class of hereditarily $\varphi $ sets" means, where $\varphi $ is some property.

  • $\begingroup$ BTW the word hereditary is also used in topology, with a different meaning: A topological space $X$ is heredtarily-P iff every subspace of $X$ is P. (With the exception when P is " disconnected". A space $X$ is hereditarily disconnected iff $X$ is disconnected and every subspace of $X$ with more than $1$ member is disconnected.)... And a property P is called hereditary iff any space that is P is also hereditarily P ... E.g. if P is "Hausdorff". $\endgroup$ – DanielWainfleet Aug 7 '18 at 8:49
  • $\begingroup$ Incidentally, you have a number of questions with good answers. Have you considered accepting answers to any of your questions? If nothing else, this will remove them from the "unanswered" queue. $\endgroup$ – Noah Schweber Aug 8 '18 at 20:45
  • $\begingroup$ Thanks Noah, i didn't know i could do that so i will now accept the answers. $\endgroup$ – Mark Kortink Aug 9 '18 at 7:32

A set $x$ is hereditarily $\varphi$ if and only if every member of its transitive closure (including $x$ itself) has property $\varphi$.

For instance, being hereditarily finite means not just that $x$ is finite, but also every member of $x$ is finite, and every member of every member of $x$, and so on.

(Note that in the absence of foundation, this is a bit peculiar. For instance, if $x=\{x\}$, then $x$ is hereditarily finite, although it does not belong to $V_\omega$.)

  • 2
    $\begingroup$ +1. To the OP, note that there is one situation where hereditariness is simpler: it's a good exercise to show that a set is hereditarily transitive (= it and every element of its transitive closure is transitive) iff it is transitive and all its elements are transitive. That is, to verify hereditary transitivity we just have to go "down one level." (Incidentally, the hereditarily transitive sets are exactly the ordinals!) $\endgroup$ – Noah Schweber Aug 7 '18 at 4:27
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    $\begingroup$ Most of the time things aren't this simple; for example, if $x=\{\{\{\omega\}\}\}$ then $x$ is finite, all elements of $x$ are finite, and all elements of elements of $x$ are finite, but $\omega$ is infinite and in the transitive closure of $x$. $\endgroup$ – Noah Schweber Aug 7 '18 at 4:29
  • $\begingroup$ Thanks, excellent, just what I wanted to know :) $\endgroup$ – Mark Kortink Aug 7 '18 at 7:06

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