
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 20th, 2019, 06:24 PM  #1 
Newbie Joined: Oct 2019 From: Las Vegas Posts: 3 Thanks: 1  Hereditary sets and posterity
Hi, My question is about hereditary sets and posterity in set theory. The definition of a hereditary set is a set whose members are a hereditary set. I am trying to wrap my mind around this definition. Perhaps I am thinking too hard on it. My understanding is that it is just what the name implies. Hereditary in English means you inherit something, or your direct descendants, whereas posterity means your direct ascendants. So let's take the number 5 for example. The hereditary set of 5 is {{0} {0,1},{0,1,2},{0,1,2,3}, {0,1,2,3,4}} and its posterity is {{6}, {6,7},{6,7,8}...n...}. Am I even close? 
October 20th, 2019, 08:19 PM  #2 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 64 Math Focus: Area of Circle 
I don't think your example is the correct one. I even doubt the definition of Hereditary set makes sense, I didn't get it and don't get it now.

October 20th, 2019, 08:41 PM  #3  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
https://en.wikipedia.org/wiki/Hereditary_set In standard set theory, all sets are hereditary. I don't think there's a concept of posterity in set theory. Where did you find it?  
October 21st, 2019, 11:53 AM  #4  
Newbie Joined: Oct 2019 From: Las Vegas Posts: 3 Thanks: 1  Quote:
 
October 21st, 2019, 12:23 PM  #5  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
In standard set theory, all elements of sets are sets themselves. So all sets are hereditary. There can only be nonhereditary sets in set theories with urelements; that is, in set theories in which sets can contains things that aren't sets.  
October 21st, 2019, 05:42 PM  #6  
Newbie Joined: Oct 2019 From: Las Vegas Posts: 3 Thanks: 1  Quote:
 
October 21st, 2019, 06:15 PM  #7  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
The terminology often comes up when talking about hereditarily finite sets, hereditarily definable sets, and so forth. If you want to define numbers, you might be interested in the von Neuman ordinals. https://en.wikipedia.org/wiki/Ordina...on_of_ordinals In von Neuman's scheme 0 is the empty set, 1 is the set containing 0, 2 is the set containing 0 and 1, and so forth. But this process is stronger than generating hereditary sets; it generates transitive sets wellordered by the membership relation; that is, ordinal numbers. If you want to learn set theory you need to use more modern sources than Frege and Russell else you'll just confuse yourself. Terminology and ideas have changed a lot since then. On the other hand if you're interested in set theory as it was in the 1880's, you might ask on https://hsm.stackexchange.com/. You are asking questions about contemporary usage but referring to ancient texts; and that is confusing. Last edited by Maschke; October 21st, 2019 at 06:49 PM.  
October 22nd, 2019, 03:13 AM  #8  
Senior Member Joined: Oct 2009 Posts: 905 Thanks: 354  Quote:
Nowadays mathematicians still define numbers but in a very different way that essentialy avoid everything Russel does.  

Tags 
hereditary, posterity, sets 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
HELP in sets  kanchinha123  Elementary Math  4  August 17th, 2016 01:06 AM 
sets  kanchinha123  Elementary Math  4  August 14th, 2016 01:22 PM 
Open sets and closed sets  Luiz  Topology  3  July 15th, 2015 06:31 AM 
Open Sets and Closed Sets  fienefie  Real Analysis  6  February 24th, 2015 03:14 PM 
minor sets sets problem  Jamsisos  Advanced Statistics  1  June 21st, 2012 02:11 PM 