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October 20th, 2019, 06:24 PM   #1
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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?
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October 20th, 2019, 08:19 PM   #2
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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.
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October 20th, 2019, 08:41 PM   #3
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Quote:
Originally Posted by infinity1010 View Post
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?
You might find this helpful.

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?
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October 21st, 2019, 11:53 AM   #4
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Originally Posted by Maschke View Post
You might find this helpful.

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?
Bertrand Russell in his intro to mathematical philosophy uses posterity and hereditary sets. Set theory uses hereditary but i am confused on the definition. I cant fi d any helpful info on it. The definition seems circular but the differece is that the definition refers to its members not to itself. If it was circular it would say a hereditary set is a set that is hereditary. It says its its members are hereditary, but what does that mean? Could it be something like a particular father has sons who are also fathers, thus making him the grandfather? This could be analougous to a particular number has preceeding numbers that also have preceeding numbers.
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October 21st, 2019, 12:23 PM   #5
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Originally Posted by infinity1010 View Post
Bertrand Russell in his intro to mathematical philosophy uses posterity and hereditary sets. Set theory uses hereditary but i am confused on the definition. I cant fi d any helpful info on it. The definition seems circular but the differece is that the definition refers to its members not to itself. If it was circular it would say a hereditary set is a set that is hereditary. It says its its members are hereditary, but what does that mean? Could it be something like a particular father has sons who are also fathers, thus making him the grandfather? This could be analougous to a particular number has preceeding numbers that also have preceeding numbers.
I think you're overthinking it. The Wiki page I linked is pretty clear. I'm not familiar with Russell's book but frankly I don't think that's a good way to be learning contemporary set theory.

In standard set theory, all elements of sets are sets themselves. So all sets are hereditary. There can only be non-hereditary sets in set theories with urelements; that is, in set theories in which sets can contains things that aren't sets.
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October 21st, 2019, 05:42 PM   #6
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I think you're overthinking it. The Wiki page I linked is pretty clear. I'm not familiar with Russell's book but frankly I don't think that's a good way to be learning contemporary set theory.

In standard set theory, all elements of sets are sets themselves. So all sets are hereditary. There can only be non-hereditary sets in set theories with urelements; that is, in set theories in which sets can contains things that aren't sets.
Thank you Maschke. I'm not reading Russell to learn set theory. I was inspired to read it after reading Gottlobb Frege's work. It's trivial to most, but I found that defining number is an interesting concept since its all we use. Its purely for general interest, nothing more. However, he focused on hereditary sets when defining the series of natural numbers, so I need to clarify what he meant. Was my example in the original post accurate then?
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October 21st, 2019, 06:15 PM   #7
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Originally Posted by infinity1010 View Post
Thank you Maschke. I'm not reading Russell to learn set theory. I was inspired to read it after reading Gottlobb Frege's work. It's trivial to most, but I found that defining number is an interesting concept since its all we use. Its purely for general interest, nothing more. However, he focused on hereditary sets when defining the series of natural numbers, so I need to clarify what he meant. Was my example in the original post accurate then?
I don't know, I'm not familiar with Russell's book. It's not correct with respect to the modern definition of a hereditary set. A set is either hereditary or not; and in ZFC all sets are hereditary. The terminology of the ancestors or descendants of a set are not in common use AFAIK but you are free to make such definitions if you like.

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 well-ordered 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.
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October 22nd, 2019, 03:13 AM   #8
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Quote:
Originally Posted by infinity1010 View Post
Thank you Maschke. I'm not reading Russell to learn set theory. I was inspired to read it after reading Gottlobb Frege's work. It's trivial to most, but I found that defining number is an interesting concept since its all we use. Its purely for general interest, nothing more. However, he focused on hereditary sets when defining the series of natural numbers, so I need to clarify what he meant. Was my example in the original post accurate then?
Defining numbers is very interesting, but Russell's way of doing this is inherently flawed. Russel eventually abandoned the project because it got too complicated.
Nowadays mathematicians still define numbers but in a very different way that essentialy avoid everything Russel does.
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