Hereditary Sets and posterity

[infinity1010]

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, 7,...n...}. Am I even close?

 

[HallsofIvy]

"The definition of a hereditary set is a set whose members are a hereditary set."

That's a rather circular definition! How do you determine whether a member of a set is a hereditary set?

And your example doesn't make much sense. "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}}"

You were talking about a "hereditary set", NOT the "hereditary set of a number"! Can you please give the precise statement of the definition of "hereditary set of a number" from your textbook?

Added: Wikipedia does define a "hereditary set" as "a set all of whose elements are hereditary sets" which still doesn't make a lot of sense! But they don't say anything about the "hereditary set of a number" and the example they give is not anything like yours. Their (only) example starts by saying that the empty set \(\displaystyle \Phi\) is "clearly hereditary" because it has NO elements. And then \(\displaystyle \{\Phi, \{\Phi\}\}\), the set whose elements are the empty set and the set containing the empty set, is also hereditary.

Yes, that is one way to create the natural numbers- we associate "0" with \(\displaystyle \Phi\), "1" with \(\displaystyle \{\Phi, \{\Phi\}\}\), "2" with \(\displaystyle \{\Phi, \{\Phi\}, \{\{\Phi\}\}\}\), etc.

 

[infinity1010]

"The definition of a hereditary set is a set whose members are a hereditary set."

That's a rather circular definition! How do you determine whether a member of a set is a hereditary set?

And your example doesn't make much sense. "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}}"

You were talking about a "hereditary set", NOT the "hereditary set of a number"! Can you please give the precise statement of the definition of "hereditary set of a number" from your textbook?

Added: Wikipedia does define a "hereditary set" as "a set all of whose elements are hereditary sets" which still doesn't make a lot of sense! But they don't say anything about the "hereditary set of a number" and the example they give is not anything like yours. Their (only) example starts by saying that the empty set \(\displaystyle \Phi\) is "clearly hereditary" because it has NO elements. And then \(\displaystyle \{\Phi, \{\Phi\}\}\), the set whose elements are the empty set and the set containing the empty set, is also hereditary.

Yes, that is one way to create the natural numbers- we associate "0" with \(\displaystyle \Phi\), "1" with \(\displaystyle \{\Phi, \{\Phi\}\}\), "2" with \(\displaystyle \{\Phi, \{\Phi\}, \{\{\Phi\}\}\}\), etc.

Im reading from bertrand Russell's work Intro to Mathematical philosophy. Im not trying to learn set theory necessarily from him since terms have changed since then, but i would like to see how he generates the natural numbers. How can we be sure that the empty set is clearly hereditary if we dont know what hereditary even means in this context? Also, doesnt each number contain each of its predecessors? 1 contains 0 and the empty set, 2 contains the empty set, 0, 1, 2. And so on?