Math Is Fun Forum

Reuel

Member

Registered: 2010-11-28

Posts: 178

Impredicativity

Hello. I have a definition question.

According to wikipedia and wolfram alpha, impredicativity is the self-reference of a set. What would be a simple example of this?

If we have a point on (x,f(x)), say, (a,b) and separate the two points as

and

so that either self-referencing result is not a function but is, at the specified value, the value itself... would this be a good example of the concept? Each result has x or f on both sides of the equals sign so that if x=a and f=b then we revert to our original values that constitute a single point.

Thanks.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Impredicativity

hi Reuel,

This sounds like it is leading to the Russell paradox.

Here's an example:

I have a large library of books.  Some of them have red covers.  I have decided to make a catalogue of all my red books, called the Red Books Catalogue.  I have decided to make this as a book, and give it a red cover.  Thus, it is listed in the catalogue, as a book with a red cover. 

Bob

ps.  Consider the other catalogue I have made of all the books which don't contain themselves as an entry.  I've made the catalogue but I cannot decide whether to list the catalogue as an entry. 

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Impredicativity

Hey bob.

Thanks for the reply.

Do recursive statements such as the self-referencing I gave in my example have any special name, or are they just called recursive statements?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Impredicativity

hi Reuel,

I wouldn't have called them by that name but I cannot see what is wrong with it.  I don't know any better name.  I would probably just have said something like: "So x is defined in a function that contains x itself." which doesn't exactly 'trip off the tongue' easily.

On-line I found these pages, which might be worth a look:

http://www.mathwords.com/r/recursive_formula.htm

http://www.wolframalpha.com/input/?i=re … lk=4&num=3

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob