# Do the ordinals form a set?

1. Mar 26, 2004

### meteor

Do the ordinals form a set?
I'm confused, I thought that they form a set, but the Burali-Forti paradox says that they are not a set, but instead a proper class.
I always thought that a set was a finite or infinite collection of things. If the ordinals are an infinite collection of things, I do not see why they can't form a set

2. Mar 26, 2004

### HallsofIvy

Take a close look at the Burali-Forti paradox, especially at their definition of "set". In "naive set theory" a set is any collection of things but that leads to problems (in particular, the Russell Paradox)- that's why it's called "naive". As soon as you start talking about "proper classes" you are using the rule that a "set" cannot have sets as members.

3. Mar 26, 2004

### DrMatrix

Sets certtaily can have sets as members. The ordinals are build up from the null set. Zero is defined as the null set and one is {0}. We have the null set as an element of one.

A set can be on the left or the right side of 'is an element of". A class can only be on the right of 'is an element of'.