# Thread: ORD subset of L; |ORD| uncountable; |L| countable --?

1. ## ORD subset of L; |ORD| uncountable; |L| countable --?

I have one or more of the following steps incorrect. Which one(s)?
(a) Gödel's constructible universe L is the minimal model for ZFC, hence countable.
(b) The set of ordinals ORD is uncountable.
(c) ORD is a subset of L, hence |ORD| is less than or equal to |L|.
The contradiction is evident. ????
Thanks.

2. Originally Posted by nomadreid
I have one or more of the following steps incorrect. Which one(s)?
(a) Gödel's constructible universe L is the minimal model for ZFC, hence countable.
(b) The set of ordinals ORD is uncountable.
(c) ORD is a subset of L, hence |ORD| is less than or equal to |L|.
The contradiction is evident. ????
Thanks.

There is no "the set of ordinals" in ZFC anymore than there is a "set of all sets"...take a peek at
Constructible universe - Wikipedia, the free encyclopedia

Tonio

3. Thanks, Tonio. Now that I have more sleep than when I posted this, I realize that all three points were incorrect. As you say, ORD is not a set, and L is not the minimal model, and ORD is a subclass of L, and talking of cardinalities of classes is nonsense. Oops.

4. Maybe Skolem paradox is relevant.

5. Thanks, emakarov. The Skolem paradox is one of those things that makes physicists think that mathematicians are crazy. My large oversight was that the minimal, hence countable model was not L but rather L_k for some ordinal k, and then there is a countable (in V) set of ordinals W= L_k intersect ORD such that W is a subset of L_k. I was mixing viewpoints: from inside and outside of L_k. This is of course the main reason Skolem's "paradox" seems paradoxical.