# What does Up to Isomorphism mean?

1. Jul 7, 2010

### Buri

What does "Up to Isomorphism" mean?

I was reading the final chapter in Spivak's Calculus and it says:

There is a complete ordered field and, up to isomorphism, only one complete ordered field.

I know what an isomorphism is and what it means when things are isomorphic. But I don't know what he means when it says "UP to..". I've also read things like this when talking of homotopy...

Any help is appreciated!!

2. Jul 7, 2010

### tmccullough

Re: What does "Up to Isomorphism" mean?

I'm not trying to be an ***, but it means what it sounds like. You can only have different pictures of what is ultimately the same thing...Does that even help?

In general, when the phrase "up to (blank)" is thrown out, it means that any two such things are the same, up to (blank). I don't know what else to say.

3. Jul 7, 2010

### Hurkyl

Staff Emeritus
Re: What does "Up to Isomorphism" mean?

He means there is only one isomorphism class. All complete ordered fields are isomorphic. (uniquely isomorphic, even! But that's a special fact about this particular case)

4. Jul 8, 2010

### Buri

Re: What does "Up to Isomorphism" mean?

No sorry, I'm not hearing the same sound as you lol

5. Jul 8, 2010

### Buri

Re: What does "Up to Isomorphism" mean?

Thanks a lot. It all makes sense now. :)

6. Jul 8, 2010

### Landau

Re: What does "Up to Isomorphism" mean?

In the same way, for example, there is a set of cardinality 2, and it is unique up to bijection. This means there is a set of cardinality 2 (like $\{1,2\}$), and for any (other) set of cardinality 2 there is a bijection between them (if $\{a,b\}$ is such a set, then a->1, b->2 is of course a bijection).

7. Jul 9, 2010

### Buri

Re: What does "Up to Isomorphism" mean?

Thanks for this example. It has helped me understand it better :)

8. Jul 9, 2010

### Buri

Re: What does "Up to Isomorphism" mean?

Just a question to make sure I got it all right. In my question and your example, the equivalence relation that are being used are the isomorphism and bijection relation. And the "up to isomophism/bijection" basically means that the property holds in the "universe" (in my example the set of complete ordered fields and yours the collection of sets of cardinality 2) where equality is defined by the equivalence relation. So if the bijection relation would have "divided" your universe into more than just one equivalence class then uniqueness wouldn't be true, but since it only "created" one equivalence class, uniqueness follows.

Am I right?

9. Jul 10, 2010

### HallsofIvy

Staff Emeritus
Re: What does "Up to Isomorphism" mean?

Yes, that's right. Since an isomorphism is, by definition, invertible, saying that two things are "isomorphic" is a an equivalence relation and so divides the set on which it is defined into equivalence classes. Two things are the same "up to isomorphism" if they are in the same equivalence class.

10. Jul 11, 2010

### Buri

Re: What does "Up to Isomorphism" mean?

Thanks for the help.