What does Up to Isomorphism mean?

[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!

 

[tmccullough]

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.

 

[Hurkyl]

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)

 

[Buri]

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?

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

 

[Buri]

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)

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

 

[Landau]

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).

 

[Buri]

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

 

[Buri]

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).

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?

 

[HallsofIvy]

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.

 

[Buri]

Thanks for the help.