- #1

- 1,030

- 4

How do you show that the surreal numbers form a proper class?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- #1

- 1,030

- 4

How do you show that the surreal numbers form a proper class?

- #2

Science Advisor

Homework Helper

- 9,426

- 4

Since there are 'more' ordinals than cardinals, and there is a proper class of cardinals (if there is a set of cardinals, C, what is the cardinality of the power set of C?), this becomes easy - if I've understood the one thing I read about surreal numbers and some link to ordinals, and that;s a big if.

- #3

- 1,030

- 4

- #4

- 355

- 3

Never mind, I was reading the Wiki article, and I apparently misunderstood some things.

Last edited:

- #5

Science Advisor

Homework Helper

- 9,426

- 4

- #6

- 1,030

- 4

But I haven't read anything on the "norm" of surreal numbers; maybe it's defined differently.

- #7

Science Advisor

Homework Helper

- 9,426

- 4

If a metric needs to be real-valued,

It does.

and if the surreals form a proper class

they do - apparently.

Checking the definitions is always a good idea.

- #8

- 1,030

- 4

I guess that answers the question about complete ordered fields.

- #9

- 355

- 3

(Rehash of my earlier post... only this time, correct)

Ignoring that the surreals form a proper class rather than a set, you'd need to prove that you can't define a metric on them that turns them into a complete ordered field.

One way to see that the surreal numbers are not a complete ordered field is to use the axioms of a complete ordered field to show that they satisfy the archimedean property, which says that there are no infinitesimals other than 0 and no infinities. Since the surreal numbers have non-zero infinitesimals and infinities, they cannot be a complete ordered field.

Ignoring that the surreals form a proper class rather than a set, you'd need to prove that you can't define a metric on them that turns them into a complete ordered field.

One way to see that the surreal numbers are not a complete ordered field is to use the axioms of a complete ordered field to show that they satisfy the archimedean property, which says that there are no infinitesimals other than 0 and no infinities. Since the surreal numbers have non-zero infinitesimals and infinities, they cannot be a complete ordered field.

Last edited:

- #10

Staff Emeritus

Science Advisor

Gold Member

- 14,971

- 26

(hint... can you find one bigger than everything in that set?)

- #11

- 1,030

- 4

- #12

Staff Emeritus

Science Advisor

Gold Member

- 14,971

- 26

Or by trichotomy...

- #13

- 1,030

- 4

How does it violate trichotomy?

- #14

Staff Emeritus

Science Advisor

Gold Member

- 14,971

- 26

You've asserted:

X is the set of all surreal numbers

{X | } is a surreal number

and it's a basic fact of surreal numbers that

{X | } is larger than every member of X

so...

Now that I think of it, I'm pretty sure it's also a theorem that every surreal number is a well-founded set.

X is the set of all surreal numbers

{X | } is a surreal number

and it's a basic fact of surreal numbers that

{X | } is larger than every member of X

so...

Now that I think of it, I'm pretty sure it's also a theorem that every surreal number is a well-founded set.

Last edited:

Share:

- Replies
- 4

- Views
- 572

- Replies
- 126

- Views
- 4K

- Replies
- 15

- Views
- 835

- Replies
- 5

- Views
- 542

- Replies
- 9

- Views
- 1K

- Replies
- 1

- Views
- 643

- Replies
- 1

- Views
- 1K

- Replies
- 3

- Views
- 1K

MHB
Do Gödel numbers can be used to derermine the usefulness of an infinite set as a complete whole?

- Replies
- 1

- Views
- 816

- Replies
- 1

- Views
- 881