Post History
#3: Post edited
by
clemens
·
2026-05-25T12:37:27Z (4 months ago)
edited to remove dangling footnote
- > Do the classes of ordinals without maximal element cover all gaps of the surreal numbers? Or are there gaps that are not represented by such a class?
- There are gaps not represented by such classes. The gap between 0 and the surreals above 0 is a clear example. If we were to represent it in von Neumann sign notation it would be a transfinitely continued sequence $+,-,-,-,-,-,-,-,-,…$. Hence it will be represented in the OP's notation by $\{\emptyset\}$. As we can see, this gap can only be represented by a class with a maximal element (in this case the maximal element is $\emptyset$).
- However, if we get rid of the condition *without maximal element*, all gaps of the surreal numbers are indeed covered!
This is clear from the definition of surreal numbers.[^1] Thus let Ω be an inaccessible ordinal, or in other words an ordinal of all "small" ordinals. Then the whole discussion in the OP works just as well for surreal numbers with birthday <Ω, replacing the term "proper class of ordinals" by the term "subset of Ω". We see that, by the definition of surreal numbers, all gaps in the surreal numbers with birthday <Ω are covered by surreal numbers with birthday Ω.- And the surreal numbers with birthday Ω have a one-to-one correspondence with subsets of Ω, as von Neumann notation illustrates. $\blacksquare$
- > Do the classes of ordinals without maximal element cover all gaps of the surreal numbers? Or are there gaps that are not represented by such a class?
- There are gaps not represented by such classes. The gap between 0 and the surreals above 0 is a clear example. If we were to represent it in von Neumann sign notation it would be a transfinitely continued sequence $+,-,-,-,-,-,-,-,-,…$. Hence it will be represented in the OP's notation by $\{\emptyset\}$. As we can see, this gap can only be represented by a class with a maximal element (in this case the maximal element is $\emptyset$).
- However, if we get rid of the condition *without maximal element*, all gaps of the surreal numbers are indeed covered!
- Thus. let Ω be an inaccessible ordinal, or in other words an ordinal of all "small" ordinals. Then we may repeat the whole discussion in the OP works, this time surreal numbers with birthday <Ω, replacing the term "proper class of ordinals" by the term "subset of Ω". We see that, by the definition of surreal numbers, all gaps in the surreal numbers with birthday <Ω are covered by surreal numbers with birthday Ω.
- And the surreal numbers with birthday Ω have a one-to-one correspondence with subsets of Ω, as von Neumann notation illustrates. $\blacksquare$
#2: Post edited
by
clemens
·
2026-05-07T18:05:23Z (4 months ago)
- > Do the classes of ordinals without maximal element cover all gaps of the surreal numbers? Or are there gaps that are not represented by such a class?
There are gaps not represented by such classes. The gap between 0 and the surreals above 0 is a clear example. If we were to represent it in von Neumann sign notation it would be a transfinitely continued sequence $+,-,-,-,-,-,-,-,-,…$. Hence it will be represented in the OP's notation by $\{\emptyset\}$. As we can see, this gap can only be represented by a class with a maximal element.- However, if we get rid of the condition *without maximal element*, all gaps of the surreal numbers are indeed covered!
- This is clear from the definition of surreal numbers.[^1] Thus let Ω be an inaccessible ordinal, or in other words an ordinal of all "small" ordinals. Then the whole discussion in the OP works just as well for surreal numbers with birthday <Ω, replacing the term "proper class of ordinals" by the term "subset of Ω". We see that, by the definition of surreal numbers, all gaps in the surreal numbers with birthday <Ω are covered by surreal numbers with birthday Ω.
- And the surreal numbers with birthday Ω have a one-to-one correspondence with subsets of Ω, as von Neumann notation illustrates. $\blacksquare$
- > Do the classes of ordinals without maximal element cover all gaps of the surreal numbers? Or are there gaps that are not represented by such a class?
- There are gaps not represented by such classes. The gap between 0 and the surreals above 0 is a clear example. If we were to represent it in von Neumann sign notation it would be a transfinitely continued sequence $+,-,-,-,-,-,-,-,-,…$. Hence it will be represented in the OP's notation by $\{\emptyset\}$. As we can see, this gap can only be represented by a class with a maximal element (in this case the maximal element is $\emptyset$).
- However, if we get rid of the condition *without maximal element*, all gaps of the surreal numbers are indeed covered!
- This is clear from the definition of surreal numbers.[^1] Thus let Ω be an inaccessible ordinal, or in other words an ordinal of all "small" ordinals. Then the whole discussion in the OP works just as well for surreal numbers with birthday <Ω, replacing the term "proper class of ordinals" by the term "subset of Ω". We see that, by the definition of surreal numbers, all gaps in the surreal numbers with birthday <Ω are covered by surreal numbers with birthday Ω.
- And the surreal numbers with birthday Ω have a one-to-one correspondence with subsets of Ω, as von Neumann notation illustrates. $\blacksquare$
#1: Initial revision
by
clemens
·
2026-05-07T17:03:25Z (4 months ago)
> Do the classes of ordinals without maximal element cover all gaps of the surreal numbers? Or are there gaps that are not represented by such a class?
There are gaps not represented by such classes. The gap between 0 and the surreals above 0 is a clear example. If we were to represent it in von Neumann sign notation it would be a transfinitely continued sequence $+,-,-,-,-,-,-,-,-,…$. Hence it will be represented in the OP's notation by $\{\emptyset\}$. As we can see, this gap can only be represented by a class with a maximal element.
However, if we get rid of the condition *without maximal element*, all gaps of the surreal numbers are indeed covered!
This is clear from the definition of surreal numbers.[^1] Thus let Ω be an inaccessible ordinal, or in other words an ordinal of all "small" ordinals. Then the whole discussion in the OP works just as well for surreal numbers with birthday <Ω, replacing the term "proper class of ordinals" by the term "subset of Ω". We see that, by the definition of surreal numbers, all gaps in the surreal numbers with birthday <Ω are covered by surreal numbers with birthday Ω.
And the surreal numbers with birthday Ω have a one-to-one correspondence with subsets of Ω, as von Neumann notation illustrates. $\blacksquare$