Post History
#4: Post edited
by
celtschk
·
2022-10-29T05:24:09Z (about 2 years ago)
Better name for the tag
#3: Post edited
by
celtschk
·
2022-10-16T09:21:06Z (about 2 years ago)
Added a textual description of the formula
- As is well known, the addition of natural numbers can be extended to the ordinal numbers in different ways. The first way is the ordinal sum, and the second is the natural or Hessenberg sum.
- Now I've been thinking about other possible sums of ordinals. For that purpose I've used the following general definition:
- > Given two ordinals $\alpha$ and $\beta$, an ordinal $\gamma$ is a sum of $\alpha$ and $\beta$ if there exists a partition of $\gamma$ into two subsets $A$ and $B$ such that $A$ is order-isomorphic to $\alpha$ and $B$ is order-isomorphic to $\beta$
- Obviously the ordinal sum $\alpha+\beta$ is the special case where you can choose $A$ and $B$ so that all elements of $A$ are smaller than all elements of $B$. Also, if I'm not mistaken, the natural sum is simply the largest sum, that is,
- $$\alpha\oplus\beta = \max_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- Now given this, there is an obvious other candidate for a specific sum:
- $$\alpha\boxplus\beta = \min_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- This minimal sum obviously exists because the ordinals are well-ordered, and moreover from the definition is it immediately obvious that the minimal sum is commutative.
- Now the obvious next thing one wants is an explicit expression for the minimal sum. Now I think if we write $\alpha = \lambda + m$ and $\beta = \mu + n$ where $\lambda$ and $\mu$ are limit ordinals or zero, and $m$ and $n$ are finite ordinals, then we have
- $$\alpha \boxplus \beta =
- \begin{cases}
- \beta & \text{if } \alpha < \mu\\\\
- \alpha & \text{if } \beta < \lambda\\\\
- \alpha +n & \text{otherwise}
- \end{cases}$$
- If this should indeed be the case, then obviously the minimal addition would also be associative, and moreover, if we restrict it to the initial ordinals, it would reduce to cardinal addition, so it would actually be a generalisation of cardinal addition to arbitrary ordinals.
- Therefore my question is:
- **Is the formula above correct, and if so, how can I prove it?**
- Also, if the formula is not correct, what would be a counterexample, and can you tell me the correct formula?
- As is well known, the addition of natural numbers can be extended to the ordinal numbers in different ways. The first way is the ordinal sum, and the second is the natural or Hessenberg sum.
- Now I've been thinking about other possible sums of ordinals. For that purpose I've used the following general definition:
- > Given two ordinals $\alpha$ and $\beta$, an ordinal $\gamma$ is a sum of $\alpha$ and $\beta$ if there exists a partition of $\gamma$ into two subsets $A$ and $B$ such that $A$ is order-isomorphic to $\alpha$ and $B$ is order-isomorphic to $\beta$
- Obviously the ordinal sum $\alpha+\beta$ is the special case where you can choose $A$ and $B$ so that all elements of $A$ are smaller than all elements of $B$. Also, if I'm not mistaken, the natural sum is simply the largest sum, that is,
- $$\alpha\oplus\beta = \max_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- Now given this, there is an obvious other candidate for a specific sum:
- $$\alpha\boxplus\beta = \min_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- This minimal sum obviously exists because the ordinals are well-ordered, and moreover from the definition is it immediately obvious that the minimal sum is commutative.
- Now the obvious next thing one wants is an explicit expression for the minimal sum. Now I think if we write $\alpha = \lambda + m$ and $\beta = \mu + n$ where $\lambda$ and $\mu$ are limit ordinals or zero, and $m$ and $n$ are finite ordinals, then we have
- $$\alpha \boxplus \beta =
- \begin{cases}
- \beta & \text{if } \alpha < \mu\\\\
- \alpha & \text{if } \beta < \lambda\\\\
- \alpha +n & \text{otherwise}
- \end{cases}$$
- Or said differently, if the ordinals are infinitely far apart, the minimal sum is just the larger of them, otherwise the finite parts add up.
- If this should indeed be the case, then obviously the minimal addition would also be associative, and moreover, if we restrict it to the initial ordinals, it would reduce to cardinal addition, so it would actually be a generalisation of cardinal addition to arbitrary ordinals.
- Therefore my question is:
- **Is the formula above correct, and if so, how can I prove it?**
- Also, if the formula is not correct, what would be a counterexample, and can you tell me the correct formula?
#2: Post edited
by
celtschk
·
2022-10-16T09:17:18Z (about 2 years ago)
- As is well known, the addition of natural numbers can be extended to the ordinal numbers in different ways. The first way is the ordinal sum, and the second is the natural or Hessenberg sum.
- Now I've been thinking about other possible sums of ordinals. For that purpose I've used the following general definition:
- > Given two ordinals $\alpha$ and $\beta$, an ordinal $\gamma$ is a sum of $\alpha$ and $\beta$ if there exists a partition of $\gamma$ into two subsets $A$ and $B$ such that $A$ is order-isomorphic to $\alpha$ and $B$ is order-isomorphic to $\beta$
- Obviously the ordinal sum $\alpha+\beta$ is the special case where you can choose $A$ and $B$ so that all elements of $A$ are smaller than all elements of $B$. Also, if I'm not mistaken, the natural sum is simply the largest sum, that is,
- $$\alpha\oplus\beta = \max_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- Now given this, there is an obvious other candidate for a specific sum:
- $$\alpha\boxplus\beta = \min_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- This minimal sum obviously exists because the ordinals are well-ordered, and moreover from the definition is it immediately obvious that the minimal sum is commutative.
- Now the obvious next thing one wants is an explicit expression for the minimal sum. Now I think if we write $\alpha = \lambda + m$ and $\beta = \mu + n$ where $\lambda$ and $\mu$ are limit ordinals or zero, and $m$ and $n$ are finite ordinals, then we have
- $$\alpha \boxplus \beta =
- \begin{cases}
- \beta & \text{if } \alpha < \mu\\\\
\alpha & \text{if } \beta < \nu\\\\- \alpha +n & \text{otherwise}
- \end{cases}$$
If this should indeed be the case, then obviously the minimal addition would also be associative, and moreover, if we restrict it to the initial ordinals, it would reduce to cardinal addition, so it would actually be a generalisation of cardinal addition to ordinals.- Therefore my question is:
- **Is the formula above correct, and if so, how can I prove it?**
- Also, if the formula is not correct, what would be a counterexample, and can you tell me the correct formula?
- As is well known, the addition of natural numbers can be extended to the ordinal numbers in different ways. The first way is the ordinal sum, and the second is the natural or Hessenberg sum.
- Now I've been thinking about other possible sums of ordinals. For that purpose I've used the following general definition:
- > Given two ordinals $\alpha$ and $\beta$, an ordinal $\gamma$ is a sum of $\alpha$ and $\beta$ if there exists a partition of $\gamma$ into two subsets $A$ and $B$ such that $A$ is order-isomorphic to $\alpha$ and $B$ is order-isomorphic to $\beta$
- Obviously the ordinal sum $\alpha+\beta$ is the special case where you can choose $A$ and $B$ so that all elements of $A$ are smaller than all elements of $B$. Also, if I'm not mistaken, the natural sum is simply the largest sum, that is,
- $$\alpha\oplus\beta = \max_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- Now given this, there is an obvious other candidate for a specific sum:
- $$\alpha\boxplus\beta = \min_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
- This minimal sum obviously exists because the ordinals are well-ordered, and moreover from the definition is it immediately obvious that the minimal sum is commutative.
- Now the obvious next thing one wants is an explicit expression for the minimal sum. Now I think if we write $\alpha = \lambda + m$ and $\beta = \mu + n$ where $\lambda$ and $\mu$ are limit ordinals or zero, and $m$ and $n$ are finite ordinals, then we have
- $$\alpha \boxplus \beta =
- \begin{cases}
- \beta & \text{if } \alpha < \mu\\\\
- \alpha & \text{if } \beta < \lambda\\\\
- \alpha +n & \text{otherwise}
- \end{cases}$$
- If this should indeed be the case, then obviously the minimal addition would also be associative, and moreover, if we restrict it to the initial ordinals, it would reduce to cardinal addition, so it would actually be a generalisation of cardinal addition to arbitrary ordinals.
- Therefore my question is:
- **Is the formula above correct, and if so, how can I prove it?**
- Also, if the formula is not correct, what would be a counterexample, and can you tell me the correct formula?
#1: Initial revision
by
celtschk
·
2022-10-16T09:13:41Z (about 2 years ago)
Is this formula for the minimal sum correct?
As is well known, the addition of natural numbers can be extended to the ordinal numbers in different ways. The first way is the ordinal sum, and the second is the natural or Hessenberg sum.
Now I've been thinking about other possible sums of ordinals. For that purpose I've used the following general definition:
> Given two ordinals $\alpha$ and $\beta$, an ordinal $\gamma$ is a sum of $\alpha$ and $\beta$ if there exists a partition of $\gamma$ into two subsets $A$ and $B$ such that $A$ is order-isomorphic to $\alpha$ and $B$ is order-isomorphic to $\beta$
Obviously the ordinal sum $\alpha+\beta$ is the special case where you can choose $A$ and $B$ so that all elements of $A$ are smaller than all elements of $B$. Also, if I'm not mistaken, the natural sum is simply the largest sum, that is,
$$\alpha\oplus\beta = \max_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
Now given this, there is an obvious other candidate for a specific sum:
$$\alpha\boxplus\beta = \min_{\text{$\gamma$ is a sum of $\alpha$ and $\beta$}} \gamma$$
This minimal sum obviously exists because the ordinals are well-ordered, and moreover from the definition is it immediately obvious that the minimal sum is commutative.
Now the obvious next thing one wants is an explicit expression for the minimal sum. Now I think if we write $\alpha = \lambda + m$ and $\beta = \mu + n$ where $\lambda$ and $\mu$ are limit ordinals or zero, and $m$ and $n$ are finite ordinals, then we have
$$\alpha \boxplus \beta =
\begin{cases}
\beta & \text{if } \alpha < \mu\\\\
\alpha & \text{if } \beta < \nu\\\\
\alpha +n & \text{otherwise}
\end{cases}$$
If this should indeed be the case, then obviously the minimal addition would also be associative, and moreover, if we restrict it to the initial ordinals, it would reduce to cardinal addition, so it would actually be a generalisation of cardinal addition to ordinals.
Therefore my question is:
**Is the formula above correct, and if so, how can I prove it?**
Also, if the formula is not correct, what would be a counterexample, and can you tell me the correct formula?