Post History
#5: Post edited
by
clemens
·
2026-02-27T02:19:32Z (7 months ago)
- (Update: this answer is not complete; I made a misstep, as you can see in footnote 2. Sorry!)
- I shall use a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.<sup>2</sup>
However, as we still have subtraction of finite integers, the situation is not unlike that which we already have with wheels and $\frac00$. More precisely, adjoining alephs to Sobociński's system gives us a proper class of new operations $\aleph_\alpha$ each of which are equal to their antipodes and dominate any operators less than or equal to themselves (i.e. $\aleph_\alpha + \aleph_alpha).The partial order $a > b \Leftrightarrow a + b = a$- ---
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
- <sup>2</sup> The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
- (Update: this answer is not complete; I made a misstep, as you can see in footnote 2. Sorry!)
- I shall use a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.<sup>2</sup>
- However, as we still have subtraction of finite integers, the situation is not unlike that which we already have with wheels and $\frac00$. More precisely, adjoining alephs to Sobociński's system gives us a proper class of new operations $\aleph_\alpha$ each of which are equal to their antipodes and dominate any operators less than or equal to themselves (i.e. $\aleph_\alpha + x = \aleph_\alpha$ whenever $x ≤ \aleph_\alpha$).
- But this answer is incomplete, as it is not fully clear how equational rules that allow addition would fit elegantly in this system (e.g. what is $\frac{\aleph_0}{\aleph_0} + \frac00$?).
- ---
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
- <sup>2</sup> The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
#4: Post edited
by
clemens
·
2026-02-27T02:16:33Z (7 months ago)
- (Update: this answer is not complete; I made a misstep, as you can see in footnote 2. Sorry!)
- I shall use a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.<sup>2</sup>
We can either use the *non-abelian group* of ordinals (and give up certain nice category-theoretic properties) or use some other system, e.g. the abelian group of surreals.- ---
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
- <sup>2</sup> The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
- (Update: this answer is not complete; I made a misstep, as you can see in footnote 2. Sorry!)
- I shall use a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.<sup>2</sup>
- However, as we still have subtraction of finite integers, the situation is not unlike that which we already have with wheels and $\frac00$. More precisely, adjoining alephs to Sobociński's system gives us a proper class of new operations $\aleph_\alpha$ each of which are equal to their antipodes and dominate any operators less than or equal to themselves (i.e. $\aleph_\alpha + \aleph_alpha).
- The partial order $a > b \Leftrightarrow a + b = a$
- ---
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
- <sup>2</sup> The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
#3: Post edited
by
clemens
·
2026-02-27T02:09:27Z (7 months ago)
corrected an error
(Update: this post is erroneous; see below. Sorry!)I see a way in which it is possible, but using a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.<sup>2</sup>
We can either use the *non-abelian group* of ordinals (and give up certain nice category-theoretic properties) or use some other system, e.g. the abelian group of surreals. I do not see any way to make the cardinals themselves into a wheel.- ---
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
- <sup>2</sup> The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
- (Update: this answer is not complete; I made a misstep, as you can see in footnote 2. Sorry!)
- I shall use a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.<sup>2</sup>
- We can either use the *non-abelian group* of ordinals (and give up certain nice category-theoretic properties) or use some other system, e.g. the abelian group of surreals.
- ---
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
- <sup>2</sup> The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
#2: Post edited
by
clemens
·
2026-02-27T02:03:37Z (7 months ago)
corrected an error
- I see a way in which it is possible, but using a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
It is not difficult to extend this to any abelian group. If we operate upon cardinals as asked in the OP, we no longer have unambiguous subtraction, but we can still use the [*Grothendieck group*](https://en.wikipedia.org/wiki/Grothendieck_group) of cardinalities, whose objects are *differences* of cardinals, e.g. $\aleph_0 - 2$.
- (Update: this post is erroneous; see below. Sorry!)
- I see a way in which it is possible, but using a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
- Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
- The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
- It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.<sup>2</sup>
- We can either use the *non-abelian group* of ordinals (and give up certain nice category-theoretic properties) or use some other system, e.g. the abelian group of surreals. I do not see any way to make the cardinals themselves into a wheel.
- ---
- <sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
- <sup>2</sup> The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
#1: Initial revision
by
clemens
·
2026-02-27T01:49:53Z (7 months ago)
I see a way in which it is possible, but using a slightly different formalization of wheel theory I read from Sobociński's excellent *Graphical Linear Algebra*<sup>1</sup>, where the numbers on the wheel are considered as *linear relations* between input and output.
Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
<sup>1</sup>(See chapter 26 of his online book, ["Keep calm and divide by zero"](https://graphicallinearalgebra.net/2015/12/14/26-keep-calm-and-divide-by-zero/).)
It is not difficult to extend this to any abelian group. If we operate upon cardinals as asked in the OP, we no longer have unambiguous subtraction, but we can still use the [*Grothendieck group*](https://en.wikipedia.org/wiki/Grothendieck_group) of cardinalities, whose objects are *differences* of cardinals, e.g. $\aleph_0 - 2$.