I believe that the solution to this problem involves surreal numbers. Here’s an extract from an email that I sent to Amanda Askell. I’m planning on writing up a full post on this soonish, but I’m also looking for jobs at the moment, so there is a bit of a conflict there. I know this needs to be formalised more though.
“Thanks for feedback on using surreal numbers.
Eddy Chen and Daniel Rubio seem to be using an approach quite similar to me. In particular, they made two key insights:
If non-standard infinities are going to be used, then surreal numbers are a much more natural non-standard class number to use than hyperreals
Sequences can also have a surreal number attached as a length
However, that presentation is not quite a complete theory. One of the biggest issues is that they argued it is invalid to re-arrange sequences, when the spatial order should not make a difference. In particular, they wanted to say that it was invalid to rearrange 1,-1/2,1/3,-1/4… is rearranged to 1,-1/2,1/3,1/5,-1/4,1/7,1/9,1/11,1/13,-1/6 as the original sequence had the same number of positive and negative terms, but the later sequence has more positive terms up to any particular point.
An informal description of my approach to resolve this works as follows:
Instead of simply attaching a single length to a sequence, we need lengths attached to all sub-sequences. If we do this, then we can take a countably infinite sequence 1,1,1… with length X (with X is surreal) and another sequence 2,2,2… with length Y and splice them together into a sequence 1,2,1,2… with total utility X+2Y. We could splice them to be 1,1,2,1,1,2… or 1,2,2,1,2,2… instead, but this won’t change the total utility as long as we keep in mind that there are X ones and Y twos.
Similarly, for the 1,-1/2,1/3,-1/4 sequence. If there are X positive terms and Y negative terms, then this will remain the same after it is rearranged
We define homogenous sequences as sequence where that the odd numbered places have the same “length” as the even numbered places and three subsequences of every third element have the same length and so on for every fourth question ect. (It’s actually a bit more complex than this)
After we have defined homogenous sequences, we can say 1,2,1,2 (length X, homogenous) is different from 1,1,2,1,1,2… (length X, homogenous) as Eddy Chen and Daniel Rubio wanted to do, but using a more formal account.
As per Eddy Chen and Daniel Rubio’s model, this will behave as expected with regards to standard changes – adding elements, deleting elements, increasing single values, decreasing single values, increasing all values, decreasing all values, multiplying all values ect. At the same time, rearrangements preserve utility.”
I believe that the solution to this problem involves surreal numbers. Here’s an extract from an email that I sent to Amanda Askell. I’m planning on writing up a full post on this soonish, but I’m also looking for jobs at the moment, so there is a bit of a conflict there. I know this needs to be formalised more though.
“Thanks for feedback on using surreal numbers.
Eddy Chen and Daniel Rubio seem to be using an approach quite similar to me. In particular, they made two key insights:
If non-standard infinities are going to be used, then surreal numbers are a much more natural non-standard class number to use than hyperreals
Sequences can also have a surreal number attached as a length
However, that presentation is not quite a complete theory. One of the biggest issues is that they argued it is invalid to re-arrange sequences, when the spatial order should not make a difference. In particular, they wanted to say that it was invalid to rearrange 1,-1/2,1/3,-1/4… is rearranged to 1,-1/2,1/3,1/5,-1/4,1/7,1/9,1/11,1/13,-1/6 as the original sequence had the same number of positive and negative terms, but the later sequence has more positive terms up to any particular point.
An informal description of my approach to resolve this works as follows:
Instead of simply attaching a single length to a sequence, we need lengths attached to all sub-sequences. If we do this, then we can take a countably infinite sequence 1,1,1… with length X (with X is surreal) and another sequence 2,2,2… with length Y and splice them together into a sequence 1,2,1,2… with total utility X+2Y. We could splice them to be 1,1,2,1,1,2… or 1,2,2,1,2,2… instead, but this won’t change the total utility as long as we keep in mind that there are X ones and Y twos.
Similarly, for the 1,-1/2,1/3,-1/4 sequence. If there are X positive terms and Y negative terms, then this will remain the same after it is rearranged
We define homogenous sequences as sequence where that the odd numbered places have the same “length” as the even numbered places and three subsequences of every third element have the same length and so on for every fourth question ect. (It’s actually a bit more complex than this)
After we have defined homogenous sequences, we can say 1,2,1,2 (length X, homogenous) is different from 1,1,2,1,1,2… (length X, homogenous) as Eddy Chen and Daniel Rubio wanted to do, but using a more formal account.
As per Eddy Chen and Daniel Rubio’s model, this will behave as expected with regards to standard changes – adding elements, deleting elements, increasing single values, decreasing single values, increasing all values, decreasing all values, multiplying all values ect. At the same time, rearrangements preserve utility.”
Thanks! Someone (maybe it was you?) pointed me to Chen and Rubio’s stuff before, and it sounds interesting.
I don’t fully understand the informal write up you have above, but I’m looking forward to seeing the final thing!