You want to conceive of this problem as “a sequence whose order-type is ω”, but from the surreal perspective this lacks resolution. Is the number of elements (surreal) ω, ω+1 or ω+1000? All of these are possible given that in the ordinals 1+ω=ω so we can add arbitrarily many numbers to the start of a sequence without changing its order type.
It seems to me that measuring the lengths of sequences with surreals rather than ordinals is introducing fake resolution that shouldn’t be there. If you start with an infinite constant sequence 1,1,1,1,1,1,..., and tell me the sequence has size ω, and then you add another 1 to the beginning to get 1,1,1,1,1,1,1,..., and you tell me the new sequence has size ω+1, I’ll be like “uh, but those are the same sequence, though. How can they have different sizes?”
Because we should be working with labelled sequences rather than just sequences (that is sequences with a length attached). That solves the most obvious issues, though there are some subtleties there
Why? Plain sequences are a perfectly natural object of study. I’ll echo gjm’s criticism that you seem to be trying to “resolve” paradoxes by changing the definitions of the words people use so that they refer to unnatural concepts that have been gerrymandered to fit your solution, while refusing to talk about the natural concepts that people actually care about.
I don’t think think your proposal is a good one for indexed sequences either. It is pretty weird that shifting the indices of your sequence over by 1 could change the size of the sequence.
It seems to me that measuring the lengths of sequences with surreals rather than ordinals is introducing fake resolution that shouldn’t be there. If you start with an infinite constant sequence 1,1,1,1,1,1,..., and tell me the sequence has size ω, and then you add another 1 to the beginning to get 1,1,1,1,1,1,1,..., and you tell me the new sequence has size ω+1, I’ll be like “uh, but those are the same sequence, though. How can they have different sizes?”
Because we should be working with labelled sequences rather than just sequences (that is sequences with a length attached). That solves the most obvious issues, though there are some subtleties there
Why? Plain sequences are a perfectly natural object of study. I’ll echo gjm’s criticism that you seem to be trying to “resolve” paradoxes by changing the definitions of the words people use so that they refer to unnatural concepts that have been gerrymandered to fit your solution, while refusing to talk about the natural concepts that people actually care about.
I don’t think think your proposal is a good one for indexed sequences either. It is pretty weird that shifting the indices of your sequence over by 1 could change the size of the sequence.