As gjm said, this article assumes that there are only digits at integer-numbered positions, and whatever gets pushed to infinity position is effectively lost (because there is no such position). Having digits at surreal-integer-numbered positions seems also interesting, but it’s not what I was trying to do.
I am bit unsure on reading whether uncountable integers are supposed to start with periodic integers or only with non-periodic integers.
Only with non-periodic, because… what you said. Otherwise, there is a finite periodic part (i.e. the sequence that gets repeated is finite), optionally followed by a finite part at the end (and optionally a finite decimal part), that is only countably many options.
I get that …9999 + 1 = …000 goes to an interesting direction and …9999 + 1 = 100...000 goes to an interesting direction. And we can assume/axiom into those directions. But it can also be taken as a claim. One could explore what “2+2=5” implies but another natural reaction is also to be disinterested because one asssume things in opposition to that. So is there multiple ways to make formal or concretise the informal intuitions and does that say something interesting how addition works?
As gjm said, this article assumes that there are only digits at integer-numbered positions, and whatever gets pushed to infinity position is effectively lost (because there is no such position). Having digits at surreal-integer-numbered positions seems also interesting, but it’s not what I was trying to do.
Only with non-periodic, because… what you said. Otherwise, there is a finite periodic part (i.e. the sequence that gets repeated is finite), optionally followed by a finite part at the end (and optionally a finite decimal part), that is only countably many options.
I get that …9999 + 1 = …000 goes to an interesting direction and …9999 + 1 = 100...000 goes to an interesting direction. And we can assume/axiom into those directions. But it can also be taken as a claim. One could explore what “2+2=5” implies but another natural reaction is also to be disinterested because one asssume things in opposition to that. So is there multiple ways to make formal or concretise the informal intuitions and does that say something interesting how addition works?
Some thought experiments have later some surprising use e.g. in physics. Other thought experiments don’t. It is probably difficult to predict.