Infinite precision intermediate arithmetic: how much would break?

In infinite-bit-width two’s complement arithmetic, you abstractly never have a “sign bit” at all; instead you have the rule that the integer has either an infinite sequence of zeroes to the left (if nonnegative) or an infinite sequence of ones to the left (if negative). This is the same thing as sign extension, just expressed as an intrinsic property of the number rather than a conversion step.