So, does 1+ω make sense? It does, for the ordinals and hyperreals only.
It make sense for cardinals (the size of “a set of some infinite cardinality” unioned with “a set of cardinality 1″ is “a set with the same infinite cardinality as the first set”) and in real analysis (if lim f(x) = infinity, then lim f(x)+1 = infinity) too.
What about −1+ω? That only makes sense for the hyperreals.
And for cardinals (the size of the set difference between “a set of some infinite cardinality” and “a subset of one element” is the same infinite cardinality) and in real analysis (if lim f(x) = infinity, then lim −1+f(x) = infinity) too.
The cardinal set difference one is not well defined. If I remove the evens from the integers, I have infinitely many left over. If I remove the integers from the integers, I have nothing.
The limits are also not well defined with addition and subtraction, as you can add a function that goes to infinity with one that goes to negative infinity and get all sorts of stuff. Hyperreals are what you get when you take the limits and try to make them well defined under that stuff.
It make sense for cardinals (the size of “a set of some infinite cardinality” unioned with “a set of cardinality 1″ is “a set with the same infinite cardinality as the first set”) and in real analysis (if lim f(x) = infinity, then lim f(x)+1 = infinity) too.
And for cardinals (the size of the set difference between “a set of some infinite cardinality” and “a subset of one element” is the same infinite cardinality) and in real analysis (if lim f(x) = infinity, then lim −1+f(x) = infinity) too.
The cardinal set difference one is not well defined. If I remove the evens from the integers, I have infinitely many left over. If I remove the integers from the integers, I have nothing.
The limits are also not well defined with addition and subtraction, as you can add a function that goes to infinity with one that goes to negative infinity and get all sorts of stuff. Hyperreals are what you get when you take the limits and try to make them well defined under that stuff.
Have rephrased as “So, does 1+ω make sense as something different from ω?”.
1+ω = ω, for the usual ordering convention for ordinal addition.
Edit: I can’t figure out how to delete my comment, but ricraz already said this.
https://www.lesserwrong.com/posts/GhCbpw6uTzsmtsWoG/the-different-types-not-sizes-of-infinity/xDfSmdiQATFF4sLPt