By Traditional Mathematics infinite set
N = {1,2,3,...} is in bejection with any infinite subset of it, so according to
https://math.stackexchange.com/ques...ss-than-the-natural-numbers?noredirect=1&lq=1 there are no infinite subsets of
N with cardinality less than |
N|.
By this reasoning it is concluded that |
N| is the smallest cardinality > than any given
n.
Please pay attention that the accurate value of |
N| is undefined and so is the case about the value of the cardinality of any infinite subset of it.
All what traditional mathematicians care is about the bijection between the mapped sets, and yet they claim that even if the accurate value of the cardinality of
N or any cardinality of any infinite subset of it is not accurately defined, one can claim that such sets are complete (all of their members are already included, exactly as the members of finite non-empty sets are already included in their sets).
In other words, traditional mathematicians ignore the essential fact that any non-empty finite set has a cardinality with an accurate value (and therefore can be considered as a complete set) where the infinite set
N or any infinite subset of it, do not have cardinalities with accurate value (and therefore can't be considered as complete sets).
So, the bijection between infinite set
N = {1,2,3,...} and any infinite subset of it, can't be used in order to conclude that they are also complete.
In that case infinite set
N = {1,2,3,...} and any infinite subset of it, can't be defined in terms of actual infinity.
So, we have left with potential infinity as the fundamental notion about infinite set
N = {1,2,3,...} and any infinite subset of it, where bijection is not the only possible matching between them, for example:
2 ↔ 1
4 ↔ 2
6 ↔ 3
8
...
In this case there are potentially infinite two sets, where the left set always has one more object.
Moreover, the same argument holds for any kind of sets, for example:
# ↔ &
% ↔ ?
@ ↔ !
*
...
