>>11530988The axiom of infinity is bound by itself, or rather the size of the set of N.
There are not different sizes of infinity.
I believe cantor was troubled by the concept of bijecting "the set of reals" with "the set of naturals", as the set of reals is not well ordered. Although the set of naturals doesn't have a largest element, it at least has a smallest element, while cantor probably couldn't begin to imagine what the smallest element of the reals was, making it difficult to wrap your head around the concept of bijecting N to R when you can't even figure out where to map 1 from N to inside of R.
But there are two notions that can be extracted from the axiom of infinity's application to N
1. the "infinite'th" decimal place exists inside N
2. it's size is arbitrarily large and undefinable by a real number
Bijecting the set of N into a new empty set K populated by means that, K, when ALL it's members are accounted for, contains a number in an undefinable index location that is essentially equivalent to = "0.000....1"
which would therefore probably have to be the first element of R given an ordered R.
Given a disordered R, the easiest way of looking at it is just bijecting N countably through arbitrary known/discoverable elements of R.
N: [1,2,3,4,5,...]
R: [0.333..., pi, e, graham's, ... ]
such that N1=0.333, N2=pi, etc.
and there's no honest reason that shouldn't work.
