0.222…[base 3] = 0.999…[base 10] = 1

0.999…[base 10] not= 1 [base 10]

Pi not= 3.14… [base 10]

1/3 not= 0.333… [base 10]

2/3 not= 0.666… [base 10]

1 not= 0.999… [base 10]

0.111... [base 2] > 0.222... [base 3] < ... < 0.999... [base 10] < ... < 1

If the Identity map of a non-finite collection does not exist, then its exact cardinality does not exist and the Natural numbers’ cardinality is |N|-Successor, because the Successor is permanently out of our desirable “complete” domain.

Let @ be |N|-Successor

If A = @ and B = @-2^@, then A > B by 2^@, where both A and B are collections of infinitely many elements.

Also 3^@ > 2^@ > @ > @-1 etc.

So as we can see, in my universe I have both non-finite collections and unique arithmetic between non-finite collections, which its result is always a non-finite collection.

My results are richer than the Cantorean transfinite universe, for example:

By Cantor aleph0 = aleph0+1 , by me @+1 > @ .

By Cantor aleph0<2^aleph0 , by me @<2^@ .

By Cantor aleph0-2^aleph0 is undefined, by me @-2^@ < @ .

By Cantor 3^aleph0 = 2^aleph0 > aleph0 and aleph0-1 is problematic.

By me 3^@ > 2^@ > @ > @-1 etc.

|{{1,1,…}+1, 1,1,1}| > |{{1,1,…}+1}| by |{1,1,1}|.

|{{1,1,…}+1,{1,1,…}+1}| = |{{1},{1}}|•@ > |{{1,1,…}+1}| by |{1}|•@ and
|{{{1,1,…}+1, 1,1,…}+1}| = |{{1},1}|•@ > |{{1,1,…}+1}| by |{1}|•@ but they have different internal structures
( {{1},{1}} and {{1},1} ).

0.111... [base 2] > 0.222... [base 3] < ... < 0.999... [base 10] < ... < 1

0.111... [base 2] < 0.222... [base 3] < ... < 0.999... [base 10] < ... < 1