Where is the contradiction in here with one of my claims?"In mathematics, an uncountable set is an infinite set which is too big to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the natural numbers. The related term nondenumerable set is used by some authors as a synonym for "uncountable set" while other authors define a set to be nondenumerable if it is not an infinite countable set."

-http://wapedia.mobi/en/uncountable

"Some sets are infinite; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by <math>\{ 1, 2, 3, 4, 5, \dots \}</math>, has infinitely many elements, and we can't use any normal number to give its size. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, of cardinality, which is the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.

To understand what this means, we must first examine what it doesn't mean.For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall.This is because we arrange things such that for every integer, there is a distinct odd integer: … ?2 ? -3, ?1 ? ?1, 0 ? 1, 1 ? 3, 2 ? 5, …; or, more generally, n ? 2n + 1.What we have done here is arranged the integers and the odd integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this branch of mathematics) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

A set is countable if: (1) it is finite,or (2) it has the same cardinality (size) as the set of natural numbers.Equivalently, a set is countable if it has the same cardinality as some subset of the set of natural numbers. Otherwise, it is uncountable."

-http://wapedia.mobi/en/countable

Again, a set is countable if it has a one to one correspondence with the natural numbers. For example, multiples of 7 are countable because you can come up with a formula which corresponds that set with the set of natural numbers. Both sets are infinite, but they are, by definition, countable.

I am sure your Ni world could exonerate your thinking from this contradiction.

Finite here is a requirement for countability. Infinite is the antonym of finite. Infinite does mean non-countable.