Actually that number can be written in fraction form.

Yes, a set that has finite length is finite.A series that never ends is infinite. 1,2,3,4,5,6,7....(and on forever) Yet, a set that does end, no matter at how high of a number ought to be called finite.

No. You just quoted the definition. Countable means one to one correspondence with the natural numbers (which is an infinite set).Thus, as for the use of the word uncountable, the only reason something would be uncountable is if it is too great to be counted.

Wow, you've totally misunderstood what you were reading. The "counting numbers" is the set of numbers [1, 2, 3, 4, ...(to infinity)]. That is an infinite set. It is a COUNTABLE infinite set. Any set that has a one to one correspondence is also called an infinite set. An example would be [5, 8, 11, 14, 17...(to infinity)] which corresponds to the counting numbers by multiplying by 3 and adding 2. That is also an infinite set that's countable. Any set that you can write a function for on a computer that corresponds to the counting numbers is countable. An example of an uncountable set is the set of all numbers from 1 to 2, because for each function you come up with, you can think of a number that throws it off.Thus, clearly, as stated in the opening paragraph, a finite set is one that has one to one relations with natural numbers. Basically, one that could be expressed in terms of basic numbers, like 4,5,6,7, and so on. No question here, these numbers as entities in themselves are finite, and perhaps the entire collection of such numbers will be as well.

@Bolded statement -- Nope.Later in the chapter (3.6), it turns out that there are two kinds of infinite sets, countable and uncountable. The only reason why an entity is uncountable is if it is too large to be counted, and this Rodgers refers to as the 'higher levels of infinity'.Yet, the countable 'infinite' sets are indeed merely very large sets.This is exactly what Rodgers in the opening paragraph stated that 'infinite' does not mean.

LOL. Mathematicians know exactly what countable means, and I can assure you they will not call it "a very large set". It means an infinite set that corresponds to the counting numbers (again, an infinite set) OR a set of finite size.In summary, from the standpoint of descriptive linguistics, infinite, by most mathematicians is regarded as a very large set, which appears to be never ending.