Certainly real wrote:Consider the following:

A = The set of all numbers (if it's a non-contradictory number, then it is included in this set)

B = The set of all numbers except the number 19

It seems that both A and B encompass an endless number of numbers.

In the English language that means that both are infinite (or "endless"). Saying that a set is infinite is merely saying that it has no last element.

Certainly real wrote:However, one cannot deny that A is greater than B in terms of quantity or cardinality. I will attempt to show: 1) B is semi-infinite in quantity, whilst A is infinite in quantity, and 2) Semi-infinites come in various sizes,

We don't have a problem with you calling some of them "semi-infinite" - meaning that the set is less than the set of all integers. We don't see

why you want to say that but it's logically ok.

Certainly real wrote:but there is only one infinity (so there aren't infinities of various sizes).

But that is not really ok.

First "infinity" isn't a quantity. "Infinity" means the first step beyond the endless. It is a target to aim for. But it is not on the map.

No set ever reaches or encompasses infinity.But also there is the issue of having more than one completely infinite set (set of all A1, A2, A3... and the set of all B1, B2, B3...). Combined infinite sets have more than merely one infinite set (obviously). And there is no limit as to how many infinite sets can be combined

so there is no "infinity".

Certainly real wrote:If you tell me "there is no end to the number of numbers that B encompasses", and I ask you "does B encompass the number 19?", you will say "no". To which I will say "clearly, there is an end to the number of numbers that B encompasses. Had you said 'excluding 19, there is no end to the number of numbers that B encompasses' I might have believed you". I say might because I'm not sure if B encompasses infinity. Either we say A is infinity (in which case B does not encompass infinity), or we say A is not infinity (in which case both A and B encompass infinity, but only A encompasses/is an infinity of numbers).

Whilst there absolutely/truly is no end to the number of numbers that A encompasses, there is an end to the number of numbers that B encompasses in an absolute sense. Having said that, the number of numbers that B encompasses is not finite in quantity (hence the term semi-infinite). Furthermore, B is one possible maximally large semi-infinite set of numbers (because it encompasses all numbers but one, and there are an endless number of semi-infinite sets that do this. A semi-infinite set that encompasses all numbers but two is smaller than the aforementioned semi-infinite set).

Hopefully, the above proves that whilst there are many semi-infinite sets of varying sizes, there is only one infinite set.

That is all merely semantics after you invented a word. But why do we care?

There are obviously sets that are more than merely the infinite set of integers. That is why James defined his "\(InfA\)" - to distinguish the set of all integers (or naturals) from any shorter or longer set. Your "set B" above would be \(InfA-1\).