Hilbert's grand hotel as infinite number of pairs

[Mark44]

Left and right as parts of one pair are bound together and you should not be able to create new pair by adding only left or only right part to it.

You really need to get past this notion that the pairing has to be unique. It is keeping you from understanding this concept.

An infinite paired set is infinite in number but its parts are limited to each other (each left shoe has as a pair a right shoe, and there are no singles available, otherwise the set wouldn't be called paired set)

This "paired set" idea that you are fixated on is no help to you. You are missing the main idea -- a set is countably infinite if there is a one-to-one mapping between the elements of the set and the set of positive integers. Period.

Regardless whether the number is limited or infinite, the pairs in the row are created and cannot be increased by adding only to one side of the pair.

Baloney.

One would argue that we don't increase infinity, because its value cannot be increased or decreased by adding or subtracting, but by not taking in account the properties of the paired infinite set we have a moment between the pairings when one of the sets is with number greater than the other.

No, this isn't true. You not grasping the idea that an infinite set is fundamentally different from a finite set.

Again, "with number greater than the other" is not correct use for infinity, so we rather call it unpaired number (one of the sides contains unpaired number of shoes). That unpaired number implies limit to both sides when we look at them as pair parts.

Unpaired numbers would be relevant if we were dealing with finite sets, but once we start talking about infinite sets, it doesn't matter in the slightest that there are some numbers that aren't included. What does matter is that we can establish a one-to-one pairing with a countably infinite set (such as the positive integers). In every example I gave, I showed you the pairing.

That's why I say that a pairing should occur only once if we don't want to create a paradox.
In Hilbert's case we have complete infinite number of pairs (no singles available) and any adding to one of the sides will break that completeness.
Hence the paradox.

Like pwsnafu said, "we don't care about what you think "should" or "should not" be doable in mathematics."

I'm not trying to change mathematics, neither am I challenging your knowledge to prove you wrong.
Don't get frustrated if you cannot answer some of my questions and logical points.

We have answered every one of your questions and have refuted all of your logical points. What is frustrating, is that you are unable to let go of your incorrect ideas about infinite sets, despite being shown that they are faulty.

Most probably that is because my points are so lame that you cannot make sense of them

 

[Mark44]

The question has been asked and answered (ad infinitum, so to speak). I am closing this thread.