Started conversation Jun 13, 2005
Posted Jun 13, 2005
What I would like to know is how they can afford all those little soaps and shampoos for their infinity-squared guests.
Infinite Grand Hotel
Posted Aug 17, 2010
This is a deep and profound topic.
The Infinite Grand Hotel and The Nature of the Infinity Boundary.
The shifting of guests and flipping between infinite set of number vs infinite set of odd or even numbers is a function that relies on the disregarding of the attributisation boundary of numeric infinite sets to work. In other words, we have to understand both cardinality (countable equivalence) and 'quality of numberness' in order to untangle the paradox.
The integer infinity boundary can be understood as an 'attribute boundary', that is the infinity of a 'single quality of number', that is the infinity of all integer numberness. But within this singular attribute boundary,the infinity of positive integer, we can also have the infinity of odd numbers and the infinity of even numbers, and the infinity of any other categorization or quality of (countable) number. If we regard them as partitioned attribute infinities, then we can't shift guests about, as 'odd' and 'even' are different partitioned attributes of number (numeric oil and water - different qualities of infinite numberness) but if we regard all subset categories of numbers (odd, even, prime, whatever...) as all part of the singular integer numberness infinity, without attribute partitioning, then the Hotel works, and the cause of the paradox, the attribute infinity boundary, is hidden.
I make these observations to show that part of the paradoxical nature of these infinities lies in the nature of the infinity boundary, which is more accurately understood as an attribute boundary such that the infinity of interger number represents the attribute boundary of the realm of all integer number. The limit is bounded and defined by the quality or kind of number, an attribute boundary - a dimension-like boundary, not, obviously, a magnitude limit.
If we disregard the significance of the attribute boundaries of other infinite qualities of numberness (odd, even, prime,...), then the paradoxical behaviour is preserved there is cardinality in these infinities and we can apparently 'swap' between them or add others.
However there may be instances where the infinity attribute boundaries are relevant.