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> ETA: For instance: One infinite set does not have "as many" or "more" or "less" members than any other infinite set. You cannot compare amounts, there are not

> ETA: For instance: One infinite set does not have "as many" or "more" or "less" members than any other infinite set. You cannot compare amounts, there are not "amounts" to compare. Not exactly true - there are different infinities. For example, the number of integers is infinite, and the number of real numbers is infinite, but there are more real numbers than integers. The number of integers is "countably infinite" and the real numbers are "uncountably infinite".

shannifin 5y ago
> "there are more real numbers than integers." This is neither true nor false as it has no meaning. Whether countable or not, infinite sets do not have sizes to compare. Granted, I know many mathematicians still prefer to understand uncountable sets to be those "that contains too many elements to be countable" which implies size. But the statement is still meaningless. Countability vs uncountability is really about something a bit more subtle with the axioms that we use to define sets in the first place.
- chimeracoder 5y ago
  > This is neither true nor false as it has no meaning. Whether countable or not, infinite sets do not have sizes to compare. Infinite sets do have cardinality, and cardinalities can be compared, in a manner which corresponds pretty well to what people mean when they colloquially say "more" or "fewer". If you want to split hairs between "size" and "cardinality", that's your choice, but even Cantor himself used the term "size" in this manner.
  - shannifin 5y ago
    > in a manner which corresponds pretty well to what people mean when they colloquially say "more" or "fewer" I would argue that it does not correspond very well to what people mean when they say "more" or "fewer", and hence the apparent paradox of Hilbert's Hotel. That is, my original point was that thinking of infinite sets as having comparable sizes is the precise confusion that leads to the supposed paradox. If you're specifically comparing cardinalities, "more" or "fewer" lose their "colloquial" meaning.

shannifin 5y ago

> "there are more real numbers than integers." This is neither true nor false as it has no meaning. Whether countable or not, infinite sets do not have sizes to compare. Granted, I know many mathematicians still prefer to understand uncountable sets to be those "that contains too many elements to be countable" which implies size. But the statement is still meaningless. Countability vs uncountability is really about something a bit more subtle with the axioms that we use to define sets in the first place.

chimeracoder 5y ago
> This is neither true nor false as it has no meaning. Whether countable or not, infinite sets do not have sizes to compare. Infinite sets do have cardinality, and cardinalities can be compared, in a manner which corresponds pretty well to what people mean when they colloquially say "more" or "fewer". If you want to split hairs between "size" and "cardinality", that's your choice, but even Cantor himself used the term "size" in this manner.
- shannifin 5y ago
  > in a manner which corresponds pretty well to what people mean when they colloquially say "more" or "fewer" I would argue that it does not correspond very well to what people mean when they say "more" or "fewer", and hence the apparent paradox of Hilbert's Hotel. That is, my original point was that thinking of infinite sets as having comparable sizes is the precise confusion that leads to the supposed paradox. If you're specifically comparing cardinalities, "more" or "fewer" lose their "colloquial" meaning.