- #1
sandReckoner
- 2
- 0
Greetings, comrades!
In a previous thread, a user articulated a common argument:
His analogy mapping knights to horses makes intuitive sense, but how can we apply this idea to two infinite sets of knights and horses? How can we treat finite and transfinite sets equal in that sense and then treat them unequal in the sense that two transfinite sets can have the same cardinality but one can be a proper subset of the other (unlike finite sets)?
Is there a robust, analytic proof of the first statement in the quote? Is it simply a definition? Is there some justification for the induction argument?
Also, as an aside, could someone please explain the general applications of transfinite cardinality? Set theory in general? My textbook always explains what to do very well but not always why.
Thanks!
~sR#j
In a previous thread, a user articulated a common argument:
Two sets of objects (let's call them A and B) are said to have the same
cardinality, if and only if, they have "1-to-1 correspondence." This is to say, if
every object in A can be attached to a unique object in B, and vice-versa. An
intuitive example would be: "Every knight has a horse, and every horse has a
knight." Let the first set, A, represent some set of horses and let the second set,
B, be represented by a set of knights. Even if we haven't counted the set of horses
or the set of knights, we can know that they have an equal number of horses and
knights if we can't mount every knight to a horse and have no horses or knights left
over.
His analogy mapping knights to horses makes intuitive sense, but how can we apply this idea to two infinite sets of knights and horses? How can we treat finite and transfinite sets equal in that sense and then treat them unequal in the sense that two transfinite sets can have the same cardinality but one can be a proper subset of the other (unlike finite sets)?
Is there a robust, analytic proof of the first statement in the quote? Is it simply a definition? Is there some justification for the induction argument?
Also, as an aside, could someone please explain the general applications of transfinite cardinality? Set theory in general? My textbook always explains what to do very well but not always why.
Thanks!
~sR#j