Name: MathMajor 2004-12-27 13:40
This past semester I took the first halves of Introduction to Analysis and Modern (Abstract) Algebra in which, naturally, we covered the basics of the cardinalities of infinities. I understand the concepts of what makes something countable verses uncountable. I simply have an intuitive problem with the fact that the rational numbers are countable. The other basics countable infinite sets (Integers, Evens, Odds, Natural Numbers, etc.) all sit well with me. For some reason, I just have a problem accepting that the Rationals are truly countable. I understand that a definition is not arguable, but it just does not feel right.
Part of my discontent lies in the fact that like the continueum, there is no smallest positive rational number. This is not true for the previously mentioned sets. I feel that the rationals are somehow set apart from this set. If I recall correctly, it was proven that "It cannot be proven nor disproven that there exists an infinite set whose cardinality is greater than the cardinality of the countable infinite sets and less than the cardinality of the continueum (the reals)."
If anyone knows where I may be able to find more information on this, or even has an answer to my dilemma, please let me know. Thanks!
-Will
Part of my discontent lies in the fact that like the continueum, there is no smallest positive rational number. This is not true for the previously mentioned sets. I feel that the rationals are somehow set apart from this set. If I recall correctly, it was proven that "It cannot be proven nor disproven that there exists an infinite set whose cardinality is greater than the cardinality of the countable infinite sets and less than the cardinality of the continueum (the reals)."
If anyone knows where I may be able to find more information on this, or even has an answer to my dilemma, please let me know. Thanks!
-Will