tag:blogger.com,1999:blog-1572154923785712186.post2087744928277226670..comments2024-01-22T12:51:39.817-05:00Comments on Professor Quibb: Infinity: The Cardinality of the ContinuumLouishttp://www.blogger.com/profile/15382160997783595665noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-1572154923785712186.post-87015483852711220902012-02-01T06:35:25.721-05:002012-02-01T06:35:25.721-05:00Thank you for your comment.
Your understanding is...Thank you for your comment.<br /><br />Your understanding is indeed correct, and it illustrates an important method for equating infinite cardinalities. <br /><br />In set theory, many cardinal equalities are established by determining that each set has a cardinality less than or equal to the other. In this case, the confirmation that |P(N)| ≤ |R| was also performed by showing that |P(N)| corresponded to the real numbers on the interval {0,1}, clearly a subset of the set of real numbers.Louishttps://www.blogger.com/profile/10746982398555711955noreply@blogger.comtag:blogger.com,1999:blog-1572154923785712186.post-3780632170464365262012-01-31T23:58:47.976-05:002012-01-31T23:58:47.976-05:00Thank you for this insightful series of posts.
I ...Thank you for this insightful series of posts.<br /><br />I actually took a discrete math course in university some years ago and recently had my interest piqued again, and I came here looking for an explanation of why the cardinality of the continuum is 2^aleph-zero. I think your post is the first one that has made any sense to me, but I want to be 100% sure about the crucial statement:<br /><br />"Using this method, every real number can be generated from a subset (finite or infinite) of the natural numbers, and the real set is the power set of the natural numbers."<br /><br />The way I'm understanding this is as such (R = the real set, P(N) = the power set of natural numbers):<br /><br />- Since you can generate any real number from a subset of natural numbers (via the binary string method discussed), the reals are "contained" in the power set of natural numbers (i.e. the cardinality of R must be less than or equal to than of P(N)).<br /><br />- In addition, every subset of natural numbers generates a real number, using the same method. There's no subset that generates an imaginary or otherwise non-real number. Therefore, P(N) is also contained in R, and |P(N)| ≤ |R|.<br /><br />Since |R| ≤ |P(N)| and vice versa, |R| = |P(N)|. Is my understanding of the logic behind this conclusion correct?Arindamnoreply@blogger.com