August 21st, 2016, 08:26 AM | #1 |
Senior Member Joined: May 2016 From: USA Posts: 758 Thanks: 303 | To Jimbo
To Jimbo on Cantor's Proof This is a non-mathematician's explanation of Cantor's proof based on some things I learned from JeffJo and Maschke. I expect Maschke or JeffJo or both may say where it is wrong in a rigorous sense. But I think that rigor may get in the way of INITIAL understanding, and anyway I cannot supply any rigor. So for a 50,000 foot view of the proof. The whole thing of course is premised on the supposition that "infinity" "exists" even though we can "construct" it only in our imaginations. (1) We give a demonstration that a particular infinite set that is NOT premised on any number system has a greater cardinality, "size", or "number" (not worrying about semantics at this point) than the set of natural numbers. (JeffJo and Maschke taught me this, whiich was in fact initially developed by Cantor himself.) If this demonstration is satisfactory, it has now been shown that there is at least one cardinality greater than the cardinality of the set of natural numbers. This is the bridge that I suspect people really are reluctant to cross psychologically. So let's get it out of the way first. (2) We show how to put that set into 1-1 correspondence with the set of all infinite sums of the form $\displaystyle \sum_{i=1}^{\infty} \left ( a_i * \dfrac{1}{2^k} \right )\ where\ a_i \in \{0,\ 1\}\ and\ \sum_{i=1}^{\infty}a_i \ne 0\ and\ \prod_{i=1}^{\infty}a_i \ne 1.$ (3) Therefore the set constructed mentally in step 2 has a greater cardinality than the set of natural numbers. (4) We then split this second set into two exhaustive but mutually exclusive subsets, one of which can be set into 1-1 correspondence with the set of natural numbers. (5) Therefore the second subset has a greater cardinality than the set of natural numbers. (6) We prove that every element of the second subset is a representation of a unique real number in (0, 1). To prove this requires analysis, but it is intuitively obvious. (For all I know, certain kinds of infinite sums may be one way to define real numbers.) (7) Therefore the number (cardinality) of real numbers in (0, 1) is greater than the number (cardinality) of all natural numbers. Without worrying about any details in each step and assuming for the moment that step 1 CAN be demonstrated, does this explanation make sense to you? It seems straight forward to me, but that's just me. |
August 23rd, 2016, 01:06 PM | #2 |
Senior Member Joined: Mar 2016 From: UK Posts: 101 Thanks: 2 |
Thank you yet. I believe I do now have it, thank you. Your explanation is also clear - and I do believe that step 1 is okay too. I will say however, that for a non-mathsy post, your definition of an infinite sequence of binary digits is, err, a little mathsy... Especially the bit that says "not all ones and not all zeros either" Last edited by skipjack; August 23rd, 2016 at 02:34 PM. |
August 23rd, 2016, 02:51 PM | #3 | |
Senior Member Joined: May 2016 From: USA Posts: 758 Thanks: 303 | Quote:
Actually I stayed away from anything to do with binary notation. In step 1 I might have used # and @. And although my sum of powers of two of course corresponds to binary notation, it strictly uses decimal notation: notice the 2. I like using the fractional approach because it ties back to the first known exploration of infinity, namely Zeno's paradox. | |