Discussion: [Article] The Higher Infinite
From the forum: Sam's Essays
This thread was started by: Glowing Face Man.
Discussion start time: 2009-11-14 16:31:23.
From the forum: Sam's Essays
This thread was started by: Glowing Face Man.
Discussion start time: 2009-11-14 16:31:23.
From: Anonymous, from the old Blog Comments
Date: October 31, 2008.
Date: October 31, 2008.
Does an infinity of things that are not numbers really stand to be counted and sized along with infinities that are?
It seems like the very largest, most mindboggling varieties of infinities there are really just semantic gamesmanship rather than concept with real mathematical merit. I mean, it's kind of interesting to think about it, but it really does sound like kids saying "infinity + 1!"
It seems like the very largest, most mindboggling varieties of infinities there are really just semantic gamesmanship rather than concept with real mathematical merit. I mean, it's kind of interesting to think about it, but it really does sound like kids saying "infinity + 1!"
From: Anonymous, from the old Blog Comments
Date: December 5.
Date: December 5.
(Incidentally, which set is bigger, the power set of the natural numbers, or the set of real numbers? Or are they the same size? It turns out this question is unanswerable. Read more at Continuum Hypothesis)
They are the same size. The Continuum Hypothesis states instead that their cardinality is the second smallest infinite cardinaly (i.e., there is no cardinality between that of the naturals and that of the continuum).
The Generalized Continuum Hypothesis states that if X is infinite, then P(X) has the next bigger cardinality. (This is clearly not true in the finite case).
From: Juan, from the old Blog Comments
Date: August 19, 2009.
Date: August 19, 2009.
I stumbled on your article by something not directly related.
I noticed several words that do not apply properly to the concept of infinite, such as "big" or "size", or the fact that there are more sizes to infitity. This is probably related to the concept of cardinality, but not when it comes to infinity. To speak of "size" or "big" one needs a limit, a boundary. Infinity, by definition has neither, and thus although thinking of infinity as an ever growing concept is somewhat common, it is not the right way to think about it. In order to grow, you cannot be infinite.
I noticed several words that do not apply properly to the concept of infinite, such as "big" or "size", or the fact that there are more sizes to infitity. This is probably related to the concept of cardinality, but not when it comes to infinity. To speak of "size" or "big" one needs a limit, a boundary. Infinity, by definition has neither, and thus although thinking of infinity as an ever growing concept is somewhat common, it is not the right way to think about it. In order to grow, you cannot be infinite.
Unfortunately there is a flaw in your argument for Cantor's Theorem: there IS a covering of the reals with intervals of length 1/n, where n ranges over natural numbers:
For example, it is easily seen that the rational numbers can be listed as q_1, q_2, q_3, ... and so on (note that the pairs of natural numbers can be listed like this). Then cover each q_1 with an interval of length 1/n. Since the rationals are dense in the reals, every real gets covered as well. QED
However, the fact remains that the reals cannot be listed as r_1, r_2 ,... , but I don't know any proof essentially different from the famous diagonal argument.
Other minor bug is that obviously e.g. ZFC proves ZFC, but from Gödel's Theorem it follows that ZFC cannot prove the CONSISTENCY of ZFC. Large cardinals entail Con(ZFC), since from a large cardinal one can construct a model of ZFC.
Also, I wouldn't call the Axiom of Choice (the ''C'' in ''ZFC'') so obvious that if it fails, then the cows fly as well...
For example, it is easily seen that the rational numbers can be listed as q_1, q_2, q_3, ... and so on (note that the pairs of natural numbers can be listed like this). Then cover each q_1 with an interval of length 1/n. Since the rationals are dense in the reals, every real gets covered as well. QED
However, the fact remains that the reals cannot be listed as r_1, r_2 ,... , but I don't know any proof essentially different from the famous diagonal argument.
Other minor bug is that obviously e.g. ZFC proves ZFC, but from Gödel's Theorem it follows that ZFC cannot prove the CONSISTENCY of ZFC. Large cardinals entail Con(ZFC), since from a large cardinal one can construct a model of ZFC.
Also, I wouldn't call the Axiom of Choice (the ''C'' in ''ZFC'') so obvious that if it fails, then the cows fly as well...
Just because the rationals are dense in the reals does not imply that every real is covered. In fact, if q_i is covered by an interval of length 1/i^2 instead of 1/i, then the reals absolutely cannot be covered, by measure theory. Not certain about the 1/i case (the measure theory argument doesn't work since the harmonic series diverges, but the intervals overlap so we still don't know whether their combined measure is finite or infinite)
It's all so scandalous, I wrote about it in an article of its own:
http://www.xamuel.com/covering-the-rationals/
BTW, it's not about covering the rationals with intervals of length 1/1, 1/2, 1/3, ...
It's about covering the rationals with intervals whose TOTAL length all together is 1/n.
It's all so scandalous, I wrote about it in an article of its own:
http://www.xamuel.com/covering-the-rationals/
BTW, it's not about covering the rationals with intervals of length 1/1, 1/2, 1/3, ...
It's about covering the rationals with intervals whose TOTAL length all together is 1/n.