tag:blogger.com,1999:blog-3722233.post7384992833624363197..comments2024-03-04T02:59:26.350-06:00Comments on Computational Complexity: Global Warming and the Axiom of ChoiceLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-3722233.post-47373487506012505062013-12-16T04:06:04.169-06:002013-12-16T04:06:04.169-06:00Here's another "paradox" of the flav...Here's another "paradox" of the flavor of Banach-Tarski that doesn't rely on the axiom of choice: You can break a ball up into an infinite number of pieces and reassemble them into a ball of twice the size. How do you do it? Well, completely shatter it into its component points and then move each point twice as far from the center.<br /><br />So now which axiom should you reject as causing a ridiculous paradox???Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-63231855212613959862013-12-16T03:42:44.715-06:002013-12-16T03:42:44.715-06:00You could similarly say that the Peano axioms for ...You could similarly say that the Peano axioms for the integers have absurd consequences, because you can place the positive integers in one-to-one correspondence with the entire set of integers. Neither that nor the Banach-Tarski paradox bothers me in the least.<br /><br />As for global warming, there are a lot more people who are completely convinced in one way another if whether it exists than have actually read any scientific paper on the subject. I guess this is OK for the layman, but this is also true for professors, who should know better. Freeman Dyson, one of the first researchers in climate change, was once a skeptic of GIGO computer models showing global warming but now believes the current evidence it exists, at least as recently as the talk of his I saw a few years ago. However, he thinks it's an open question whether global warming is bad. I have no opinion on either matter, however, since I have not studied the issue sufficiently to be entitled to an opinion.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-57197534695397896932013-12-12T05:20:14.312-06:002013-12-12T05:20:14.312-06:00The Banach-Tarski paradox does NOT say that "...The Banach-Tarski paradox does NOT say that "you can break a ball into 5 pieces, reassemble, and get twice the volume" anymore than it says that you can well-order the reals or that you can pick the corresponding choices from infinite collections. Now way "you can", actually you and I can't. It says that "there is" a way of blowing up the volume, there is a way of well-ordering the reals, and there is a way of having infinitely many representatives of each set in an infinite collection in a single set, in the same way that math says as well that "there are" inaccessible ordinals or, for that matter, infinite graphs or natural numbers like A(10,10) (Ackermann's I mean) or even simply ten to the ten to the ten. All these things "exist" like unicorns do (whether pink or invisible or both), no one of them is real, not even high enough natural numbers that would not fit within the reachable universe. The point is, for working on that unreal but, at the end of the day, actually enormously useful realm, I have no problem in admitting that such weird things exist. To me, this is pretty different from negating the problem of global warming!<br />Balquihttps://www.blogger.com/profile/09857844891905542749noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-18811645598150710992013-12-09T10:09:55.322-06:002013-12-09T10:09:55.322-06:00It wasn't global warming; it was because she r...It wasn't global warming; it was because she ran off to California with that movie guy.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-47562328562664106472013-12-07T22:36:32.818-06:002013-12-07T22:36:32.818-06:00I'm not a set theorist, but from what I've...I'm not a set theorist, but from what I've read, the strongest objection to AC is that it is unnecessary for any statement that pertains to the natural numbers. The power of AC is only needed when reasoning about uncountable sets, and we don't have a standard model for uncountable sets the same way that we do for natural numbers. So why have AC at all? It's often convenient, but it also often proves unintuitive results.Lukehttps://www.blogger.com/profile/08867175989749027625noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-40251368761719145362013-12-03T19:10:56.874-06:002013-12-03T19:10:56.874-06:00I would very much like to understand the pushback ...I would very much like to understand the pushback against AC. Do those who object it also object the Axiom of Infinity? For otherwise one can create a model of PA within ZF such that a nonhalting Turing machine halts nonetheless -- (albeit at an infinite stage).<br /><br />I mean, if we are calling things absurd based on some physical intuition then we should also reject the ordinal numbers -- no? <br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-5283954136307750782013-12-03T11:23:24.450-06:002013-12-03T11:23:24.450-06:00Is there anything von Neumann *didn't* do? The...Is there anything von Neumann *didn't* do? The guy is like a modern Euler...thecodhttps://www.blogger.com/profile/10717993079747767784noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-19374457933971427102013-12-02T15:35:57.460-06:002013-12-02T15:35:57.460-06:00--------------
GASARCH asks "Who was the firs...--------------<br /><b>GASARCH</b> asks "Who was the first scientist to warn of Global Warning?"<br />--------------<br /><br /><b>Answer</b> John von Neumann, in his chapter "Can we survive technology?" (1955), which appeared in <b><a href="http://www.amazon.com/The-Fabulous-Future-America-1980/dp/B000KEKCL8" rel="nofollow"><i>The Fabulous Future: America in 1980</i></a></b>:<br /><br />--------------<br />"All major weather phenomena are ultimately controlled by the solar energy that falls on the earth. […] The carbon dioxide released into the atmosphere by industry's burning of coal and oil — more than half of it during the last generation — may have changed the atmosphere's composition sufficiently to account for a general warming of the world by about degree Fahrenheit. <br /><br />Intervention in atmospheric and climatic matters will come in a few decades, and will unfold on a scale difficult to imagine at present. […] Such actions would be more directly and truly worldwide than recent, or presumably, future wars, or the economy at any time."<br />--------------<br /><br />Best wishes for Happy Holidays are extended to all readers of <i>Computational Complexity</i> and thanks are extended to Lance and GASARCH for sustaining this fine forum!<br /><br />------------<br />@incollection{Author = {J. Von Neumann},<br />Booktitle = {The Fabulous Future: America in 1980}, <br />Pages = {33--48},<br />Publisher = {E. P. Dutton {\&} Company}, <br />Title = {Can we survive technology?},<br />Year = 1955}John Sidleshttps://www.blogger.com/profile/16286860374431298556noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-24316744192270156522013-12-02T12:52:52.419-06:002013-12-02T12:52:52.419-06:00About the axiom of choice: It may be that mathemat...About the axiom of choice: It may be that mathematicians believe it because it makes the trans-finite "universe" much "nicer". As far as I know without it would seems that more questions would have an answer of "it depends...". Also, reasonable competing axioms such as "all subsets of [0,1] are measurable" may lead to biting a bullet on other topics (the axiom of choice guarantees for example that "bigness" is always comparable).<br /><br />As for global warming, my guess is that the existence of known lies causes skeptics to doubt all claims first and ask questions later. Also the usual proposed "solutions" (not really solutions because the US is not the one culprit) are far from "moderately republican" so a middle ground does not exist.<br /><br />But on an optimistic note, usually humanity bands together on the right side when the ecological impact is bad enough. Examples are CFC gases (now almost gone) and the paper industry (it used to be a major threat for forests). An example close to my home is the preservation of wild vegetation in Israel (nobody picks wild flowers in Israel).Eldar Fischernoreply@blogger.com