Comments on Comparing InfinitiesTypePad2011-12-12T05:06:41ZMark Thomahttps://economistsview.typepad.com/economistsview/tag:typepad.com,2003:https://economistsview.typepad.com/economistsview/2011/12/comparing-infinities/comments/atom.xml/mfasano commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e2015438582898970c2011-12-15T18:00:34Z2011-12-15T18:00:34ZmfasanoI don't think this is correct. The dedekind cut splits rational numbers less than the one irrational number on the...<p>I don't think this is correct. The dedekind cut splits rational numbers less than the one irrational number on the cut and the rational numbers greater than the irrational number on the cut.</p>
<p><a href="http://en.wikipedia.org/wiki/Dedekind_cut" rel="nofollow">http://en.wikipedia.org/wiki/Dedekind_cut</a></p>jh commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e2015438484b5b970c2011-12-14T12:48:19Z2011-12-14T12:48:19ZjhCareful... In the correspondence sense, you have the same number of descriptions and numbers (not "more descriptions than numbers"), despite...<p>Careful... In the correspondence sense, you have the same number of descriptions and numbers (not "more descriptions than numbers"), despite multiple descriptions for certain numbers (like 1.000... and 0.999...). Since the set of such numbers is countable, it doesn't 'increase' the power of the set of descriptions.</p>cm commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdc4f80f970d2011-12-14T03:01:36Z2011-12-14T03:01:36ZcmYes, the set of descriptions is countable. OTOH, every real number is trivially "described" by its potentially infinite decimal representation...<p>Yes, the set of descriptions is countable. OTOH, every real number is trivially "described" by its potentially infinite decimal representation - and then you even have more descriptions than numbers, e.g. 0.99999.... being the same as 1.<br />
</p>cm commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eb8d731970b2011-12-14T02:57:48Z2011-12-14T02:57:48ZcmBut both "algorithms" will result in the same number of balls at step #n, for every (finite) n. The catch...<p>But both "algorithms" will result in the same number of balls at step #n, for every (finite) n. The catch is of course assuming an infinite number of steps with a finite fractional second. But he probably explained this part too.</p>Adam Stephanides commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdc1ab79970d2011-12-13T20:43:26Z2011-12-13T20:43:26ZAdam Stephanideshttp://completelyfutile.blogspot.com"The cardinality of the set of irrational numbers, however, is not the same as that of the set of all...<p>"The cardinality of the set of irrational numbers, however, is not the same as that of the set of all integers. It's aleph-one"</p>
<p>The cardinality of the set of irrational numbers is indeed greater than the cardinality of the set of integers, but it's not necessarily aleph-one. To say that it's aleph-one means that there are no infinite sets whose cardinality is greater than that of the set of all integers and less than that of the set of all irrational numbers (which has the same cardinality as the set of all reals). This is the famous Continuum Hypothesis, which it is known can neither be proven or disproven from the generally accepted axioms of set theory: that is, there are models of set theory in which the cardinality of the set of irrationals is aleph-one, and models in which it isn't.</p>Jacob Davies commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20154383e0d97970c2011-12-13T17:51:50Z2011-12-13T17:51:50ZJacob DaviesConfidently stating something as fact does not make it reality either. The question of whether the universe is bounded is...<p>Confidently stating something as fact does not make it reality either. The question of whether the universe is bounded is far from settled right now and most cosmologists would tell you they think it is unbounded. Hence Greene's point above.</p>kaleberg commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eb3dc94970b2011-12-13T17:34:27Z2011-12-18T00:31:54Zkaleberghttp://profile.typepad.com/kalebergIt all comes down to the finite age (and therefore finite size) of our universe, though it seems I should...<p>It all comes down to the finite age (and therefore finite size) of our universe, though it seems I should have said our "causal universe" or "observable universe". The issue is still the fact that the potential source of light is finite, not infinite, as that Olber's Paradox article notes. </p>
<p>(Are areas of the universe that by definition we cannot observe and cannot affect us causally actually part of our universe? It is more and more common to argue that they are, so I should start being more careful in my phrasing.)</p>
<p>For a really good treatment of the history of the problem, grab a copy of Harrison's Darkness at Night.</p>Jacob Davies commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdbffb13970d2011-12-13T17:21:22Z2011-12-13T17:21:22ZJacob DaviesNo. http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html<p>No.</p>
<p><a href="http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html" rel="nofollow">http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html</a></p>cm commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eb3b706970b2011-12-13T17:13:11Z2011-12-13T17:13:11ZcmI didn't describe this state of affairs specifically, but it is included in "sufficiently dense". Aside from that, even with...<p>I didn't describe this state of affairs specifically, but it is included in "sufficiently dense". Aside from that, even with infinitely many stars you may not get good illumination if they are spaced far enough apart for the falloff in light energy to cancel out the numbers of far away stars. Basically a number of stars per unit of space give you an average light energy per unit. This argument doesn't account for light from very far away stars accumulating over time. But wouldn't that be the case in a finite universe too?<br />
</p>kaleberg commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdbbfdbe970d2011-12-13T05:02:26Z2011-12-18T00:31:54Zkaleberghttp://profile.typepad.com/kalebergYou are probably right. I should have said arbitrarily long descriptions, not infinite descriptions. The idea is that if you...<p>You are probably right. I should have said arbitrarily long descriptions, not infinite descriptions. The idea is that if you can put the descriptions into one to one correspondence with the integers, as with an enumeration of all computer programs or textual descriptions, you just don't have enough of them to cover the reals.</p>
<p>This was discussed in a blog post from Mark Chu Carrol (Good Math, Bad Math).<br />
</p>kaleberg commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543839ebf0970c2011-12-13T04:57:04Z2011-12-18T00:31:54Zkaleberghttp://profile.typepad.com/kalebergIt isn't the density of stars that matters if the universe is actually infinite. Even a very sparse, but infinite,...<p>It isn't the density of stars that matters if the universe is actually infinite. Even a very sparse, but infinite, array of stars would result in a light sky at night. It's only if there is a limited number of stars in an otherwise infinite universe that you'd have a dark night sky. That would correspond to a density of zero.<br />
</p>cm commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543839e5cf970c2011-12-13T04:49:31Z2011-12-13T04:49:31ZcmOnly if sufficiently densely populated with stars.<p>Only if sufficiently densely populated with stars.</p>cm commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdbbeeca970d2011-12-13T04:45:26Z2011-12-13T04:45:26ZcmAs some people here have pointed out there are different types of infinities.<p>As some people here have pointed out there are different types of infinities.</p>cm commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdbbe9bc970d2011-12-13T04:38:51Z2011-12-13T04:38:51ZcmI'm not sure your point about infinite descriptions is accurate, but then I wouldn't consider something that is infinite a...<p>I'm not sure your point about infinite descriptions is accurate, but then I wouldn't consider something that is infinite a "description" to begin with.<br />
</p>jh commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543839db4c970c2011-12-13T04:36:35Z2011-12-13T04:36:35ZjhWhat you say is right, but you're not really correcting a. He wasn't talking about rationals (or irrationals, per se)...<p>What you say is right, but you're not really correcting a. He wasn't talking about rationals (or irrationals, per se) -- just the set of all reals, and the set of all reals in an interval. You're both right.</p>a commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eafa520970b2011-12-13T04:03:59Z2011-12-13T04:03:59Zaa decent undergraduate course in set theory usually covers this material (and more) but its possible in most universities to...<p>a decent undergraduate course in set theory usually covers this material (and more) but its possible in most universities to major in mathematics (and of course physics and economics) without ever taking such a course and encountering the continuum hypothesis or the axiom of choice to take two important notions ... </p>kaleberg commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eaf0929970b2011-12-13T01:52:12Z2011-12-18T00:31:54Zkaleberghttp://profile.typepad.com/kalebergThe dark night sky belies this. If the universe were infinite in any dimension, spatial or temporal, the sky at...<p>The dark night sky belies this. If the universe were infinite in any dimension, spatial or temporal, the sky at night would be extremely bright. (Interestingly, Edgar Allen Poe, of all people, was the first to point this out.)</p>kaleberg commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdbb2fdd970d2011-12-13T01:46:38Z2011-12-18T00:31:54Zkaleberghttp://profile.typepad.com/kalebergActually, there are an infinite number of rational numbers and an infinite number of irrational numbers between any two rational...<p>Actually, there are an infinite number of rational numbers and an infinite number of irrational numbers between any two rational numbers. It's just that there are infinitely more irrational ones than rational ones.</p>
<p>One of my favorite fun facts is that most numbers cannot be described, even with infinitely long descriptions. There are only as many such descriptions (or computer programs) as there are integers, but there are a lot more rational numbers than that. Most numbers are indescribable.</p>Min commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543838d1a1970c2011-12-13T01:10:50Z2011-12-13T01:10:50ZMin"Any mathematician will tell you the problem lies in infinity being a situation not a number." You should talk to...<p>"Any mathematician will tell you the problem lies in infinity being a situation not a number."</p>
<p>You should talk to John Conway. ;)</p>neteroo commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eae9bd5970b2011-12-13T00:48:23Z2011-12-13T00:48:23Zneteroohttp://www.reconart.com/case-managementYes, but what are the odds that you are selected and actually make it on the stage with Monty to...<p>Yes, but what are the odds that you are selected and actually make it on the stage with Monty to even have such a problem to consider...? :)</p>RW commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdbabe4f970d2011-12-13T00:40:27Z2011-12-13T00:40:27ZRWJust an observation: Read an article about a conference at the Santa Fe Intitute some years ago in which several...<p>Just an observation:</p>
<p>Read an article about a conference at the Santa Fe Intitute some years ago in which several sessions were shared by Economists and Physicists. Physicists came away impressed by the amount of math Economists used -- they had no idea how math-intensive the discipline had become -- and the Economists came away rather shocked at the casual way math was used by the physicists.</p>
<p>This reminded me of my college science adviser, even more years ago than that, telling me to take the math courses set up for science majors and avoid the courses (covering many of the same topics) in the math major, saying, "too many axioms and proofs can give a scientist bad habits, we don't deal in ideal cases."</p>ScentOfViolets commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdbab4c3970d2011-12-13T00:32:13Z2011-12-13T00:32:13ZScentOfVioletsHeh. Martin Gardner was popularizing this sort of thing in his Scientific American column half a century ago. Here's one...<p>Heh. Martin Gardner was popularizing this sort of thing in his Scientific American column half a century ago. Here's one of his oldies but goodies:</p>
<p>Start some time before 11:00 p.m. with two balls in a bucket labelled 1 and 2 . One hour before midnight, the ball labelled with the smallest number, number 1 in this case, is removed from the bucket and two more balls labelled 3 and 4 are added. Half an hour before midnight, the ball labelled with the smallest number, number 2 in this case, is removed from the bucket and balls 5 and 6 are added. Continue in this fashion a quarter of an hour before midnight, an eighth of an hour, etc, each time removing the ball labelled with the smallest number from the bucket and adding two more balls labelled with the next two consecutive integers: 7 and 8, 9 and 10, etc.</p>
<p>Question: at midnight, how many balls are in the bucket? The answer, Gardner shows, is zero! Equally astounding to anyone encountering this sort of thing for the first time is that by simply changing the procedure so that the ball with the largest number in the bucket is removed rather than the ball with the lowest number results in an infinite number of balls in the bucket at midnight.</p>
<p>Good times.</p>Greg commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eadfb68970b2011-12-12T22:32:56Z2011-12-12T22:32:56ZGregThe universe is bounded, in the four large dimensions anyway. Thinking about infinities can provide a lot of insight, and...<p>The universe is bounded, in the four large dimensions anyway.</p>
<p>Thinking about infinities can provide a lot of insight, and it is very difficult, but don't mistake it for reality.</p>anne commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e2015438372550970c2011-12-12T21:27:49Z2011-12-12T21:27:49ZanneClever.<p>Clever.</p>Julio commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e2015438371a9f970c2011-12-12T21:23:12Z2011-12-14T23:12:06ZJuliohttp://profile.typepad.com/juliokI cannot tell which confident ignoramus you're referring to, which is itself telling.<p>I cannot tell which confident ignoramus you're referring to, which is itself telling.</p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eac29d5970b2011-12-12T20:13:01Z2011-12-12T20:13:01Zpainei remember the first time i saw a demonstration of a triangle of infinite length but only finite area or...<p>i remember the first time i saw a demonstration <br />
of a triangle of infinite length but only finite area </p>
<p>or the first time i went over the derivation<br />
of pi by viete </p>
<p>or fermats area under the curve of x to the n or ...<br />
gauss and his contruction<br />
with euclidian instruments only <br />
of the till then unknown 17 sided polygon</p>
<p>beauty... wonder ....the sublime <br />
</p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb83d93970d2011-12-12T20:03:12Z2011-12-12T20:03:12Zpaineno there are countable infinites of different sizes and more basically any chunk of the real number line contains an...<p>no there are countable infinites of different sizes</p>
<p>and more basically<br />
any chunk of the real number line contains <br />
an uncountable number of real numbers <br />
hence it is larger <br />
try it you'll see </p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb830be970d2011-12-12T19:57:31Z2011-12-12T19:57:31Zpainemore importantly greene here is wrong when he writes "there simply is no absolute answer to the question of which...<p>more importantly greene here is wrong<br />
when he writes</p>
<p>"there simply is no absolute answer to the question of which of these kinds of infinite collections are larger"</p>
<p>as you suggest <br />
these examples are all the same size </p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543836181c970c2011-12-12T19:50:54Z2011-12-12T19:50:54Zpainei agree with anne the wonder of its counter intuitivity oughta still swell your soul this "seen that done that...<p>i agree with anne </p>
<p>the wonder of its counter intuitivity <br />
oughta still swell your soul </p>
<p>this "seen that done that " macho</p>
<p>sucks damp socks </p>
<p>though i must say</p>
<p>banging on mark seems off<br />
its mr greenspleens here that wrote all of this</p>
<p><br />
that mark may not have taken a number theory course<br />
or delved behind the limit rules to their first layer of why's...</p>
<p>let he without sin cast the first stone here </p>
<p>we all cut corners somewhere down the chain of levels of comprehension </p>
<p>and to think there are no foundations .down below ...<br />
ah but that really chastens the blow hards eh ??</p>PeterE commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eabfbf4970b2011-12-12T19:50:38Z2011-12-12T19:50:38ZPeterEYou might find these Wikipedia articles instructive: http://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals http://en.wikipedia.org/wiki/Cardinal_number<p>You might find these Wikipedia articles instructive:</p>
<p><a href="http://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals" rel="nofollow">http://en.wikipedia.org/wiki/Ordinal_number#Arithmetic_of_ordinals</a></p>
<p><a href="http://en.wikipedia.org/wiki/Cardinal_number" rel="nofollow">http://en.wikipedia.org/wiki/Cardinal_number</a></p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eabf1f1970b2011-12-12T19:45:42Z2011-12-12T19:45:42Zpaineya under limits and series sums ...maybe<p>ya under limits and series sums ...maybe </p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb816a8970d2011-12-12T19:44:10Z2011-12-12T19:44:10Zpaineyes the fraction matrix shows this nicely the major diagonal is of the form n/n ie equals 1 the matrix...<p>yes the fraction matrix shows this nicely</p>
<p>the major diagonal is of the form n/n ie equals 1 <br />
the matrix is on equal size to either side of this diagonal </p>
<p>of course its al beautifully counter intuitive </p>
<p>no wonder you gotta stick to <br />
the assumed set of rules <br />
to find your way thru </p>
<p>its a wonderland like ...errrr...<br />
real business cycle theory </p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eabe46e970b2011-12-12T19:38:45Z2011-12-12T19:38:45Zpaineap math high school stuff<p>ap math <br />
high school stuff </p>paine commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb80aed970d2011-12-12T19:38:21Z2011-12-12T19:38:21Zpainehopelessly wrong all countable infinites can be ranked and this over confident ignoramous has not only just shown us countable...<p>hopelessly wrong</p>
<p>all countable infinites can be ranked <br />
and this over confident ignoramous<br />
has not only just shown us countable infinites</p>
<p> these are all countable infinities of exactly the same size </p>Jacob Davies commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eab850c970b2011-12-12T19:01:55Z2011-12-12T19:01:55ZJacob Davies"the universe is considered to be spatially finite" ... no it isn't. The question of finiteness is unresolved but most...<p>"the universe is considered to be spatially finite"</p>
<p>... no it isn't. The question of finiteness is unresolved but most cosmologists I think would tell you that they think it is infinite. In fact they're more likely to consider time finite (since it looks a lot like it has one constraint at the Big Bang) than space.</p>
<p>Or there's David Deutsch's take, "other times are just special cases of other places".</p>Daniel commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543834f861970c2011-12-12T17:42:02Z2011-12-12T17:42:02ZDanielYou should have stopped reading Brian Greene's passage after the very first sentence "Comparisons involving infinitely large numbers are notoriously...<p>You should have stopped reading Brian Greene's passage after the very first sentence</p>
<p>"Comparisons involving infinitely large numbers are notoriously tricky,"</p>
<p>in which he claims there's such a thing as an infinitely large "number." Any mathematician will tell you the problem lies in infinity being a situation not a number. You can't add infinities, nor multiply them, as you do with numbers.</p>
<p>Physicists often refer to the cancelation of infinities, but this is a cancelation of terms which lead to infinities when uncanceled; or we refer to absorption of infinities by renormalization schemes, but these don't involve mathematical processes on infinities as if they were numbers. We use lazy language for rigorously developed mathematical concepts; but when pressed can clearly distinguish an infinity from a number.</p>Min commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eaac6e8970b2011-12-12T17:24:22Z2011-12-12T17:24:22ZMinBTW, the fact that the universe is expanding, and indeed, accelerating in its expansion, means that it may not be...<p>BTW, the fact that the universe is expanding, and indeed, accelerating in its expansion, means that it may not be so that every possible outcome will be realized infinitely many times.</p>Seth commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543834bc8f970c2011-12-12T16:59:14Z2011-12-12T16:59:14ZSethA deep question ... Brian Greene talks about this because (for example) quantum field theory is full of integrals which...<p>A deep question ...</p>
<p>Brian Greene talks about this because (for example) quantum field theory is full of integrals which diverge and have to be 'renormalized' by figuring out which kind of infinity is 'bigger' -- it's a reflection of the subtle tensions among our intuitions of space, time, energy, etc. It is generally resolved by using symmetries to arrive at plausibly 'natural' or reasonable ways to carry out the calculations. </p>
<p>A more homely, economic, example might be that 'infinite horizon' NPV calculations -- such as those presented to claim that the welfare state is "broke" -- are (intellectually) bankrupt. How big a number "all future liabilities of Medicare" actually is, requires a lot of information you simply do not have. Information like future rates of return on securities, growth of the economy, population growth, technological change, impact of climate change, possible impacts of wars, etc. etc. </p>
<p>To be clear, the 'bankruptcy' lies in pretending no choices were made about how to make the calculation. Physicists at least have to justify how they arrive at a computation and usually have far less impenetrable mysteries to contend with.<br />
</p>Min commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb6b274970d2011-12-12T16:50:49Z2011-12-12T16:50:49ZMinI found Greene's writing puzzling. Usually popularizers do not aim to confuse their readers. ;) As others have pointed out,...<p>I found Greene's writing puzzling. Usually popularizers do not aim to confuse their readers. ;) As others have pointed out, Cantor provided a solution over a century ago. I can only suppose that infinite cardinality does not answer the questions that physicists have about the measure problem.</p>
<p>Fortunately the Web has led me to a definition of the measure problem: </p>
<p>"We must also solve a long-standing fundamental problem in cosmology: how to define probabilities in an infinite universe where every possible outcome, no matter how unlikely, will be realized infinitely many times. This "measure problem" is inextricably tied to the quantitative prediction of the cosmological constant."<br />
( <a href="http://physics.berkeley.edu/index.php?option=com_dept_management&act=events&Itemid=451&task=view&id=735" rel="nofollow">http://physics.berkeley.edu/index.php?option=com_dept_management&act=events&Itemid=451&task=view&id=735</a> )</p>
<p>And indeed, as mathematicians well know, probabilities for actual infinite sets are not well defined. You have to make additional assumptions. </p>
<p>Since the universe is considered to be spatially finite, I suppose that they are assuming infinite time. The common sense solution, which is to take the probabilities of a finite interval of time as representative of the probabilities for all time, runs into the objection that the probabilities may be significantly different in the next interval, and so on. The probability that the sun will rise tomorrow is almost 100%. The probability that it will rise forever is zero.</p>Rick Schaut commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eaa769a970b2011-12-12T16:25:01Z2011-12-12T16:25:01ZRick Schaut"The really interesting seemingly counter-intuitive results arrive when you realize that any interval (a,b) has the same cardinality as the...<p>"The really interesting seemingly counter-intuitive results arrive when you realize that any interval (a,b) has the same cardinality as the entire real line!"</p>
<p>That's not quite right. The cardinality of the set of rational numbers in any given interval (a,b) (doesn't matter whether the interval is inclusive or non-inclusive) is the same as the cardinality of the set of all integers.</p>
<p>The cardinality of the set of irrational numbers, however, is not the same as that of the set of all integers. It's aleph-one, which means that, given an interval consisting of any two rational numbers, there is an uncountably infinite number of irrational numbers. Now, that's a concept that's difficult to wrap one's head around.</p>anne commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675eaa56da970b2011-12-12T16:10:21Z2011-12-12T16:10:21ZanneThe fact that Mark Thomas's and Brian Greene's conjectures about comparisons between infinite numbers stimulates interest in "high level" economics...<p>The fact that Mark Thomas's and Brian Greene's conjectures about comparisons between infinite numbers stimulates interest in "high level" economics blog is a result of the lack of rigorous math and science teaching beginning in elementary schools and continuing through Post Graduate Studies....</p>
<p>[Actually the conjectures are fascinating and deserving of significant consideration and discussion. By the way, to my delight, Brian Greene's discussion of physics and math on public television keep coming up on casual conservations I happen to listen to.]</p>Jacob Davies commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb66047970d2011-12-12T15:47:15Z2011-12-12T15:47:15ZJacob DaviesI note that none of the comments here dismissing this as kid stuff seem to understand why this is important...<p>I note that none of the comments here dismissing this as kid stuff seem to understand why this is important to physics or offer any practical advice for dealing with it. And I never found the basic treatment of infinities (high school level in the UK...) as anything other than some hesitant first step in explaining the various mysteries.</p>
<p>The problem is that our everyday experience of reality indicates that probability determines what events follow what other events. But in an infinite universe where all possibilities are realized an infinite number of times, how do you make comparisons like this? If I flip a coin many times, what should I expect to experience happening? Simple counting doesn't work, because occurrences in which the coin strictly alternates heads and tails or always comes up heads happen "just as many times" as the random outcome we expect to actually experience.</p>
<p>But by all means dismiss this as just another "bull session". I mean, it's just the question of how reality actually works - just a bunch of bull, basically.</p>Dirk van Dijk commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e2015438344de9970c2011-12-12T15:31:10Z2011-12-12T15:31:10ZDirk van Dijkinfinity = infinity = infinity not that hard to understand<p>infinity = infinity = infinity not that hard to understand</p>Jason Dick commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb61ee2970d2011-12-12T14:58:56Z2011-12-12T14:58:56ZJason DickIn defense of Mark Thoma, I have a Ph.D. in physics, and I never saw this particular mathematical situation inside...<p>In defense of Mark Thoma, I have a Ph.D. in physics, and I never saw this particular mathematical situation inside a classroom. It's one of those things that I learned instead from other students just chatting randomly about neat things in math. Given that physics is vastly more math-intensive than economics, it wouldn't surprise me in the least that even a successful tenured professor in economics may have missed it.</p>
<p>@eric,<br />
No, the even numbers aren't actually larger. If you only take a finite subset of the envelopes, this is true. But if you're taking the thought process to its infinite extreme, the set of all even numbers and the set of all natural numbers are exactly the same.</p>
<p>There are other fun things you can do with infinities. One classic example is Hilbert's Grand Hotel. If we have a hotel with infinite rooms, and every single room is filled, we can still add new people by asking everybody starting at room A to switch one room to the right. The new person goes in the room A which is left vacant, and now we have one more person and nobody is left without a room, even though the hotel was full to begin with.</p>Bill Jefferys commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb61432970d2011-12-12T14:47:54Z2011-12-12T14:47:54ZBill Jefferyshttp://bayesrules.netHow can it make a difference? In either case you end up with aleph_null dollars, since by the rules of...<p>How can it make a difference? In either case you end up with aleph_null dollars, since by the rules of transfinite arithmetic, 2*aleph_null=aleph_null+aleph_null=aleph_null.</p>mfasano commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675ea9d846970b2011-12-12T14:30:19Z2011-12-12T14:30:19ZmfasanoI agree with a. I think the number of subsets of the set of natural numbers is considered a higher...<p>I agree with a. I think the number of subsets of the set of natural numbers is considered a higher order of cardinality.</p>eric commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e2015438337098970c2011-12-12T12:40:53Z2011-12-12T12:40:53Zeric"the question of which of these kinds of infinite collections are larger"... is a dumb question, no? If they're infinite,...<p>"the question of which of these kinds of infinite collections are larger"...</p>
<p>is a dumb question, no? If they're infinite, they're infinite, and so the same; if they're not, then obviously the even numbers will be larger. The more interesting question is: why would this bug you all day?</p>David commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e2015438335fb0970c2011-12-12T12:22:27Z2011-12-12T12:22:27ZDavidIf the question is truly only the game show, then the answer is to take the doubled envelopes. You have...<p>If the question is truly only the game show, then the answer is to take the doubled envelopes. You have no position that the second choice will leave you worse of. The confusion of whether it is better or not does not matter, only the question of whether it leaves you worse off. The business world is full of similar examples (though not involving infinity).</p>foosion commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb53c0f970d2011-12-12T11:57:11Z2011-12-21T01:17:40Zfoosionhttp://profile.typepad.com/deltmilesGeorg Cantor analyzed these issues about 140 years ago and his results are very well known (at least at the...<p>Georg Cantor analyzed these issues about 140 years ago and his results are very well known (at least at the college math level).</p>John M commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb53649970d2011-12-12T11:50:32Z2012-01-17T05:42:40ZJohn Mhttp://profile.typepad.com/johnm307I agree with Bob. More generally, if you can't find some way to eliminate (or hide) the infinity, the problem...<p>I agree with Bob. More generally, if you can't find some way to eliminate (or hide) the infinity, the problem is physically impossible. So because of the finite time to open an envelope, you only have a finite number of envelopes regardless of the official rules of the game.</p>
<p>GA has a point as well. In fact, all the commentators here have valid points.</p>
<p>Electrodynamics (classical as well as quantum) has an interesting problem with infinities. As a charged particle moves, it drags its electromagnetic field along with it. The field exerts a drag force on the particle whenever it accelerates. The result resembles a mass for the particle, and necessarily has to contribute to the overall measured mass of the particle.</p>
<p>So the electromagnetic energy of the field is included in the mass of the particle -- and it doesn't matter that the electromagnetic energy is infinite.<br />
</p>GA commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675ea8c93a970b2011-12-12T11:07:45Z2011-12-12T11:07:45ZGAI think this question needs a monetarist riposte. If the participant won an infinite amount of money, it would presumably...<p>I think this question needs a monetarist riposte. If the participant won an infinite amount of money, it would presumably hit inflation, after all. What's the point of winning money if you can't spend it without causing ... Weimar?</p>reason commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201543832e728970c2011-12-12T10:56:57Z2011-12-12T13:27:17Zreasonhttp://profile.typepad.com/6p00e3982108db8833Why is this important?<p>Why is this important?</p>Narwhal commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb4ec04970d2011-12-12T10:56:45Z2011-12-12T10:56:45ZNarwhalI agree with Lee, a and David: The fact that Mark Thomas's and Brian Greene's conjectures about comparisons between infinite...<p>I agree with Lee, a and David:</p>
<p>The fact that Mark Thomas's and Brian Greene's conjectures about comparisons between infinite numbers stimulates interest in "high level" economics blog is a result of the lack of rigorous math and science teaching beginning in elementary schools and continuing through Post Graduate <br />
Studies.</p>
<p>As implied by the above comments this is the kind of discussion that we had in overnight bull sessions that I participated during my freshman or sophomore year in engineering school nearly 50 years ago. </p>
<p> </p>Bob commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675ea8b4af970b2011-12-12T10:47:18Z2011-12-12T10:47:18ZBobMath Shmath the envelopes may be infinite but your time on earth isn't. Take the deal that gives you the...<p><br />
Math Shmath the envelopes may be infinite but your time on earth isn't. Take the deal that gives you the most $$$ in each envelope. Besides paper cuts hurt and you'll minimize your blood loss with the higher value envelopes.</p>David commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb4728e970d2011-12-12T09:10:21Z2011-12-12T09:10:21ZDavidWasn't this covered in your introductory calculus class?<p>Wasn't this covered in your introductory calculus class?</p>a commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e201675ea83732970b2011-12-12T09:02:40Z2011-12-12T09:02:40ZaI think the standard argument is that any set that you can put in a 1-1 correspondence with the set...<p>I think the standard argument is that any set that you can put in a 1-1 correspondence with the set of natural numbers has the same cardinality as the set of natural numbers ("aleph-null"). this is just saying that all countably infinite sets have the same cardinality (and this cardinality is referred to as "aleph-null")</p>
<p>In your instance, applying this notion the set of even numbers (or odd numbers for that matter) all have the same cardinality as the set of natural numbers, viz. aleph-null.</p>
<p>The really interesting seemingly counter-intuitive results arrive when you realize that any interval (a,b) has the same cardinality as the entire real line!</p>
<p><br />
</p>Lee commented on 'Comparing Infinities'tag:typepad.com,2003:6a00d83451b33869e20162fdb446a2970d2011-12-12T08:46:31Z2011-12-12T08:46:31ZLeeIsn't this basic algebra taught in introductory real analysis or abstract algebra courses, which are almost mandatory for application for...<p>Isn't this basic algebra taught in introductory real analysis or abstract algebra courses, which are almost mandatory for application for economics Phd programs today?</p>
<p>I am a bit surprised that you, a prominent economist, did not ponder about this issue decades ago when you were 20. </p>
<p>It seems that the training for young economists was a lot different in your time compared to today's.</p>