wtf

You struggle with logarithms but want to solve the Riemann hypothesis?

Yes that's a good point. In fact on checking the Wiki article my proof sketch is a little inaccurate. When I said to consider all the well-ordered subsets, the question is, which well-order? The answer is that we actually Zornify on the collection of all possible well-orders on all subsets of the reals. It's a minor but important modification of what I initially said. The OP missed a chance to call me out on a nontrivial inaccuracy but didn't. Wonder if he's gone. Too bad, I love a nice chat about Zorn's lemma, well-ordering, the axiom of choice, and other aspects of nonconstructive math. But yes, as you note, the well-order of the reals is decidedly NOT the usual order.

People who "really know this stuff" know that you're wrong, since they know the proof of the well-ordering theorem. That leaves those of us who know this stuff but also enjoy sparring with cranks alternative thinkers. So post your proof. Or don't. It's wrong regardless. But speaking for myself, I'd like to see your proof. Especially if it's only four lines. By the way here's the proof sketch of the well-ordering theorem. Consider the collection of well-ordered subsets of the reals. Verify that this collection satisfies the hypotheses of Zorn's lemma. Zorn then asserts the existence of a maximal well-ordered subset of the reals;, and this must in fact be the entire set of reals. Done.

It's by no means a mathematical question. Mathematically, zero is finite. This is a semantic question, concerning how people, even mathematicians who know better, use words in casual conversation.

Not understanding the question. The successive differences of a polynomial are eventually all zeros. I can't relate this to the subject of the thread, which is the casual semantics of distinguishing zero from finite.

No. The empty set is finite since it's not in bijection with any proper subset of itself. That's because it has no proper subsets. So it's Dedekind-finite. And it's in bijective correspondence with a natural number, namely zero. So it's finite in the classical (non-Dedekind) sense. It's of interest that absent the axiom of choice, there are infinite sets that are Dedekind-finite. That is, they're not bijectable to any natural number, and they're not in bijection with any proper subset of themselves. These are very weird sets. The only way we can say zero is finite is by abuse of terminology in casual usage. In fact in any written work we shouldn't even use the the incorrect terminology to make sure nobody is confused.

At first I said, "Of course zero is finite." But people gave good examples of casual usage in which finite means nonzero. I realized that when something has a small but nonzero probability, I'll say it has a "finite probability," meaning it's not zero. Interesting semantic point!

I apologize if my attempt at lighthearted hotel-related mathematical humor struck you as diminishing or denigrating the importance of your project. But if a guy named Cantor shows up, look out.

When I saw the title I thought this was going to be about Hilbert's hotel. This becomes a more interesting problem if you have infinitely many arrivals and you don't want to force guests to move too much as new guests arrive.

I'm curious as to why someone who knows forcing would ask this question on this site rather than, say, math.stackexchange where it will get knowledgable responses. And secondly, surely you know that V is not a model of ZFC. V is a proper class, hence not a model of anything. And if we knew ZFC had a model then we'd have a consistency proof for ZFC, which we haven't got. A strange post.

AC applies to finite collections of sets, but it's not needed for finite collections of sets. That is: If we have one nonempty set, we may choose an element from it, since the set's nonempty. If we have two nonempty sets, we may choose an element from each. We can always choose a an element from each of a finite collection of sets. (The individual sets themselves may be finite or infinite, makes no difference). It's only in the case of an infinite collection of sets that we need AC to prove that we can simultaneously choose an element of each set. In other words AC applies to finite collections, but it's not needed. It's using dynamite to kill a fly. Effective, but overkill.

There's an instruction in a thread called "All about quotes" or something like that, saying to click an icon that supposedly appears right about the B in the markup row, but that icon doesn't seem to be there anymore. Is there a way to edit at the BBCode level?

This working? $e^{i \pi} + 1 = 0$ [math]e^{i \pi} + 1 = 0[/math] [tex]e^{i \pi} + 1 = 0[/tex] Aha. [math]e^{i \pi} + 1 = 0[/math] 'math'

Yes, here is the bottom-up proof you are looking for. Modern math rests on first-order predicate logic. [All references below] Above that, we assume the axioms of Zermelo-Fraenkel set theory (ZF). In particular, we assume the Axiom of Infinity which says that there is an infinite, inductive set. Given this foundation, we can let 0 be the empty set, and for any set X, the successor of X is defined as X unioned with {X}. The resulting collection is an inductive set as given by the Axiom of Infinity, and it's a model of the Peano axioms of the natural numbers. Part of your confusion is that the Peano axioms are NOT the axioms for the real numbers. They don't apply to the real numbers. There are no successors in the real numbers. You can't apply the Peano axioms to the real numbers. Also, the Peano axioms are not strong enough to model the real numbers as there are no infinite sets in the Peano axioms. There are infinitely many natural numbers 0, 1, 2, 3, ... but there is no completed set of them. For that, we need the Axiom of Infinity. In fact it's precisely the Axiom of Infinity that makes the conceptual leap from Aristotle's potential to actual infinity. The Peano axioms give us potential infinity. The sequence 0, 1, 2, 3, 4, ... is never completed, but can always be extended. It's the Axiom of Infinity that says we may collect ALL the natural numbers into a completed infinity. Thus the real numbers require more powerful axiomatic principles than do the natural numbers. Once we have the natural numbers modeled within ZF, we can construct the integers (positive and negative naturals and 0). Then we can construct the rationals via a construction known as the fraction field of an integral domain. Then we construct the real numbers as the set of limits of sequences of rationals. This is traditionally done via a construction like Dedekind cuts or equivalence classes of Cauchy sequences. There are alternate constructions besides these. The point of all this is that we can build up a model of the real numbers within ZF. That is, the reals are an example of a complete, ordered field. Complete in this context means that there are no "holes" in the real numbers. Every nonempty set that's bounded above has a least upper bound. It's the completeness property that makes .999... = 1 work. In fact 1 is provably the least upper bound of the set {.9, .99, .999, .9999, ...}. Once we do that, the traditional proof via infinite series given by Uncool a few posts back is valid and is now proven from first principles. That is, if you'll grant me the empty set and the axioms of ZF, along with the inference rules of first-order predicate logic; along with the definition of the limit of a sequence, and the definition of the sum of an infinite series; I can then prove rigorously from first principles that .999... = 1 One more thing I wanted to mention. There is no smallest positive real number. Indeed if you claim that x is such, I'll just give you back x/2, which is a smaller positive real. Likewise the expression .000...1 makes no sense. The reason is because of the definition of decimal expressions. Each decimal position is indexed by a positive integer. We have decimal places 1, 2, 3, 4, ... There is no "last" decimal place for the exact same reason that there is no last positive integer. You can always take the successor of a natural number. There's no largest natural number and there's no "last" decimal position, since the natural numbers index the decimal positions. https://en.wikipedia.org/wiki/First-order_logic https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.wikipedia.org/wiki/Axiom_of_infinity https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Field_of_fractions https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach https://en.wikipedia.org/wiki/Least-upper-bound_property

Factually false, read the two threads. People who understood me may be in the minority but the set's not empty. I won't be posting further on this topic.

I'd like to hear from the moderator, not just the critic. I get that the critic stated his case. I am curious to know why the moderator bought it. Thanks for the kind words. If I had it to do over again I'd have either made my point in a much more literal way, or else said nothing at all.

Of course, and you had no problem understanding that. I HAD to. The subject was monkeys. I wanted to make a point that evolution managed to produce a primate who typed out the works of Shakespeare. To do so I had to speak casually and conflate primates and monkeys and ignore the technical details of evolution. I had to ignore the technical point in order to make the real point. That evolution did what probability never could. Someone missed my point so completely, and took my post so literally, that they fired off an indignant message to a moderator, who decided to go along with the deliberate misreading of my post. I have no use for people who are either so literal they can't understand the larger point; or who PRETEND to be so literal in order to be able to reel off a sequence of gotchas, none of which are on point to the sense of what I wrote. People like that, I have no interest in responding to. My estimation of that individual stands as stated. I'd like to hear from the moderator who moved my post. Did you honestly think I was suggesting a specific mechanism for evolution? That that was the *point* of my post? As opposed to simply noting the interesting fact that evolution provides a far better mechanism for writing plays than randomness does? Why did you choose to elevate the complainant's disingenuous literalism to forum policy?

Nice talkin' to y'all over the years on this site. I'm probably going to get banned by humorless scolds who can't read through a tongue-in-cheek post to the deeper more serious point, which I then explained in detail. All the best.

You're an idiot.

Math markup works fine for me on this site. [math]\mathbb C[/math] renders as [math]\mathbb C[/math]. Doesn't work for you?

I recently made the observation, in a thread about the odds of monkeys eventually typing out the works of Shakespeare, that this has in fact already happened. That human beings, according to the theory of evolution, are in fact monkeys who evolved to write the complete works of Shakespeare. My post was moved to the Speculations forum. A post from one of the moderators says: Please note that all posts that are baseless in scientific fact or that are outside of mainstream physics can and will be moved to the Speculations forum. Make sure that you think about the nature of your post before you hit the "post" button. I must say that I am puzzled. Evidently some moderator on this site feels that discussion of evolution is "baseless in scientific fact." I would like to know, so that I can avoid running afoul of the forum rules in the future, what exactly is this forum's position on evolution? If I say that primates on earth used to live wild in the woods and eventually, over millions of years, evolved to write the complete works of Shakespeare, is that idea regarded as being outside of mainstream science? If so, I apologize and I affirm that the world was created in six days by God, and that He impregnated a woman who gave birth to the Baby Jesus, to whom I swear my total and undying devotion. I reject evolution and all heretical notions that man evolved from monkeys. I trust that I am now back in the good graces of the moderator who labeled as "baseless speculation" my suggestion that humans are monkeys who learned to type. It's not my fault, I learned these heretical ideas in the public schools, from which religion has been ruthlessly expunged by the Godless minions of Satan and the United States Supreme Court. See Engel v. Vitale, 370 U.S. 421 (1962), in which the court "ruled it is unconstitutional for state officials to compose an official school prayer and encourage its recitation in public schools." https://en.wikipedia.org/wiki/Engel_v._Vitale. May they all burn in everlasting hell. Yours in the Brotherhood of Christ and against the evil Darwin and his deluded followers.

I must say I'm puzzled that my post about evolution would be moved to Speculations. Did this become a Creationist website? All praise the Baby Jesus then, and God who created the world in six days and rested on Sunday. So if everyone is so sure my thought experiment is so unrealistic, then how to you explain the fact that it already happened? Are you saying it's too big a leap to start the monkeys on typewriters, and we should start them out making marks in the sand with a stick? That's probably how we got started. But that's not an objection to my point, it's just a refinement to the experiment. Ok we start them out with sticks and reward them for making meaningful marks. Eventually they'll invent typewriters, just like we did. What, you think our typewriters fell from the sky? How does everyone suppose the primates Homo Sapiens managed to write the complete works of Shakespeare in only 14 billion years since the big bang? I'm simply astonished to find people on a science forum taking exception to the idea that a group of monkeys would evolve written language. Have the public schools gotten this bad?

Jeez man that's not right. What is the dimension of [math]\mathbb R^2[/math]? If you're thinking of [math]\mathbb R^{\mathbb N}[/math], the direct product of countably many copies of the reals; or else perhaps [math]\oplus_{\mathbb N} \mathbb R[/math], the direct sum of countably many copies of the reals, both of those are infinite-dimensional vector spaces. The direct sum is the set of functions on the naturals that are zero at all but finitely many places. The direct product doesn't have that restriction. It's like the distinction between the collection of formal polynomials versus the collection of formal power series. The direct sum has countably infinite dimension. The direct product can not possibly have countably infinite dimension. Why is that? Because linear combinations are defined as being finite. So the obvious basis [math]\{e_i\}[/math] doesn't work for the direct product, while it does for the direct sum.

I just skimmed through the topic of one-dimensional random walks on the integers over at Wiki and although I don't have any more insight into any of this, I did discover a whole lot more things I'll never know anything about. https://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk

I had this crazy thought and this is a thread hijack so please feel free to not read this. But I had this thought. You get some monkeys and some typewriters. Or even better, an early version of a direct neural interface! The next generation of brain sensors and software is going to be incredible. So you feed the monkeys a boring, bare subsistence diet. But anytime they type (or think) a coherent word, or even a word fragment, they get a reward. A banana chip or a strawberry or something. And now here is the idea. We always think of monkeys typing forever as if the monkeys are immortal. But instead, what if the monkeys are allowed to breed too? We have a whole colony of monkeys, and they each have an eight hour work shift, an eight hours sleep cycle, and eight hours of home life. Cable tv, sex, whatever monkeys do. You'll note that what I described is no different than the life of the average middle class worker in our society. But I didn't come here to talk about sociology so I'll get on with it. Now: Over multiple generations of breeding, what happens? Well, the monkeys who get more treats have higher status in the tribe, and therefore higher reproductive likelihood. So we would breed monkeys to type word fragments ... then words ... then sentences. The question is, Do you think the monkeys would get smarter over the years, and in a few generations or a few hundred, the would learn how to think abstractly ? Just like us. Oh thanks for listening to my idle thoughts for the day. But really ... this is the deeper meaning of my earlier question. How did a monkey named Shakespeare evolve to create his works? What are the odds of that? And what does it mean that it happened to us? That's the essence of the question about the monkeys. It's about how we ourselves managed to type out the complete work of Shakespeare in only 14 billion or so years since the big bang.