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aleph_0

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Everything posted by aleph_0

  1. I have no idea what you're talking about. Whatever it is, it's certainly not what I was talking about.
  2. I know of none, but I also know of no reason to suppose that the universe did not have a first moment.
  3. If, by "will", you mean nothing other than an uncaused cause, then what about this concept implies the possession of consciousness? I grant that, if humans have free will (in any interesting sense of "will"), it must not be caused by external factors. However, that does not imply that every uncaused event is thereby an act of will like that of human will--unless you have a further argument to this effect. The problem seems to be that you start off by defining "will" in some terms which do not necessarily involve a conscious entity. You then proceed in your proof, now using "will" to imply the existence of a consciousness. So it would be better not to confuse the issue with use of the term "will". This would be like me saying, "By the word 'will', I just mean a rock. And there is clearly a rock, therefore there is clearly a will which is a rock, and therefore rocks have willpower." This is obviously an ill-formed argument, owing to my having started the proof using one definition of "will" and ended it using another. Just try to argue the case, starting from the assumption of an uncaused beginning, and concluding in the fact that this uncaused event implies the existence of a consciousness. Since you use the word "will" to just mean "an uncaused event", you shouldn't need the word "will" in the first place. But my point was that it is nonsense to claim that, if the universe did not have a will power, it would have remained static, because we cannot talk about reasons for why the universe is the way that it is. This is, as I've claimed, tantamount to giving reasons for why there is something rather than nothing. Such reasons are impossible, and claims about why the universe is the way that it is are just as impossible. The universe just is; and the universe just is the way that it is. There are no intelligible questions or reasons antecedent to these facts. I cannot see the relevance of the quote, nor do I see the force of your argument. Firstly, I do not believe there is a problem with thinking that there are infinitely many things in existence, contrary to what most people on this forum seem to believe. But even if we assume that there are only finitely many things in existence at any one time, that does not mean that there cannot exist infinitely many times at which each of these things exist. The only sense in which "effects exist" is in the sense that an event occurred in virtue of certain facts just before the event. For instance, when I say that the effect of the ball was the breaking of the class, I am not saying that there is a ball, and some glass, and an effect. I'm just saying there is a ball, and some glass, and at one time the ball was headed toward the glass, and at another time the ball passed through the glass. But there are only two objects--the ball and the glass--in this conversation, not three--the ball, the glass, and the effect. I don't accept that counting effects is a way of counting things that exist in the universe (in the way same way that a ball exists in the universe). So I'm afraid that I still don't accept that there must be a first point in time, or a beginning of the universe.
  4. Any book on abstract algebra will cover those. Dummit and Foote have a highly-regarded one, and one very, very smart friend of mine swears it's the only way to do algebra. I used Fraleigh and it was horrible, so don't use that one. My professor liked Ardin's book, though I wasn't thrilled about it, and Serge Lang's is very intense, but if you like a small, dense collection of proofs then that's the way to go. It's probably a good idea to use a variety of them.
  5. Without getting into another argument about the (lack of a) relationship between identity and infinity, this is similar to what I was saying.
  6. Right, I thought that might be the passage referred to. I find it a little indirect since she speaks of actual quantities being finite, which might imply that she's speaking of quantities at a time, and so this would not have bearing on the question of whether time can fail to have a beginning point.
  7. To use the multiquote feature, click the button corresponding to the message(s) you would like to quote in your response. You can even browse to other pages in the thread, and you can even browse to other threads, and the multiquote feature will keep tracking you. When you finally click "Add Reply" at the bottom of any thread, you will be taken to a new text editor box, and the quotations will already appear in the editor with "quote tags" around each message you've decided to quote. If you want to insert extra quote tags, you can simply put text around the string of symbols [ q u o t e ] and [ / q u o t e ], where all spaces are omitted. To play with this, in order to see how it works, see what happens when you post the following text, removing the appropriate spaces: This is a [ q u o t e] quote [ / q u o t e ] Note that you can always post something and then edit it within a few minutes of having posted it, so in order to keep the forum tidy, please edit out your experiments with the quote tags and only leave your response.
  8. Oh well, mods can remove the topic, then. Still, I'm surprised at the largely non-negative reaction to the article.
  9. I don't see why, depending on what you count as a "thing". What is "self-destructive" about it? I take it you mean either self-contradictory or leading to an infinite regress, when you say "self-destructive", but I'm not sure what is so bad about an infinite regress in certain contexts. Clearly we cannot have an infinite regress when we talk about reason because man's mind is limited. If reasons for action or belief were infinitely supported by other antecedent reasons, then no person would ever have any reason and the term would be meaningless. In the case of time, however, I see no reductio ad absurdum in supposing that time regresses infinitely into the past (and future). If this is what they say, I would be interested to see a quote from Ayn Rand, but that's mere curiosity. In any case, without some additional argument, I don't accept this premise. I'll make a subtle point, but here is the only thing that I take to fit loosely what you are talking about: The fact that the universe exists is something that does not need justification. This is, in some sense, a bottom-line beyond which any reasoning makes no sense. We cannot talk about the cause for the fact of existence, and so in some sense this is uncaused--but this point makes no claim about time, and still recognizes the possibility of an infinite past. In the limited sense of an "uncaused" existence that I described above, this does not follow, so I still reject the thought that there was any kind of "act" involved. By the very notion of the fact that it is uncaused, this implies that nothing caused it to do anything. This means that no kind of willpower could have caused the universe to act, since presumably this willpower would be a thing which causes the universe to act, contradicting the supposition that it is uncaused. To this you might reply that willpower isn't really a thing in the universe, but at this point I think, "Then what's the difference between this statement, and saying that the willpower just doesn't exist at all?" Now if you just mean, by "will", the act of doing something which is not at all caused or incited by any antecedent facts, then this might be called an act of will, but there is no reason to think that it was a conscious will, i.e. that this quote-unquote "willing act" is anything like the willpower that humans disputably possess, since there is nothing to suggest that this act of "will" was guided by reason or conscious act. This is the question of, why is the universe the way that it is, rather than some other way? As Ayn Rand herself stated, and I agree with her: This question deserves the same answer that the question, "Why is there something rather than nothing?" Namely, all reason is based on the fact of existence, and not just any existence but this existence. We cannot step outside of reality in order to answer questions about reality. The existence, and the form of existence, of the universe is primitive, from which other questions may be answered. Hope this has helped clarify, or at least provided some new thoughts.
  10. Well if you want to get snippy about it, to be honest, your argument is to blame. Write me when you have something intelligible, and lacking so many errors. If this is intended as a response to me, I'm sorry, but I'm not going to read something that makes its point in 10+ paragraphs of children's historical fiction, when it could be made in a paragraph of argument. If this isn't directed at me, then nevermind this post.
  11. Note the acknowledgments: The authors are Neera Badhwar and Roderick T. Long, neither of whom I am familiar with, and both of whom have tangential interest in Rand, though I don't think either are actually Objectivists.
  12. I just saw that about a month or two ago, SEP added this article!: Ayn Rand This has been in the works for some time, I think as long as a year ago there was a promise on the website of adding this article. Note that SEP is a highly regarded collection of topic-specialized articles, written by prominent modern philosophers. So what they say about her is probably going to rouse Objectivist ire.
  13. I'm not convinced of this, myself. I don't see why we should identify this first act as an act of will, if there were a first event. What is there that is special about an act of will that allows it to be a first-event when other inanimate events cannot be? Why could there not have been merely a first moment of time, and everything followed from it, lacking any act of will the whole time? I'm not sure why it couldn't be an entity. I'm not sure why it couldn't be both an entity and the collection of all entities.
  14. The original poster seemed confused about the matter, thinking it of philosophical significance. And indeed, while the most immediate answer to the question is purely mathematical and not philosophical, the answer gives rise to a natural philosophical question. Well, that depends on what you think the usual 0 and 1 are. If you mean the integers, then in a sense I would agree with you. As I've said repeatedly, the natural numbers are for counting (finite quantities). However, I think most mathematicians think of 0 and 1 as complex numbers. So it kind of depends. What do you mean, "need"? The point is that you can approximate it, and that this relatively trivial example extends to less trivial cases like .333..., .314..., and others. If you don't have any need for approximating it, then don't. A great thing about math is, you not do it if you don't want to. Don't care what 0 + 0 is? Fine. Don't add them. However, since the real numbers are often used to measure, it can be quite valuable to know and easy to calculate the limit of a sequence. For instance, you may know that the function f(x) describes the acceleration of an object very well, and you would like to know [f(x) - f(x_0)]/[x - x_0] at some point x_0, for values of x arbitrarily close to x_0, i.e. the derivative of the function at the point, which describes the instantaneous velocity, you will want to take the limit as x goes to x_0. This can be understood as finding the value of the fraction at points x_1, x_2, x_3, ... where this sequence approaches x_0. Perhaps x_0 is 1 in this scenario, and good choices for x_1, x_2, x_3, ... happen to be .9, .99, .999, ... Well that's no problem, because luckily we now know that .999... converges to 1. Whew! Good for us. I suspect you don't understand that the equality is not a convention, it's a genuine result. And I'm not sure what "precision" you're referring to. If you mean the approximation of 1 by a sequence, even the approximation is precise in the sense that it converges to the number 1, and even does so monotonically and we have very precise bounds for the difference between the nth term of the sequence and the point to which it converges. But you seem to be talking about precision in meaning, and on that account everything is perfectly precise. So I'm not sure what you want. Every term is defined, every operation explicit. [Edit: added my penultimate paragraph.]
  15. I'm not sure how this suggests that we don't have a firm grasp of what the real numbers are.
  16. Oh noes, it's Evader Hollyfiled! No, I'm not protesting, I'm just mocking you to show the silliness of your tactic. The tactic is as I stated: You cannot support your claims, and so you call someone an "evader". You might as well just say, "Nuh-uh! Noo! NNNOOOOO!" If you cannot provided an argument--and I'm certain you can't--then don't ruin the forum by resorting to name-calling and ad hominem accusations. Now stop evading your evasion and recognize that you're an evader! Darth Evader. That wasn't even the point at issue, and at best is done in the surreal numbers, though even then it's not clear that's what is going on. The point at issue was why .999... = 1. We've now side-tracked into whether there can be infinitely many things, but I have explained what it means to say that a thing has an infinite quantity--namely, there being a one-to-one correspondence with the natural numbers--so the only question is whether this is possible, and I've seen no statement that indicates a negative answer. Calculus has been alternately constructed using infinities and infinitesimals to produce all of the exact same results, i.e. there is an isomorphism between the mathematical structures, so the difference between the two is name only. Moreover, other branches of mathematics have employed the use of infinities and infinitesimals. Whoa, going off the deep end a bit? Are you ready to deny the law of excluded middle in order to hand on to the idea that the universe is finite? Here is the exact definition: Standing in a injective relationship with the set of the closure of 0 with the successor function, i.e. that there is an injection from the natural numbers to the other set (the set that we are calling infinite). This has the property that, for any natural number n, there is no injection from the set {1, ..., n} to the infinite set, nor is there a surjection from the infinite set to the set {1, ..., n}. It has the property that, given a homomorphic map, injectivity does not imply surjectivity or vice versa (this is not the case with finite sets of equal size). It has many others. I'm not sure if there's one in particular that you're looking for. Arithmetic is defined only on the natural numbers, and aleph_0 is not a natural number, so you would have to extend the system of arithmetic in order to account for such a number. So arithmetic is used only for operations on finite quantities. This in no way implies that infinite quantities do not exist. However, here are some properties that may answer your question: For any set S with cardinality aleph_c which is greater than or equal to aleph_0, and any element r which is not in S, {r} union S has cardinality aleph_c. (This is analogous to addition since the new set has "one extra".) A similar statement is made for any element s in S and the set S \ {s}, which is analogous to subtraction. However, the power set of any set is strictly larger, thus the cardinality of P(S), which is analogous to raising 2 to the power of the cardinality, is aleph_c+1.
  17. This later argument is perhaps the most persuasive, though I am not certain we will never find evidence that the universe is infinite. For instance, we have evidence to suggest that the cosmic background radiation has a kind of reverberation due to the Big Bang, and the direction of the ripples indicates the place where the bang occurred. If we were to find that the reverberation had the same frequency at every point in the universe, this could plausibly suggest that the "beginning" of the universe is infinitely far away from every point, and from this we could possibly argue--if we satisfied certain other logical conditions for the argument--that there are at least aleph_0 electrons in the universe. But until any such evidence is found, I am merely stating that we cannot rule out the possibility, which is what has been at issue all along. I can deny it--it's intended, here, to understand why they should be identified with each other in mathematics. I chose the phrase "suggestive notation" because we use the phrase all the time in mathematics, such as when we discuss some particular field like integers modulo 5, and denote 0 as the additive identity and 1 as the multiplicative identity. Even though we are not dealing with the usual 0 and 1, we suggest that they share a kinship because when you "add" the "0-element" of the system to any other element, your operation just returns the other element. Here the notation suggests that .999... (where this is understood as the limit of a sequence) can be approximated by arbitrarily many decimals, and that the limit of a particular sequence is 1. There is nothing sneaky about it. Why would this be true? Why couldn't there be an infinity of many things? Why can I not simultaneously hold the assertion "A equals A", which is the statement of the law of identity, and the assertion "There is an infinity of electrons," which is a very different statement with seemingly no logical connection between them? The connection between the two is distant at best, so to treat it as if the latter is somehow obviously false because of the former is plainly ridiculous. Yes, a thing's identity has to be defined, but nobody has given any reason to suspect that infinity is not defined any less than "finite" is defined. Note that we are not talking about a jar when we talk about the universe because the meaning of "a jar" precludes its being infinite and so having finitely many things with some finite volume (I assume electrons have some non-zero volume, but if not we could easily amend the example to talk about marbles). Part of the concept of a jar is that it encompasses a finite space. The universe is just the collection of all things that exist, and there is nothing inherent in that notion which precludes an infinite quantity of electrons contained therein. Also note that finite does not mean "no specific amount". It means "without limit to its amount", and so implies that there is an injection from the set of natural numbers to any infinite set. No notion here is ill-defined, every term is specific. I was hoping to avoid this discussion by talking about the number of some particular kind of object like electrons or marbles.
  18. So are they entities or do they refer to entities? And in either case, how did you come about this information? Now perhaps if I were claiming that I have some evidence that the universe is infinite, which I don't, you could claim that I'm making something like the error you accuse me of. But I'm not, and nothing you've said has yet indicated the impossibility of infinite quantity. To illustrate: I observe that all ravens are black. I come to the reasonable judgement that all ravens are black, though I realize that I might one day find a non-black raven. None of this, however, even suggests that a white raven is impossible.
  19. I'm not so sure about that. Some of the people here seem to think that mathematical entities are real, or intrinsically refer to some real-world objects and therefore mathematics itself must be finite. But if that's not the case, all the better. No, you're evading! And if you say you're not, then you're evading that you're an evader! ... This tactic is childish. I've asked for an argument, and I've received two responses. First, an attempt at an induction, and I pointed out how this fails. Providing a counter-argument is not evading anything--it's pointing out that the argument provided is insufficient. If you insist on evading this fact, so be it. You evader. The second response was just a re-insistence that nothing can be infinite, but this is obviously not an argument. I am talking about mappings between natural numbers and existents. Here is an instance of such a mapping: From the set of numbers {1, 2, 3} I map 1 to me, 2 to you, and 3 to Grames. In fact, this is a one-to-one correspondence with the set {me, you, Grames} because it is surjective and injective, and thus the two sets have the same size, i.e. the quantity of things in each set is the same. Thus when I speak of the the universe being infinite, I speak of the set of, say, electrons being in a one-to-one correspondence with the set of natural numbers. There's nothing that anybody has said that makes the coherence of such a notion even dubious. Aleph_0, which is the cardinality of the natural numbers, is less than aleph_1 (and aleph_1 < aleph_2 < ...), so problem solved. Maybe not since I have no idea what "pointing" has to do with anything, but I have a clear idea of what "infinity" means and there is nothing contradictory about the hypothesis that the universe contains infinitely many objects. You're not just an evader, you're a circular evader! (Sorry, I just find the Objectivist tendency to yell "evader!" when they can't support their claims to be extremely, extremely hilarious. And only something dirty evaders would do.) Again, this is just not the case, as I've explained with another example of the limit of a sequence of elements. Just because a thing is the limit of a sequence does not mean that the thing is itself somehow infinite. It is not as though we have a piece of clay and we add on infinitely more, infinitely smaller pieces of clay to it. The limit of a sequence is a single thing, and there are just an infinite number of other elements that are arbitrarily close to it. Just because we represent this element by a suggestive notation that contains, in some sense, an infinite string of characters does not mean that this element is any less a number or well-defined element in a mathematical system. Now the real number system is not a system that is appropriate for quantifying things, in any sense other than the fact that the natural numbers "live inside" the real numbers. But that does not mean they're not numbers, and it doesn't mean they're ill-defined.
  20. Unless the universe is infinite. Nothing contradictory about that.
  21. No, quantity just means "how many", or to be more mathematical about it, to be in an equivalence class defined by standing in a bijective relationship. There's nothing contradictory about the idea that I could put the natural numbers in a one-to-one correspondence with some collection of physical objects.
  22. I see that the thumbnails posted here don't look like the current text editor, so I was wondering if there is anything currently in the text editor that produces more non-standard (and, in particular, mathematical and logical) symbols. I'd like to point out, if there isn't already something which handles this, that www.mathhelpforum.com uses a text-editor which takes commands like those in LaTeX, and since that is so ubiquitous in the sciences, and even for some who are not in the sciences... You know. That kind of thing could be free. Would be nice. Just sayin. Below, I'm going to experiment a bit with some attempts at doing just this, so ignore it. ∞ ∞ What is a marquee? &#10132
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