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GrandMinnow

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Everything posted by GrandMinnow

  1. You probably mean not to include that "within" clause. Every bounded set has a least upper bound (the least upper bound does not have to be a member of the set).
  2. I didn't object to his terminology. I just wished to say that, as far as I understand Objectivism, it doesn't reject the notion of sets. While, of course, as I've said, I do understand the point that Objectivism would not accept that such abstractions as sets have existence independent of consciousness. And, yes, I quite understand your own point in that regard too.
  3. I read where she uses the term 'mental entity' and says that concepts are mental entities. I don't wish to insist that Objectivist terminology does or does not prefer 'mental existent', but reading Ayn Rand in a straightforward manner would seem to allow using the term 'mental entity', just as she mentioned that concepts are mental entities.
  4. (1) Not an attempt at a "gotcha". So that the poster may respond to clarify, I merely highlighted that the poster's own argument seems to go in the opposite direction he seems to intend. (2) I've already said that, as far as I understand, Objectivism holds that a mental entity is not independent of consciousness. (3) I'm not a mathematician. (4) I don't know what contradiction you believe you have found in my posts.
  5. According to Ayn Rand, concepts are mental entities. If, as you say, sets are concepts, then sets are mental entities.
  6. I already mentioned that it is not at issue that Objectivism does not allow mental objects to be taken as independent of consciousness. But non-Objectivist mathematicians too would ordinarily allow that sets are mental objects or conceptual objects and not concrete objects, and even some mathematicians may deny that sets are platonic objects. So I don't see the point in saying that Objectivsts deny that there are sets when talking about the comparison between the ordinary mathematical notion of sets with an Objectivist notion of set (unless one take "ordinary mathemtatics" to include a platonic commitment, which is sticky question, since probably most mathematicians would be platonists or lean toward platonism if they were finally pressed to take a philosophic stance, though one might take the position that doing mathematics does not require a decision on that philosophical question let alone a decision in favor of platonism).
  7. From what point of view? From the point of view of ordinary mathematics, the answer is that the cardinality of the set of points in the interior of a unit disc is the cardinality of the set of real numbers. I don't know what the answer would be for an Objectivist.
  8. I'm just referring to the Objectivist acceptance of the notion as described in such quotes as in the Lexicon: There is a use of [the concept] “infinity” which is valid, as Aristotle observed, and that is the mathematical use. It is valid only when used to indicate a potentiality, never an actuality. Take the number series as an example. You can say it is infinite in the sense that, no matter how many numbers you count, there is always another number. You can always keep on counting; there’s no end. In that sense it is infinite—as a potential. But notice that, actually, however many numbers you count, wherever you stop, you only reached that point, you only got so far. . . . That’s Aristotle’s point that the actual is always finite. Infinity exists only in the form of the ability of certain series to be extended indefinitely; but however much they are extended, in actual fact, wherever you stop it is finite.
  9. In Objectivist writings there are occasional uses of the word 'set' in the everyday sense of a collection. I don't know why you say Objectivism rejects that there are such things, even if those things are what you call 'epistemic artifacts'. Is there a passage in Objectivist writing (especially OPAR or ITOE) where I can see that Objectivism rejects that there are sets? And, again, I understand that Objectivsm would reject that there are such things without consciousness to conceive them. That point is not at stake.
  10. I have no objection to there being sets of greater and greater infinite cardinality. But, as far as I can tell, Objectivists tend not to look kindly on the notion. First of all, Objectivism rejects that there are actually infinite sets, let alone actually infinite sets of greater and greater cardinality. But even more fundamentally, it's likely that an Objectivist notion of 'set' differs from how set theorists regard the notion of 'set'. (On the other hand, Steven Simpson is an Objectivist and a very prominent mathematician and logician who works with classical mathematics (which includes set theory with its actually infinite sets and sets of greater and greater infinite cardinality). From my oral conversation with him, it's not clear to me how he reconciles Objectivism with set theory; and indeed he mentioned that he thinks there is work to be done to produce a more robust Objectivist account of mathematics.)
  11. In most succinct terms, your claim that an Objectivist's acceptance of the law of excluded middle demands acceptance of greater and greater infinities is incorrect. First, the technical matter: Here is the ordinary proof of Cantor's theorem: ['P' for 'power set of' and '~ ' for 'it is not the case that'.] Claim (Cantor's theorem): For any set S, there is no function from S onto PS (i.e. there is no surjection from S onto PS). Proof: Let f be any function from S into PS. Let D = {x | x in S & ~ x in f(x)}. Suppose D is in the range of f. Then, for some x in S, we have f(x) = D. But if x is in D then x is not in D, and if x is not in D then x is in D, a contradiction. Therefore, there is no x such that f(x)=D, i.e., D is not in the range of f. But D is in PS. So f is not onto PS. Since f was an arbitrary function from S into PS, there is no function from S onto PS. No use there is made of the law of excluded middle or double negation or intuitionistically invalid proof by contradiction. Rather, the intutitionistically valid form of proof by contradiction was used. And of course, it follows that card(S) < card(PS), since there is the obvious injection from S into PS but (as just proved) no surjection from S onto PS (perforce no bijection between S and PS). (Recall that card(S) < card(PS) if and only if there is an injection from S into PS but no bijection between S and PS.) In the proof, we let P stand for "D is in the range of f", then the argument is of the form: Suppose P. Then a contradiction. Therefore ~P. And that is intuitionistically valid. Or, suppose we wish to put it in terms of card from the start: Claim: card(S) < card(PS). Proof: Recall that cad(S) < card(PS) if and only if there is an injection from S into PS and there is no bijection between S and PS. Now, there is the obvious injection from S into PS. So we need only show that there is no bijection between S and PS. By the proof above in this post, there is no surjection from S onto PS. Therefore there is no bijection between S and PS. Therefore, card(S) < card(PS). Again, intuitionistically valid. aleph_1, on 08 Feb 2013 - 10:37: "However, this is not the point of this post. Since objectivists reject concepts that have no reduction, it follows that objectivists must reject as philosophically invalid the Infinity Axiom" I don't contest one way or another that Objectivists may wish or need to reject the axiom of infinity. More fundamentally, Objectivists may reject even the very framework of set theory itself so that the axiom of infinity would be rejected perforce along with the entire set theoretic framework (but that is for Objectivists to decide). What was incorrect in your previous claims is that to avoid that there are greater and greater infinities or Cantor's theorem, et. al, Objectivists (or anyone) would have to reject the law of excluded middle, and incorrect is your claim that the law of excluded middle logically demands Cantor's theorem or that there are greater and greater infinities. An Objectivist (or anyone) may quite happily and with perfect consistency accept the law of excluded middle while rejecting Cantor's theorem or that there are greater and greater infinities. To reject Cantor's theorem while still holding to the law of excluded middle can be done by, for example, (1) rejecting the very set theoretic framework of Cantor and his more formal successors, (2) rejecting the premises of Cantor's argument, such as the axiom schema of separation (not an axiom schema of Cantor himself, but an instance is used when we couch Cantor's argument in a formal axiomatic theory). And to reject that there are greater and greater infinities while still holding to the law of excluded middle can be done by, for example, (1) rejecting the very set theoretic framework of Cantor and his more formal successors, (2) (as you mentioned) rejecting the axiom of infinity, or rejecting the power set axiom. In most succinct terms, your claim that an Objectivist's acceptance of the law of excluded middle demands acceptance of greater and greater infinities is incorrect. aleph_1, on 08 Feb 2013 - 10:37: "Any mention of "potential infinities" is a reference to a concept equivalent to god." I think you meant mention of 'actual infinity'? If I recall correctly, Objectivism, per Ayn Rand herself, does accept the notion of potential infinity while, of course, rejecting actual infinity.
  12. I don't opine as to the Objectivist notions here (though, do I recall incorrectly that with Objectivism, abstractions also are existents (or is it "entities"?), indeed that to deny that abstractions (such as concepts?) are existents (or should I say "entities"?) is to suffer from being "concrete bound"?). But what I am saying is merely in context of ordinary mathematics in which points are objects as much as any other mathematical object. For example, the point in the real plane <0 1> is no less an object than the number 2 or the unit circle or any other mathematical object. So, if a point is a "nothing" than so are other mathematical objects such as the number 2 or the unit circle. Indeed, a line itself is a certain kind of set of points, and thus an abstraction, and if points are "nothings" than a line too is a "nothing" and indeed it is "nothing" set of points that are themselves "nothings". Actually, I don't even care to press a case as to "objecthood" (I don't care to press the case for any particular ontological notion of mathematics) EXCEPT the sense that IF points are "nothings" then so are such "things" (whatever their ontological or even merely linguistic "figure of speech" status, or possibly in Objectivism "concepts of method"?) as numbers, lines, and circles, complex numbers, the complex system itself, etc., especially given that in, for example, the real plane, a point is merely an ordered pair of real numbers. Particularly, that a point itself doesn't have a distance (along one of the two dimensions) is not good grounds for saying that a point is a "nothing". To illustrate (but not to argue) by analogy: Age difference is a relation between people and is not a relation a person has onto himself. It makes sense to ask "what is the age difference between Mike and Jim?" but makes no sense to ask "What is Jim's age difference?". But we don't infer then that Jim is a "nothing". It makes sense to ask, "what is the distance between the point <2 4> and the point <7 15>?" but no sense to ask "what is the distance of the point <7 15>?". Yet it seems, at best, odd to then infer that the point <7 15> is a "nothing".
  13. Additional points in response to certain mentions here: (1) As well as the law of excluded middle not being necessary to prove the various theorems mentioned, it is not the case that the law of excluded middle is sufficient to prove the various theorems mentioned. Those theorems are proven from certain set theoretic principles or axioms and the theorems are not provable merely by application of the law of excluded middle or even application of the law of excluded middle along with other logical principles or axioms. It is NOT the case that "The existence of distinct infinite cardinalities is a logical consequence of LEM." (2) The reason Brouwer and the intuitionists rejected the law of excluded middle is not Cantor's theorem or the various theorems about greater and greater infinite sets. As mentioned, the law of excluded middle is not required for those results. Rather, the reason Brouwer and the intuitionists rejected the law of excluded middle is that it is non-constructive in the sense that it allows proving that there exists an object with a certain property but without also pointing out any specific such object. A constructive proof is one such that if it proves that there exists an object with a certain property, in then the proof itself may be used to further specify a particular object that has the property. By the way, intuitionists do not reject the law of excluded middle if it is used only for arguments pertaining to finite domains. If only finite domains are countenanced, then intuitionists DO accept the law of excluded middle, since with finite domains, a finite search will produce a specific object having a certain property if there does exist such an object. It is only where infinite domains are allowed that the intuitionist rejects the law of excluded middle. (3) Points usually do not have dimensions (size or distance in different dimensions) themselves with regard to the system of dimensions in which the points are points. However, that does not make points "nothings". Points are mathematical objects (whether they be pairs of real numbers or merely unspecified as geometric points in a given geometry, or whatever, per the system considered) and have properties. That in the system the points do not themselves have distances within them (since usually distance is a function of pairs of points, not a function of a point onto itself) does not vitiate points so that they are "nothings".
  14. There are some inaccuracies that have been stated in this thread about Cantor's theorem (and other theorems) with regard to the law of excluded middle. Note: The following is an account of certain basics in set theory and mathematical logic. These points might not be accepted in Objectivism, but they are correct in the context of Cantor and set theory, as these have become a subject in this discussion. The terminology used here is with regard to set theory and mathematical logic and is not necessarily in accord with Objectivist terminology, but one cannot discuss these matters about Cantor and set theory that have been raised in this thread without keeping clear terminology within the context of set theory and mathematical logic itself. Cantor's theorem was originally proven by Cantor in his set theory that was not formalized at that time. Subsequently, set theory has been formalized in a number of ways, most commonly in certain variations of Z set theory such as Z, ZF, and ZFC or in class theories such as NBG. And Cantor's theorem may be formally proven in those formalizations also. Cantor's theorem is that there is no surjection ('surjection', 'injection' and 'bijection' are not Cantor's own terminology, but they work just as well in this context) from a set S onto the power set of S (thus, there is no bijection (i.e. 1-1 correspondence) between a set and its power set). Cantor proved similar theorems, such as that there is no surjection from the set of natural numbers onto the set of denumerable binary sequences (thus no surjection from the set of natural numbers onto the set of real numbers). These proofs do NOT rely on the law of excluded middle and these proofs do NOT rely on non-intuitionistic proof by contradiction. The logic used for these proofs is intuitionistically and constructively valid. There are two forms of proof by contradiction. One is intuitionistically valid and does not require the law of excluded middle, while the other is not intuitionistically valid and does rely on the law of excluded middle: (1) Assume P. Show a contradiction. Infer the negation of P. This is intuitionistically valid and does not use the law of excluded middle. The sentential analogue of this is: If P then [something false], therefore P is false. This is intuitionistically valid and does not use the law of excluded middle. (2) Assume the negation of P. Show a contradiction. Infer P. This is not intuitionistically valid and does use the law of excluded middle. The sentential analogue of this is: If the negation of P then [something false], therefore P is true. This is not intuitionistically valid and does use the law of excluded middle. Cantor's proofs, if we couch them as proofs by contradiction, may be given as form (1) above, which is intuitionistically valid and does not use the law of excluded middle. Cantor's proofs, of course, rely not only on the logic used but on certain premises. However, the minimal premises used to prove Cantor's theorem are (as far as I know) acceptable both for intuitionists and predicativists. (Even though the axiom schema of separation may not be in general acceptable to the predicativist, the particular instance used in proof of Cantor's theorem is, as far as I know, acceptable to the predicativist.) However, there are further conclusions drawn, such that there are infinite sets that are not in bijection and that there are "greater and greater infinite sets" (i.e., that for any infinite set S there is an infinite set T such that S injects into T but S does not surject onto T (thus there is no 1-1 correspondence between S and T). These conclusions do rely on certain premises (general principles if informal; axioms if formal), such as the power set axiom and the axiom of infinity that the intuitionist may reject. But these conclusions do NOT rely on the law of excluded middle.
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