'When Are Relations Neither True Nor False?
in [Math]
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From: Michael Stemper on 22 Mar 2010 13:02 In article <ho40c402sgc(a)news5.newsguy.com>, "J. Clarke" <jclarke.usenet(a)cox.net> writes: >On 3/20/2010 7:07 PM, Nam Nguyen wrote: >> Stefan Ram wrote: >>> Always. A relation is a set of pairs, and a set is >>> never true nor false. Therefore, every relation is >>> neither true nor false. >> >> Technically speaking, what you said isn't quite true. A false >> relation is an empty set, while a true one isn't an empty set. > >A relation is not a "set" at all, Well, except that's exactly how "relation" is defined. A relation is a set of ordered pairs. More specifically, a relation from X to Y is a set of ordered pairs (x,y) such that x in X and y in Y. Or, as Mathworld says: "A relation is any subset of a Cartesian product." > it establishes some rule by which >elements of one set may be associated with elements of another set. Relations *do* that, yes. >> A relation - a set - > >Which are you talking about, a relation or a set? A relation is a specific type of set. -- Michael F. Stemper #include <Standard_Disclaimer> If you take cranberries and stew them like applesauce, they taste much more like prunes than rhubarb does.
From: Aatu Koskensilta on 22 Mar 2010 13:10 mstemper(a)walkabout.empros.com (Michael Stemper) writes: > In article <ho40c402sgc(a)news5.newsguy.com>, "J. Clarke" <jclarke.usenet(a)cox.net> writes: > >> it establishes some rule by which elements of one set may be >> associated with elements of another set. > > Relations *do* that, yes. On the set theoretic conception, most relations don't establish any rule in the ordinary sense of the word. It is a fundamental part of this conception (famously described as "quasi-combinatorial" by Bernays) that sets are arbitrary extensional collections, not necessarily given by a any rule or definition. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Transfer Principle on 22 Mar 2010 19:30 On Mar 22, 7:50 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > And the best opinions in the world do not, IMHO, rise to the level > > of the worst evidence. I'm calling round 1 for TP here. > Bah! Even barely educated guesses rise far above the level of the worst > evidence. Just as I suspected would happen, the standard theorists are reluctant to accept the poll to which I linked as valid. On one hand, they're right in that anyone can set up a poll, and anyone, regardless of their qualifications, can participate. A Google search, in fact, reveals many different polls about whether 0.999...=1, with vastly different results. But, of course, no one's going to take what, say, Worlds of Warcraft gamers (one of the results!) seriously, but we do care about what _physicists_ say, since we're trying to find a theory that axiomatizes for math for the _sciences_. On the other hand, the poll to which I linked wasn't any arbitrary poll, but a poll given at the Metamath website. Once again, here's a link to that Metamath page: http://us.metamath.org/mpegif/0.999....html One could argue that the participants in the poll aren't really physicists and so the poll is invalid -- but then again, why would Metamath choose to link to it if they thought that the poll was meaningless? And recall that Metamath isn't a "crank" site, but an automatic theorem prover -- after all, the proof in ZF of 0.999... is given right below the link. The fact that the web maintainers at Metamath thought that the poll was worth mentioning does give some validity to the poll. > Bah! Even barely educated guesses rise far above the level of the worst > evidence. Aatu writes that he prefers educated guesses to random "evidence." I wonder what educated guess Aatu himself would give as the proportion of physicists who believe that 0.999...=1. So far, the only poster who has given any educated guess is Rotwang, himself a physicist. Rotwang's guess above is that a supermajority (approaching unanimity) of physicists do accept 0.999...=1. Even if Rotwang is correct, I doubt that physicists would accept _all_ of the theorems of ZFC as being true -- since for one thing, most of them have never heard of ZFC. Of course, we should leave out theorems that are not directly related to physics (such as the existence of, say, a fixed point of aleph). MoeBlee hints that physicists might not even have an opinion about most of these. But say we stick to, say, the real numbers R, and mathematical objects that do play a role in physics. So my question is, does there exist a statement regarding such an object that a significant number of physicists (say at least a third of them) believe is false, yet the statement is provable in ZFC (assuming that ZFC is consistent)? I believe so -- even if 0.999...=1 isn't such a statement. To me, it seems hard to believe that a supermajority of physicists -- who have never even heard of ZFC -- would be in complete agreement with _all_ of the theorems of ZFC that are related to physics (including real numbers). And if such a statement exists, then I believe that a theory in which its negation is provable (and isn't trivially inconsistent) is worth considering. And if the proportion of physicists who disagree with the statement isn't just a sizeable minority but an actual majority, then I'd argue that there's a theory _better_ than ZFC at axiomatizing math for the sciences. Indeed, going back to the original topic of this thread, I wonder whether a supermajority of physicists would accept the concept of vacuous truth (but some might point out that this is an issue of FOL, not ZFC).
From: Jesse F. Hughes on 22 Mar 2010 19:56 Transfer Principle <lwalke3(a)lausd.net> writes: > On Mar 22, 7:50 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> Marshall <marshall.spi...(a)gmail.com> writes: >> > And the best opinions in the world do not, IMHO, rise to the level >> > of the worst evidence. I'm calling round 1 for TP here. >> Bah! Even barely educated guesses rise far above the level of the worst >> evidence. > > Just as I suspected would happen, the standard theorists are > reluctant to accept the poll to which I linked as valid. That's a pretty remarkable prescience you got there. It must be because the standard theorists are stodgy old farts. The fact that your poll is a pathetic attempt at proving that 40% of physicists doubt 0.99... = 1 is merely coincidental. > On one hand, they're right in that anyone can set up a poll, and > anyone, regardless of their qualifications, can participate. A Google > search, in fact, reveals many different polls about whether > 0.999...=1, > with vastly different results. But, of course, no one's going to take > what, say, Worlds of Warcraft gamers (one of the results!) seriously, > but we do care about what _physicists_ say, since we're trying to > find a theory that axiomatizes for math for the _sciences_. > > On the other hand, the poll to which I linked wasn't any arbitrary > poll, > but a poll given at the Metamath website. Once again, here's a link > to that Metamath page: > > http://us.metamath.org/mpegif/0.999....html I see you're incapable of reading web pages as well. It's not merely Usenet that gives you problems. ,---- | Description: The recurring decimal 0.999..., which is defined as the | infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 10^1 + 9 / 10^2 + 9 / | 10^3 + ..., is exactly equal to 1, according to ZF set | theory. Interestingly, about 40% of the people responding to a poll at | http://forum.physorg.com/index.php?showtopic=13177 disagree. `---- Hint: Metamath has a pointer to a different website with that poll. It is not "*given* at the Metamath website" (emphasis mine). (Note as well that Metamath is not foolish enough to make a claim about 40% of physicists. That's original with you.) > One could argue that the participants in the poll aren't really > physicists and so the poll is invalid -- but then again, why would > Metamath choose to link to it if they thought that the poll was > meaningless? The poll isn't wholly meaningless. Here's what it means: 40% of the people responding to this poll think 0.999... < 1. Which is, of course, pretty much what Metamath said, too. > And recall that Metamath isn't a "crank" site, but an automatic > theorem prover -- after all, the proof in ZF of 0.999... is given > right below the link. The fact that the web maintainers at Metamath > thought that the poll was worth mentioning does give some validity > to the poll. What silliness! First, Metamath made no claims about the reliability of the poll. Second, *even if* they had, who cares? Metamath is not, as far as I know, run by statisticians or pollsters. -- "These mathematicians are worse than communists, as how do you explain their behavior? I *am* the American Dream, fighting for what should be mine, having to get past weak-minded academics who are fighting to block my success. But I shall prevail!!!" -- James S. Harris
From: Transfer Principle on 22 Mar 2010 20:00
On Mar 21, 8:15 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: > On 3/21/2010 10:13 AM, Jesse F. Hughes wrote: > > "J. Clarke"<jclarke.use...(a)cox.net> writes: > >> However, if the definition of a "relation" is "a set of n-tuples", then > >> by definition the empty set is not a "relation" [...] > > Nonsense! The empty set is a set of n-tuples for *every* n. Every > > element of the empty set is an n-tuple (no matter the value of n). > > The empty set is a relation. > I thought I had you killfiled. > Well, back you go. Notice that J. Clarke here is expressing similar ideas regarding the empty set and vacuous truth as the OP of this thread, Newberry. Although I don't necessarily agree with the idea of killfiling someone merely because one disagrees with them, Clarke is only doing the same thing to Hughes that Daryl McCullough, another poster in this thread, has done to me. And Clarke has killfiled Hughes for the exact same reason that McCullough has killfiled me -- because both Hughes and I have repeatedly posted statements with which Clarke and McCullough, respectively, disagree. The so-called "cranks" and the standard theorists have more in common with each other than either would like to admit. When Newberry first attacked vacuous truth in this thread, I didn't feel that his ideas were worth defending. But now that a second poster (Clarke) has expressed agreement with Newberry, I'm starting to consider defending their ideas after all. So we can attempt to come up with a theory (that isn't trivially inconsistent) in which the intuitions of Newberry and Clarke are provable. But in doing so, we must keep the words of Marshall Spight in mind, and recall that any theory that's "less powerful" and "more work to use" than ZFC, just to avoid a "harmless" counterintuition such as vacuous truth, is a waste of time. So we must seek out a theory that's either at least as powerful as, or at worst as hard to use as, ZFC. (Of course, "more work to use" is highly subjective. In another thread, some computer scientists argued that any set theory such as ZFC is _more_ work to use than a theory of arithmetic without _sets_.) So we know that in ZFC, if phi(x) is a one-place predicate of the form Ayex (psi(y)) for some one-place predicate psi, then phi(0) must hold by vacuous truth. There are two ways to avoid this. The first would be to change the laws of inference of FOL in order to avoid vacuous truth, and the other would be to change the axioms of ZFC in order to prevent the empty set 0 from existing. Given a choice between changing logical laws of inference and changing set theoretic axioms, I'd change set theoretic axioms every time. And so we seek out a set theory in which the empty set doesn't exist (yet might be still as powerful as ZFC). So instead of the Empty Set Axiom, we could have its negation as an axiom of the new theory: Ax Ey (yex) Obviously, we need to change the Separation Schema in order to prevent the empty set from existing. Now the poster zuhair has already considered set theories without empty sets. IIRC, he came up with something like: Ewez (phi(w)) -> Ex Ay (yex <-> (yez & phi(y))) What would really be elegant (and would strengthen the justification of avoiding the empty set) would be to replace Separation with a schema such that if we added it to ZFC-Separation Schema (including empty set existence) would result in, say, Russell's Paradox, but if we assume that no empty set exists, then the contradiction would be avoided. Indeed, zuhair attempted to find such a schema (and failed). |