'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 25 Mar 2010 21:55 Daryl McCullough wrote: > > Ultimately, the people on this newsgroup who object to standard > mathematics are really objecting to the idea that there can be > such a thing as a counter-intuitive result. That's a grossly erroneous over-generalization. It's many of those who defend standard mathematics who erroneously object to the counter-intuitive _but real nature_ of mathematics: relativity of truth and provability. To date they can't cite any absolutely true formula, without the formula being false in another similar context, and yet they'd believe such "absolute" truth is intuitive. > The ultimate logic > would be one in which it is impossible to prove any result that > you couldn't already guess was true. The ultimate logic is one which is relativistic.
From: Transfer Principle on 25 Mar 2010 23:07 On Mar 25, 10:40 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > I'm just a simple housewife (with somewhat hairy legs). It would be poetic justice if AP were to quote this sentence in his ..sig, with only the name of its author "Jesse F. Hughes" appearing in the sig, with neither AP nor any other name appearing anywhere in the post.
From: Jesse F. Hughes on 25 Mar 2010 23:16 Transfer Principle <lwalke3(a)lausd.net> writes: > On Mar 25, 10:40 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> I'm just a simple housewife (with somewhat hairy legs). > > It would be poetic justice if AP were to quote this sentence in his > .sig, with only the name of its author "Jesse F. Hughes" appearing > in the sig, with neither AP nor any other name appearing anywhere > in the post. Why should that bother me? He should certainly feel free to do so, so long as his name appears on the "From" line of the post headers. -- "But remember, as long as one human being follows the rules of mathematics, then mathematics as a human discipline survives. Right now I'm that one human being, so mathematics survives." -- James S. Harris
From: Transfer Principle on 26 Mar 2010 00:03 On Mar 22, 5:18 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > Notice that J. Clarke here is expressing similar ideas regarding the > > empty set and vacuous truth as the OP of this thread, Newberry. > No, he's not. He's claiming that the empty set is not a relation. He > is not claiming that vacuously true statements are neither true nor > false. > Why you think this is even remotely related to Newberry's pet project > is beyond me. There must be some reason that Clarke claims that the empty set is not a relation, in contradiction to ZFC and FOL. Let's go back to a post made by Rotwang: "And if a physicist accepts all the axioms of ZFC, as well as the usual rules of FOL, then he must accept all of the resulting theorems whether he realises it or not." Applying contapositive to Rotwang's statement, we see that if a physicist _doesn't_ accept all the theorems of ZFC/FOL, then he doesn't accept at least one of the axioms of ZFC/FOL. Of course, we don't know whether Clarke or Newberry are even physicists, since after all: > [1] I'm not saying that Newberry or Clarke is untrained in > mathematics [or physics, for that matter]. I have no idea of their > backgrounds, of course. But still, I have yet to see why the contrapositive to Rotwang's statement can't be generalized to any poster. So any poster who doesn't accept all of the theorems of ZFC/FOL must reject at least one of the axioms of ZFC/FOL. And since Clarke rejects "the empty set is a relation," which is a theorem of ZFC/FOL, then we conclude that he rejects at least one of the axioms. And I'd like to know which axiom that is (to allow for the possibility of a theory with takes the rejected axiom and replace it by, say, its negation -- a theory which hopefully would be more acceptable to Clarke). Let's look at an actual proof of "the empty set is a relation." Once again, we go back to the Metamath proof. This proof consists of three steps: http://us.metamath.org/mpegif/rel0.html We notice that in the list of axioms used by the proof (given at the bottom of the page), there is only one set theoretic axiom used in the proof -- the Axiom of Extensionality. Believe it or not, the Empty Set Axiom isn't used in the proof, despite the theorem actually referring to "the empty set"! Perhaps this is because even if 0 is merely a proper class, it would still be a relation anyway. We don't need 0 to be a set in order for it to be a relation (since there are many relations, such as =, which clearly aren't sets). All the other axioms used are actually rules of FOL. Thus, if Clarke accepts the Axiom of Extensionality, then he must reject some rule of FOL. Now Step 2 of the proof gives a definition of "relation." We note that if V is the class of all sets, then VxV is the class of all ordered pairs, and a relation is defined here as a subclass of VxV. In other words, a relation is simply a class of ordered pairs (which is the version of the definition Clarke and Hughes appear to be using). Step 1 of the proof asserts that 0 is indeed a subclass of VxV -- simply because 0 is a subclass of _any_ class. Step 3 of the proof uses a form of Modus Ponens -- from "if a class is a subclass of VxV, then it is a relation" and "0 is a subclass of VxV," conclude "0 is a relation." QED We know that Clarke rejects at least one of these steps -- but which one is it? One could argue that it's step 2 -- since neither he nor Hughes explicitly refers to subclasses of VxV. But I believe that as the definition they give is equivalent to Metamath's, my guess is that step 1 is the objectionable step instead. So we look at Step 1 in more detail. The proof that 0 is a subclass of any class is also a three-step proof: http://us.metamath.org/mpegif/0ss.html Step 1 looks straightforward -- it merely states that the empty set has no elements. Step 3 is a form of the definition of subclass, which also looks uncontroversial. But Step 2 states that any element of the empty set must be an element of any class -- which sounds like vacuous truth. And when we click on this step, we see a rule of inference: from ~phi, conclude phi -> psi -- which sounds just like the Third Paradox of Material Implication, mentioned elsewhere in this thread: http://us.metamath.org/mpegif/pm2.21i.html And why was that Third Paradox mentioned? It was mentioned in the context of _Newberry_ -- the paradoxes to which Newberry objects. And thus I claim that there is indeed a connection between Clarke and Newberry after all -- both of them object to theorems of ZFC/FOL which use the Third Paradox of Material Implication. > Why you think this is even remotely related to Newberry's pet project > is beyond me. So Hughes doubts my claim -- and we already know that the most common criticism of me by far is misinterpretation and incorrect guessing of what others are saying. So how can I reduce this bad habit of mine, and avoid misinterpreting what's in Clarke's post. We know that Clarke objects to _some_ axiom of ZFC/FOL, but how can I find out which one? I could try asking Clarke directly. But I must do so in a way so as not to appear to be a standard theorist or a "bully." I'm not trying to _interrogate_ Clarke in order to criticize him, but to find out which axiom he rejects in order to come up with an alternative theory, without the standard theorists accusing me of misinterpreting posts. Also, if I start giving formal symbolic proofs of whichever theorem of ZFC/FOL is being rejected (such as the proof of "the empty set is a relation") given above, this also tends to turn off the standard theorists' opponents -- since most of them seem to prefer informal English to formal symbolic language. But I could give a description of the proof in informal English -- and indeed, I've already done so in this very post. And so I can ask Clarke about his rejection of the empty set as a relation without interrogating him like a standard theorist or otherwise alienating him (lest me find me too hostile, and he killfiles me, just as he killfiled Hughes). So let me do so right now -- this rest of this message is therefore directed to Clarke. You say that you reject the empty set as a relation, yet I've just given a three-step proof in standard theory, ZFC/FOL, that the empty set is indeed a relation. Since you reject the conclusion of the proof, to which of its three steps do you also reject?
From: Marshall on 26 Mar 2010 00:25
On Mar 25, 11:00 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > Ultimately, the people on this newsgroup who object to standard > mathematics are really objecting to the idea that there can be > such a thing as a counter-intuitive result. The ultimate logic > would be one in which it is impossible to prove any result that > you couldn't already guess was true. QFT |