Counterintuitions and the well-ordering theorem
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From: Nam Nguyen on 16 Oct 2009 01:23 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >>>> Put it differently, it's quite possible it's impossible to know >>>> which inequality would hold - in all models of reals. It's therefore >>>> counter- intuitive to talk about the well-ordering of the whole set >>>> of reals. >>> Yes, I think it would be counterintuitive to speak of "the" >>> well-ordering, but surely people only speak of "a" well-ordering? >>> >> You missed the point: M(Pi) < M(e) could be relativistic in any model >> of reals. It's therefore counterintuitive to talk about any well-ordering >> of reals when it's _impossible_ to know if M(Pi) < M(e) is true or false. >> >> Put it differently, how could you talk about the order of numbers if it's >> _impossible_ to tell which of the 2 numbers is greater in the order >> ladder? > > It's not impossible. It just depends on which well-ordering we're > speaking of. Let M be any well-ordering of R, i.e., M:R -> |R|. > Define a new well-ordering M' by > > > M'(pi) = min{M(pi), M(e)} > M'(e) = max{M(pi), M(e)} > M'(x) = M(x) for all x distinct from pi and e. > > Now, it is perfectly clear that M' is a well-ordering of R and > M'(pi) < M'(e). So, your claim that it is impossible to compare pi > and e in a well-ordering of R is simply false. > Your analogy with M'(pi) < M'(e) is incorrect. My point is very _specific_: "M(Pi) < M(e) could be relativistic" in any given model of R, hence in any well-ordering of R. That specificity has nothing to do with your M'(pi) < M'(e).
From: Rupert on 16 Oct 2009 08:12 On Oct 16, 4:16 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Jesse F. Hughes wrote: > > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >> As just mentioned, of course you have knowledge about which one is the > >> case _in principle_. So nothing is counterintuitive here. > > >> In brief, your analogy is close but not quite correct. Let me give a > >> more precise analogy. Let's consider the statement: > > >> (1) In each of the _infinite_ number of universes there exists a planet > >> with biological life during the planet's lifetime. > > >> Can you see the truth or falsehood of (1), even _in principle_? > > >> Of course (1) is either true or false and not in between. But there's > >> a genuine _impossibility_ to know which truth value be the case. And, > >> again, consequently the assertion that it be true or false is completely > >> relativistic, depending which value one would _choose_. > > > No, that clause beginning "consequently" does not follow. > > > I'll assume, as you say, that either (1) is true or false, and we > > don't know (and can't know) which. > > Right; although it should be reminded that there's a distinction between > "don't know" and "can't [possibly] know": the later logically implies > the former, but not the other way around! The case for (1) is the > "can't [possibly] know" case; and that would make a difference. > > > It simply doesn't follow that the > > truth value of (1) depends on whether I choose to believe it or not. > > If, given an underlying reasoning framework, a truth of a statement can't > be possibly known then the truth is _undecidable_, notwithstanding the > fact it's either true or false. And whence it's undecidable, one is > *free to _decide_* which way it be! > > > Rather, either I choose correctly or I don't, but (1) is true or false > > independently of my choice. > > Once something is undecidable, there's *no _correct_* choosing or deciding! > > > > >> Closer to home, in FOL reasoning, there's an analogous situation. Whether > >> or not it's true the last digit of a extraordinarily large prime number P > >> is 1 is analogous to your "nickel in my drawer" case: we do know the truth > >> _in principle_ because the proof of it can be done in _finite_ steps. > > >> But the arithmetic truth of say GC could be something else all together. > >> It's either true or false of course (so LEM is not broken) but it could > >> be impossible to know (since its "proof" could be infinite). So its truth > >> would be relativistic, while LEM is not broken. > > > Again, either GC is true of the natural numbers or it is not. > > Right. A formula is either true or false in a model of a T, just as > the simultaneity of 2 events is either true or false in a frame of > reference. But which frame are you talking about? Similarly, which > "natural numbers" are you referring to? The one in which GC is true? > Or the one in which cGC (hence ~GC) is true? > We are referring to the unique model of second-order Peano arithmetic.
From: Jesse F. Hughes on 16 Oct 2009 10:40 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>>>> Put it differently, it's quite possible it's impossible to know >>>>> which inequality would hold - in all models of reals. It's therefore >>>>> counter- intuitive to talk about the well-ordering of the whole set >>>>> of reals. >>>> Yes, I think it would be counterintuitive to speak of "the" >>>> well-ordering, but surely people only speak of "a" well-ordering? >>>> >>> You missed the point: M(Pi) < M(e) could be relativistic in any model >>> of reals. It's therefore counterintuitive to talk about any well-ordering >>> of reals when it's _impossible_ to know if M(Pi) < M(e) is true or false. >>> >>> Put it differently, how could you talk about the order of numbers if it's >>> _impossible_ to tell which of the 2 numbers is greater in the order >>> ladder? >> >> It's not impossible. It just depends on which well-ordering we're >> speaking of. Let M be any well-ordering of R, i.e., M:R -> |R|. >> Define a new well-ordering M' by >> >> >> M'(pi) = min{M(pi), M(e)} >> M'(e) = max{M(pi), M(e)} >> M'(x) = M(x) for all x distinct from pi and e. >> >> Now, it is perfectly clear that M' is a well-ordering of R and >> M'(pi) < M'(e). So, your claim that it is impossible to compare pi >> and e in a well-ordering of R is simply false. >> > > Your analogy with M'(pi) < M'(e) is incorrect. My point is very > _specific_: "M(Pi) < M(e) could be relativistic" in any given model > of R, hence in any well-ordering of R. That specificity has nothing > to do with your M'(pi) < M'(e). I guess I haven't a clue what you mean, then. What is the meaning of M(pi)? I assumed that it was the mapping of pi under a given well-ordering of R. But there is no canonical well-ordering, as far as I can see, so this claim that it is unknowable is false. It depends on the well-ordering we're considering. In particular, starting with a given well-ordering M, I can construct a well-ordering M' (possibly equal to M) in which M'(pi) < M'(e). I don't see how any of this has anything to do with particular models of R. -- Jesse F. Hughes "You know that view most people have of mathematicians as brilliant people? What if they're not?" -- James S. Harris
From: Nam Nguyen on 17 Oct 2009 18:29 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Jesse F. Hughes wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> As just mentioned, of course you have knowledge about which one is the >>>> case _in principle_. So nothing is counterintuitive here. >>>> >>>> In brief, your analogy is close but not quite correct. Let me give a >>>> more precise analogy. Let's consider the statement: >>>> >>>> (1) In each of the _infinite_ number of universes there exists a planet >>>> with biological life during the planet's lifetime. >>>> >>>> Can you see the truth or falsehood of (1), even _in principle_? >>>> >>>> Of course (1) is either true or false and not in between. But there's >>>> a genuine _impossibility_ to know which truth value be the case. And, >>>> again, consequently the assertion that it be true or false is completely >>>> relativistic, depending which value one would _choose_. >>> No, that clause beginning "consequently" does not follow. >>> >>> I'll assume, as you say, that either (1) is true or false, and we >>> don't know (and can't know) which. >> Right; although it should be reminded that there's a distinction between >> "don't know" and "can't [possibly] know": the later logically implies >> the former, but not the other way around! The case for (1) is the >> "can't [possibly] know" case; and that would make a difference. >> >>> It simply doesn't follow that the >>> truth value of (1) depends on whether I choose to believe it or not. >> If, given an underlying reasoning framework, a truth of a statement can't >> be possibly known then the truth is _undecidable_, notwithstanding the >> fact it's either true or false. > > Right. > >> And whence it's undecidable, one is *free to _decide_* which way it >> be! > > I don't see how that follows. If one isn't free to decide which way it be then it's not undecidable. Don't you think? What do you think "undecidable" mean? > >>> Rather, either I choose correctly or I don't, but (1) is true or false >>> independently of my choice. >> Once something is undecidable, there's *no _correct_* choosing or >> deciding! > > You already agreed that the statement (1) is either true or false, > regardless of our choice. Where did I agree that (1) is true or false "regardless of our choice"? > If (1) is true and we've chosen to believe > (1), then we've chosen correctly. In the same circumstances, if we've > chosen to believe (1) is false, then we've chosen incorrectly. > > What could be clearer? You simply don't understand a clear simple fact: one simply can't apply LEM to (1) - without a context. And the context is one's assumption that it be true, or false. In other words, (1) is a statement that _has no absolute truth_! There are statements (scientific or not) that appear to have absolute truth but they don't. "God exists" is a non-scientific example, while (2) is a scientific one, where (2) is: (2) "Two particular events e1 e2 happen simultaneously". We're 100+ years after 1905 when SR was presented! So again, you should make it clear in your mind that (1) is not a kind you could claim it's true or false without a (relativization) context. > > We both agree that, whichever we choose (assuming that we do indeed > choose), we will not know whether that choice is the correct choice or > not. I simply can't imagine why you think this implies there is no > correct choice. That's because you're too quick to contemplate on what "impossibility" means, to the foundation of reasoning. Could you tell me what correct choice you'd have for (2)? > Assuming that the statement (1) is meaningful and > either true or false, then it is clear what we mean when we say that > we've chosen to believe correctly -- regardless of whether we will > ever know that this is the case. Your misconception here is a meaningful statement must have an absolute truth value. (2) is an example this is not the case. >> But which frame are you talking about? Similarly, which >> "natural numbers" are you referring to? The one in which GC is true? >> Or the one in which cGC (hence ~GC) is true? > > The natural numbers. The smallest set satisfying the Peano axioms. Which "smallest set satisfying the Peano axioms" are you talking about? the one in which GC is true? Or the one in which ~GC is true? >>> This fact doesn't depend on what we choose to believe. >> Which _undecidable_ fact are you referring to? The "natural numbers"-fact >> in which GC is true? Or the "natural numbers"-fact where it's false? > > My comment refers to the fact that either GC is true or not. How do you even know that "absolute fact" when you're under the assumption that it be _impossible_ to know the truth value of GC? > >>> Your claims about relativism are simply wrong. >> Rather, you don't seem to understand how mathematical _impossibility_ is >> equated to mathematical relativity. > > Well, you're right there. I certainly don't understand this odd claim. I'm almost certain you know something about SR and have heard of (2) in one form or another. Why don't you reflect on those scientific matters for a moment, in contemplating about similar issue: relativity in mathematical reasoning. I'll give you a hint that I think would help you here. In the thread "Nonfirstorderizability" on August 7, 2005, Torkel Franzen mentioned something to the effect that there are intuitive concepts that can't be translated into a FOL statement (hence the phrase "Nonfirstorderizability"). Now, let me ask you a question about the reverse of "Nonfirstorderizability": would you think there is any statement than can be formulated as a FOL formula but which can't be "modellizable" in a collection of models? If your answer is that there isn't any, then could you show your meta proof to back your claim? If not then you got to allow the possibility that such "nonmodellizability" could happen to GC, cGC, ~GC, ~cGC, in the collection of all that are perceived as arithmetic models - including the naturals.
From: Aatu Koskensilta on 18 Oct 2009 10:33
Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Aatu Koskensilta wrote: > >> That aside, I was wondering specifically about the claim that the >> well-ordering theorem itself is counter-intuitive, as alluded to in >> e.g. the famous quip >> >> The axiom of choice is obviously true, the well-ordering theorem >> obviously false -- and who can tell of Zorn's lemma? >> >> Just what intuitions are contradicted by the well-ordering theorem? > > Imho, the quip tries to express something completely different. > It's not an expression of mathematician's intuitions about sets. > Mathematicians don't care about sets - they care about mathematics. Sure. What do you take the quip to (try to) express? > And a general purpose foundation for mathematics should ideally not > turn something into an indispensable truth that is, for mathematics, > quite dispensable. I'm not sure what you have in mind here. > Btw, personally i find Freyling's argument quite appealing. But i also > think that AD and AC are both true, so you should better not take my > opinions too seriously. What are your reasons for thinking AD true? > (Not that there was any danger that you'd do that, anyway.) I take nothing too seriously. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |