Torkel Franzen on truth
in [Logic]
|
From: Nam D. Nguyen on 4 Jan 2008 17:44 herbzet wrote: > > MoeBlee wrote: >> herbzet wrote: >>> MoeBlee wrote: >>>> herbzet wrote: >>>>> MoeBlee's assertion that it is a theorem that every structure is a >>>>> model of some theory is a new thought to me. >>>> What is difficult about it? >>>> (1) By definition, M is model of a set of sentences G iff M is a >>>> structure for the language of G and every member of G is true in M. So >>>> every model is a structure, by definition. >>> Right. >>> >>>> (By the way, in another post I think I might have allowed G to be a >>>> set of formulas. But I think G should be a set of sentences.) >>> OK. >>> >>>> (2) Every structure M is a model of the set of valid sentences in the >>>> language that M is a structure for. >>> Yup. >>> >>>> So every structure is a model. >>> Of any validity, yes. Can we say further that every structure is a >>> model of a theory (other than the null theory containing only validities)? >> I'm not familiar with the expression 'null theory' to refer to the set >> of validities, but I'll go along with it. > > I probably just made it up, although it seems right. It's the theory > with no non-logical axioms (ie only validities as theorems). It seems > to me that this is a degenerate case of the word "theory": usually > I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic > and to have some non-logical axioms/theorems. > >> As to the question, I'd have to think about it. > > Bingo. That's what's "a new thought to me": that an arbitrary structure > is a model of some non-null theory. I don't happen to know whether > that's true. > >>> Shall we agree as to what sets of sentences constitute "a theory" first? >> For simplicity, let's confine to classical first order. >> >> Authors such as Enderton take a theory to be any set of sentences >> closed under entailment (which, thanks to the completeness theorem, is >> a set of sentences closed under provability). Authors such as Chang & >> Keisler take a theory to be any set of sentences. And some authors >> take a theory to be a pair <S E> where S is a set of sentences and E >> is the set of sentences entailed by S (which, thanks to the >> completeness theorem, is the set of theorems of S). >> >> I adopt Enderton's definition. > > OK. > >>> I usually think of "a theory" as having a recursive (or at least r.e.) >>> set of axioms. >> That is a recursively axiomatized theory. There are theories that are >> not recursively axiomatized. > > Yes, they seem a little imaginary to me, but I'm willing to go along > with it. > >>> And as being consistent (i.e. having a model)! >> Those are consistent theories. There are theories that are not >> consistent. > > It would seem that there is but one such theory over any signature: > the theory that consists of all sentences in the language. This > seems like another degenerate case of "theory". I'm willing to > accept the informal locution "inconsistent theory"; I'm willing > to accept an "inconsistent theory" as existing under the Enderton > definition of "theory". > > But the extremal cases of "theories" that contain only validities, > or that contain every sentence, do seem a little silly to me. I > guess it's the price you pay for definitional elegance. > >>>>> Even if it is so, >>>>> it does seem to me to be ill-advised to use "structure" and "model" >>>>> interchangably (although I'm as slack as anyone else on using >>>>> "structure" and "interpretation" more or less interchangeably). >>>> In such contexts as I mentioned, what is the harm of using 'model' and >>>> 'structure' interchangably, especially the context I mentioned?: >>> If I recall correctly, this started when you said to Nam: >>> >>>>> Okay, in a technical sense, '2=1+1' is true relative to models because >>>>> it's true in some models but not in others. >>> To which George objected: >>> >>>> No, true in some interpretations but not in others. >>> which might seem like a rather pedantic distinction (or, as you seem >>> to think, a false distinction), but I think that in this area with >>> Nam you have to be very precise, and the distinction is merited. >> Anyone may state definitions and explicate a discussion on the basis >> of those definitions. Meanwhile, what I said is precisely correct >> given ordinary defintions: >> >> '2=1+1' is true in some models and not true in other models. >> >> That is precisely correct. >> >> What is cloudy is the terminology 'true relative to models', which is >> Nam's terminology. I don't use that terminology. All I said (or meant >> to convey) in the passage you mentioned is that I could see a sense in >> which that terminology could be understood. >> >> So, again, to be clear: >> >> My terminology is 'true in a model' and it is precisely correct that >> '2=1+1' is true in some models and not true in other models. Nam's >> terminology is 'true relative to a model', and though I do not in any >> way claim to arbitrate what HE means by that, my point was just to say >> that IF he means 'true in a model', then yes, of course, '2=1+1' is >> true in some models and not true in other models. Then, there were the >> rest of my remarks. > > Yuh, I guess the argument turns on whether you want to affirm > > a) 2 = 1 + 1 is false in no model (of PA) > > or > > b) 2 = 1 + 1 is false in some models (of the language of PA). > > It seems to be a simple ambiguity to be resolved. I personally would > prefer (a) and would assume that you are using "model" in the sense > of (a) in the absence of an explicit definition otherwise. I concede > that it is possible that Nam uses the word "model" in some sense > other than (a). For what it's worth, I actually don't use a different definition of the word "model" here, in FOL context. The phrase that seems difficult to some to understand is: (1) 'true relative to a model' Now suppose you believe PA is consistent and let M be a model of PA, then the following meta level statement would be true: (2) '2=1+1' is true in M. The question though, if we change the inference rules and come up with a different reasoning framework, say (FOL)', then would (2) still be true? The answer is "Not necessarily", depending on what changes of rules of inference that have taken place of course. Now if we consider the formal system PA as just a *collection of axioms* then syntactically PA is the same. But what we consider as M might or might not be an absolute model of PA, right? In that context then, '2=1+1' is being true in M is relative to the fact M might or might not be a model, depending on who's doing the reasoning. In that context, then the following statement would make sense: (3) '2=1+1' is true, relative to a model. which is not different in nature from the statement: (3') the speed of the train is 100 km/hour, relative to the framework M. At any rate, '2=1+1' is no more absolute than 'the speed of the train is 100 km/hour'. Those who believes otherwise would not realize the relativity nature of human mathematical reasoning (through FOL at least). Now there is another "more technical" way to demonstrate the model-relativity connoted in (3), and I did allude to this way a couple of times in the past. Basically, if we consider a FOL system sPA ("super" PA) whose languages contains *infinite* symbols: L(0, S,+,*,<, S',+',*',<', S'',+'',*'',<'', S''',+''',*''',<''', ...) In other words, besides symbol 0, the rest would be grouped together and each group, when coupled with 0, would form a language we could use to formalize a theory we'd name "PA". [This alone would signify the relativity of "PA" and associated models - and (3). Wouldn't it?] The (infinite number of) axioms of sPA would be the union of those individual axioms per each ("PA") group mentioned above. Now let's examine the anatomy of a model M of a general FOL formal system T. In a nutshell, the major components of M are: c1: a set S of individuals of which certain n-ary relations would exist. c2: a collection of n-ary relations, each of which would correspond to an n-ary symbol of the language. c3: collection of (subjective) interpretations, each of which would predicate a theorem-formula as true. Now MoeBlee suggested above: "Anyone may state definitions and explicate a discussion on the basis of those definitions." So in this context here and now, let's call S a "structure" (and temporarily forget if some text books reserve this word for something else). Then it's not hard to see that the relativity of a model M would come from component c3! For example, if the formula is "a < b", S = {a',b'}, and the n-ary relation is {(a',b')}, I still could at my subjective willingness interpret (or predicate) "a < b" as false while you or any other as true. And so relative to whose predicating or interpreting, M would be or *not* be a model of say T = {a < b}, or for that matter of T = {~(a < b)}. Now back to sPA, let S be the structure (i.e. the _set_) of individuals of a model of the integers (i.e. not the natural numbers). The long and the short of it is out of S, we know there exist _uncountably many_ "successor" functions S()'s, hence uncountably many n-aries "addition", "multiplication", and "less-than". Put if differently, not only S is a structure for sPA, it would be the very same structure for _uncountably many_ models of each "PA" theory (written in L(sPA)). But it's not hard to demonstrate that due to the subjective interpretation in c3, if one interpret S as a model of a "PA", others might disagree and (re)interpret S as *not* a model this "PA". Of course when we talk about "PA" we typically talk about it outside the context of this orangutan sPA system. But it should not matter! Given *any* L(PA), one could consider it as part of L(sPA). And given a model - over a structure S - of *any* "PA" theory one could interpret this structure S as a non-model of this "PA". In summary, a model is always (at least) *relative* to: - which exact theory in what exact language that's under consideration - which exact *subjective interpretation* (which would make a formula true). I know it's a bit long way to explain all this, and I don't think "typical" text books would care to give a discussion. But unless one could come up with some credible counter arguments, I'd think we'd have no choice but accept the relativity nature of reasoning in general.
From: MoeBlee on 4 Jan 2008 19:16 On Jan 4, 2:44 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > For what it's worth, I actually don't use a different definition of the > word "model" here, in FOL context. The phrase that seems difficult to some > to understand is: > > (1) 'true relative to a model' > > Now suppose you believe PA is consistent and let M be a model of PA, > then the following meta level statement would be true: > > (2) '2=1+1' is true in M. We don't need 'belive' there. We could just say: If M is a model of PA, then '2=1+1' is true in M. > The question though, if we change the inference rules and come up with > a different reasoning framework, say (FOL)', then would (2) still be > true? Yes. The inference rules don't alter what sentences are true in what models. What the inference rules change is what theories we get. First order PA is DEFINED to be the set of sentences that are entailed in classical first order logic from the first order PA axioms. With a set of inference rules that doesn't yield the same theorems as classical first order logic you get a DIFFERENT theory from PA, you get HA (if the rules are intuitionistic logic) or whatever you want to name each DIFFERENT theory depending on a different logic. > The answer is "Not necessarily", depending on what changes of > rules of inference that have taken place of course. Now if we consider > the formal system PA as just a *collection of axioms* then syntactically > PA is the same. But we DON'T consider PA as just a collection of axioms. We consider PA to be the set of sentences entailed by that collection of axioms. (Or some people say it is the pair <A T> where A is the collection of axioms and T is the set of sentences entailed by A.) > But what we consider as M might or might not be an absolute > model of PA, right? What does "absolute model" mean? > In that context then, '2=1+1' is being true in M is > relative to the fact M might or might not be a model, depending on who's > doing the reasoning. In that context, then the following statement > would make sense: I'd grant that it depends on how we're doing the reasoning in the META- theory, since 'true in the model M' is defined in a meta-theory for first order PA and our proofs of whether something is true in the model are either done in the meta-theory or we rely upon the soundness theorem (which also is proven in the meta-theory) to infer that what is proven in the object theory is true in every model of the axioms of that object theory, as well as it is in the meta-theory that we prove that axioms are true in whatever models we claim the axioms to be true in. But that's not what you're talking about. Your present line of argument is, as usual, confused and ill-premised. > (3) '2=1+1' is true, relative to a model. > > which is not different in nature from the statement: > > (3') the speed of the train is 100 km/hour, relative to the framework M. > > At any rate, '2=1+1' is no more absolute than 'the speed of the train is > 100 km/hour'. > > Those who believes otherwise would not realize the relativity nature of > human mathematical reasoning (through FOL at least). > > Now there is another "more technical" way to demonstrate the model-relativity > connoted in (3), and I did allude to this way a couple of times in the past. > Basically, if we consider a FOL system sPA ("super" PA) whose languages > contains *infinite* symbols: > > L(0, S,+,*,<, S',+',*',<', S'',+'',*'',<'', S''',+''',*''',<''', ...) > > In other words, besides symbol 0, the rest would be grouped together and > each group, when coupled with 0, would form a language we could use to > formalize a theory we'd name "PA". [This alone would signify the relativity > of "PA" and associated models - and (3). Wouldn't it?] NO! That's silly! Just CALLING something 'PA' doesn't show any relativity other than the quite prosaic sense we all already know that if you say "Sally Fields played the role of James Bond" then that is true if by 'Sally Fields' you are referring to the person Sean Connery. In some given overall mathematical context, such as a particular set theory to serve as a meta-theory, we DEFINE PA to be a certain exact mathematical object: a certain exact set of finite strings of symbols. Once we make that definition, it's just silly to worry about what happens if you say, "Oh, we get something different if we use 'PA' to stand for some other thing." > The (infinite number of) axioms of sPA would be the union of those individual > axioms per each ("PA") group mentioned above. > > Now let's examine the anatomy of a model M of a general FOL formal system T. > In a nutshell, the major components of M are: > > c1: a set S of individuals of which certain n-ary relations would exist. Okay, a non-empty set. > c2: a collection of n-ary relations, each of which would correspond to an > n-ary symbol of the language. Hmm, I suppose that's okay. But I'd rather say, a function that assigns to each n-ary predicate symbol of the language an n-ary relation on S. > c3: collection of (subjective) interpretations, each of which would predicate > a theorem-formula as true. YOU want c3 for whatever odd reason you do. I have no use for c3. Through c2 you had a nicely rigorous mathematical definition going, but then you obliterated it with the UNDEFINED terminology "subjective interpretations" and "would predicate a theorem-formula as true". > Now MoeBlee suggested above: > > "Anyone may state definitions and explicate a discussion on the basis > of those definitions." Sure. Though that was not intended to disallow that we may find certain definitions and explications to clash so strongly with ordinary terminology as to be irritating to work with. > So in this context here and now, let's call S a "structure" (and temporarily > forget if some text books reserve this word for something else). So, a 'structure' is defined by you now to be what we ordinarily call 'the universe' or 'the domain of discourse' of a structure. What is the advantage of you so confusingly switching these defintions? > Then it's > not hard to see that the relativity of a model M would come from component c3! Since c3 is UNDEFINED NONSENSE, it's hard to see what comes from it! > For example, if the formula is "a < b", S = {a',b'}, and the n-ary relation > is {(a',b')}, I still could at my subjective willingness interpret (or predicate) > "a < b" as false while you or any other as true. So who the L cares about that? We already have a formal mathematical definition of 'true in the model', but you'd rather we use some junky UNDEFINED nonsense about "subjective interpretation" instead. What possible advantage comes from that? > And so relative to whose > predicating or interpreting, M would be or *not* be a model of say T = {a < b}, > or for that matter of T = {~(a < b)}. > > Now back to sPA, let S be the structure (i.e. the _set_) of individuals of a > model of the integers (i.e. not the natural numbers). The long and the short > of it is out of S, we know there exist _uncountably many_ "successor" functions > S()'s, hence uncountably many n-aries "addition", "multiplication", and "less-than". > Put if differently, not only S is a structure for sPA, it would be the very same > structure for _uncountably many_ models of each "PA" theory (written in L(sPA)). > But it's not hard to demonstrate that due to the subjective interpretation in c3, > if one interpret S as a model of a "PA", others might disagree and (re)interpret S > as *not* a model this "PA". Forget about whatever involves c3. If you'd have me adopt your c3, then I might as well have you adopt: A sentence is true in a model M iff one of the snails in my garden leaves, on the walkway, a trail more than 4 inches on the second Wednesday of the next month. > Of course when we talk about "PA" we typically talk about it outside the context > of this orangutan sPA system. But it should not matter! Given *any* L(PA), No, there is no "any" L(PA). L is an OPERATION. For each theory T, there is exactly ONE object that is L(T). For each theory, there is exactly one object that is THE language of that theory. > one could > consider it as part of L(sPA). No problem with L(sPA) as different from L(PA) if sPA and PA are different and happen also to have different languages. > And given a model - over a structure S - of *any* > "PA" theory one could interpret this structure S as a non-model of this "PA". > > In summary, a model is always (at least) *relative* to: In summary, apply c3, then check the snail trails each first Wednesday of the month, then take the Boolen product, then toss a coin for an arbitrary number of trials, then disregard having done all that, then declare the "relativity of predicating", then take deep inhalations from a bottle of model airplane glue and forget about everything whatsoever. MoeBlee
From: herbzet on 5 Jan 2008 02:28 MoeBlee wrote: > herbzet wrote: > > MoeBlee wrote: > > > I'm not familiar with the expression 'null theory' to refer to the set > > > of validities, but I'll go along with it. > > > > I probably just made it up, although it seems right. It's the theory > > with no non-logical axioms (ie only validities as theorems). It seems > > to me that this is a degenerate case of the word "theory": usually > > I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic > > and to have some non-logical axioms/theorems. > > You may consider it a degenerate or trivial case, but it is still a > theory by such ordinary defiinitions of 'theory' that I mentioned. Indeed it is. > > > As to the question, I'd have to think about it. > > > > Bingo. That's what's "a new thought to me": that an arbitrary structure > > is a model of some non-null theory. I don't happen to know whether > > that's true. > > I'd have to clear some technicalities, but I suspect it is true. But > in any case, every structure is a model of the set of valid sentences > (in the language that the structure is a structure for). And that set > of valid sentences is a theory, even if only trivially so. Therefore, > every structure is a model of some theory, even if only of the theory > that is the set of valid sentences in the language that the structure > is a structure for. For any structure S in a language L the structure assigns a truth-value to every sentence of L. For every sentence phi of L, one of phi and ~phi will be true and one will be false in S. On the assumption that not all sentences of L are validities and their negations, then S will be a model of some sentences that are not validities, and hence will be a model of some non-null theory of language L. Perhaps you would like to clean that up a bit. The further question would be whether every structure is a model of some non-null recursively axiomatized theory. > > > > Shall we agree as to what sets of sentences constitute "a theory" first? > > > > > For simplicity, let's confine to classical first order. > > > > > Authors such as Enderton take a theory to be any set of sentences > > > closed under entailment (which, thanks to the completeness theorem, is > > > a set of sentences closed under provability). Authors such as Chang & > > > Keisler take a theory to be any set of sentences. And some authors > > > take a theory to be a pair <S E> where S is a set of sentences and E > > > is the set of sentences entailed by S (which, thanks to the > > > completeness theorem, is the set of theorems of S). > > > > > I adopt Enderton's definition. > > \> > OK. > > > > > > I usually think of "a theory" as having a recursive (or at least r.e.) > > > > set of axioms. > > > > > That is a recursively axiomatized theory. There are theories that are > > > not recursively axiomatized. > > > > Yes, they seem a little imaginary to me, but I'm willing to go along > > with it. > > They are famous and important. Most famous in that way is perhaps the > theory that is the set of sentences true in the standard model for the > language of PA. It is a famous theorem that that theory is not > recursively axiomatized (or 'not recursively axiomatizable' if you > prefer). That there are theories that are not recursively axiomatized > (or 'not recursively axiomatizable' if you prefer) is a famous and > important subject in mathematical logic. Yes, I've heard of that theory. It still seems a little imaginary to me, not being recursively axiomatizable. > > > > And as being consistent (i.e. having a model)! > > > > > Those are consistent theories. There are theories that are not > > > consistent. > > > > It would seem that there is but one such theory over any signature: > > the theory that consists of all sentences in the language. > > Right. > > > This seems like another degenerate case of "theory". > > If you prefer to think of it that way. > > > I'm willing to > > accept the informal locution "inconsistent theory"; I'm willing > > to accept an "inconsistent theory" as existing under the Enderton > > definition of "theory". > > > > But the extremal cases of "theories" that contain only validities, > > or that contain every sentence, do seem a little silly to me. I > > guess it's the price you pay for definitional elegance. > > Whether it seems silly, it still is indeed the case, from the > definitions. Right. Thanks for pointing out these consequences of the definitions. > > > My terminology is 'true in a model' and it is precisely correct that > > > '2=1+1' is true in some models and not true in other models. Nam's > > > terminology is 'true relative to a model', and though I do not in any > > > way claim to arbitrate what HE means by that, my point was just to say > > > that IF he means 'true in a model', then yes, of course, '2=1+1' is > > > true in some models and not true in other models. Then, there were the > > > rest of my remarks. > > > > Yuh, I guess the argument turns on whether you want to affirm > > > > a) 2 = 1 + 1 is false in no model (of PA) > > > > or > > > > b) 2 = 1 + 1 is false in some models (of the language of PA). > > I affirm both. But not, I hope, without the parenthetical qualifications. That would be very confusing. > But the particular remark I made didn't mention PA. True. That is perhaps an unwarranted assumption on my part as to what was being talked about. > > It seems to be a simple ambiguity to be resolved. I personally would > > prefer (a) and would assume that you are using "model" in the sense > > of (a) in the absence of an explicit definition otherwise. I concede > > that it is possible that Nam uses the word "model" in some sense > > other than (a). > > There is no conflict among a), b), and my use of the word 'model'. As > to Nam's nottions, that's a whole other matter. > > > I acccept that the literature allows the usage of "model" in the > > sense of both (a) and (b). But, as I said before, this seems ill- > > advised, in that it allows this ambiguity to arise. > > There is not an ambiguity in what I said. > > '2 = 1 + 1' is true in some models and false in other models No doubt it is false in some models of some theories, hence it is false in some models, QED. [...] > > Actually, I find it even more annoying when someone is being a jerk > > while being right! > > I guess that's a joke or something. Of course, it's subjective what > annoys a person. But it does seem disconnected to me to be more > annoyed by someone who is at least correct. I can assure you from long experience, I'd rather play chess with a poor loser than a swaggering, ungracious winner. -- hz -- Posted via a free Usenet account from http://www.teranews.com
From: Newberry on 5 Jan 2008 11:19 On Jan 4, 10:30 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Newberry <newberr...(a)gmail.com> writes: > > On Dec 8, 2:04 pm, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > >> Newberry <newberr...(a)gmail.com> writes: > >> > The cornerstone of TF's argument is that the consistency of ZFC is > >> > provable in ZFC + an axiom of infinity, which is no longer manifestly > >> > true. We hesitate as we go higher up in the chain of theories, hence - > >> > according to TF - we are not better than any machine because we cannot > >> > say about an arbitrary system that its Goedel formula is true. Yet he > >> > is absolutely sure that PA and ZFC are consistent. > > >> Where did TF claim that he was "absolutely sure that ... ZFC [is] > >> consistent"? > > >> I see no evidence that that was his view in the cite you posted > >> earlier in the thread. > > > He has an entire chapter in his book "Skepticism and Confidence", > > where he refutes the skeptics. > > Having dug out the book in question, I now have the context > of the following. > > > "And given that the axioms of ZFC are so utterly compelling, so > > obviously true in the world of sets, we can do no better than to adopt > > these axioms as our starting point. Since the axioms are true, they > > are also consistent." [p.105] This is a clear affirmative statement. He believes that the axioms are manifestly true. > The very next sentence says: > > "Again, the point at issue is not whether such a view of the axioms of > ZFC is justified, but whether it makes good sense to appeal to the > incompleteness theorem in criticism of it." > > The context is exactly a critique of the argument that suggests > that *if* someone has certain knowledge of the truth of the axioms > of a system *then* they will run into trouble from the incompleteness > theorem. He is aruing that Goedel's theorem cannot be used to doubt the truth of ZFC.It is not mutually exclusive with the belief that the axioms of ZFC ARE manifestly true. Basically he is saying that EVEN IF you do not believe that the axioms of ZFC are manifestly true you cannot use Goedel's theorem to doubt ZFC. Read it more carefully. His book is not exactly clear in the sense that he jumps from aruing one point to another without making it explicitly clear. There is no one single thread of reasoning. > So the view expressed are *hypotheical*, not TF's at all. > And having set up the context clearly, TF reminds the reader > of the context afterwards. > > > "if the axioms of ZFC are manifestly true, they are obviously > > consistent" [p.105] > > As I said before, this is a conditional statement. > Why on earth do you think it expresses a commitment > to the antecedent being true? > > At this rate someone will have to give us "The uses and abuses > of TF's writings". > > (And, while I'm here, he does not refute scepticism in this chapter.) What do you think he is doing? > -- > Alan Smaill- Hide quoted text - > > - Show quoted text -
From: Alan Smaill on 5 Jan 2008 11:48
Newberry <newberryxy(a)gmail.com> writes: > On Jan 4, 10:30�am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> Newberry <newberr...(a)gmail.com> writes: >> > On Dec 8, 2:04 pm, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> >> Newberry <newberr...(a)gmail.com> writes: >> >> > The cornerstone of TF's argument is that the consistency of ZFC is >> >> > provable in ZFC + an axiom of infinity, which is no longer manifestly >> >> > true. We hesitate as we go higher up in the chain of theories, hence - >> >> > according to TF - we are not better than any machine because we cannot >> >> > say about an arbitrary system that its Goedel formula is true. Yet he >> >> > is absolutely sure that PA and ZFC are consistent. >> >> >> Where did TF claim that he was "absolutely sure that ... ZFC [is] >> >> consistent"? � >> >> >> I see no evidence that that was his view in the cite you posted >> >> earlier in the thread. >> >> > He has an entire chapter in his book "Skepticism and Confidence", >> > where he refutes the skeptics. >> >> Having dug out the book in question, I now have the context >> of the following. >> >> > "And given that the axioms of ZFC are so utterly compelling, so >> > obviously true in the world of sets, we can do no better than to adopt >> > these axioms as our starting point. Since the axioms are true, they >> > are also consistent." [p.105] > > This is a clear affirmative statement. He believes that the axioms are > manifestly true. Only if you ignore the context!! In context, it is completely clear that he envisages a *what if* scenario, and is not presenting his own opinion. >> The very next sentence says: >> >> "Again, the point at issue is not whether such a view of the axioms of >> ZFC is justified, but whether it makes good sense to appeal to the >> incompleteness theorem in criticism of it." >> >> The context is exactly a critique of the argument that suggests >> that *if* someone has certain knowledge of the truth of the axioms >> of a system *then* they will run into trouble from the incompleteness >> theorem. > > He is aruing that Goedel's theorem cannot be used to doubt the truth > of ZFC. Yes (in a particular context) -- that doesn't mean he *is* arguing that ZFC is true, which is your claim. > It is not mutually exclusive with the belief that the axioms of > ZFC ARE manifestly true. Agreed; but, contrary to your claim, he does *not* make the claim that they are true, which is a whole different situation. > Basically he is saying that EVEN IF you do > not believe that the axioms of ZFC are manifestly true you cannot use > Goedel's theorem to doubt ZFC. More accurately, he's saying that Goedel's theorem gives no help to a sceptic arguing against someone who thinks that the axioms of ZFC are clearly true. > Read it more carefully. I have read it carefully; there are simnply no grounds there for saying that *TF* believed that the axioms of ZFC are clearly true, as you suggest. > His book is not exactly clear in the sense > that he jumps from aruing one point to another without making it > explicitly clear. There is no one single thread of reasoning. He works hard to make clear what is at issue at any moment, and it's clear here that the statement you quoted is in the context of a temporary assumption that someone takes the axioms to be manifestly true, in order for TF to see what the consequences are. >> �So the view expressed are *hypothetical*, not TF's at all. >> And having set up the context clearly, TF reminds the reader >> of the context afterwards. >> >> > "if the axioms of ZFC are manifestly true, they are obviously >> > consistent" [p.105] >> >> As I said before, this is a conditional statement. >> Why on earth do you think it expresses a commitment >> to the antecedent being true? >> >> At this rate someone will have to give us "The uses and abuses >> of TF's writings". >> >> (And, while I'm here, he does not refute scepticism in this chapter.) > > What do you think he is doing? He's saying that the sceptics are not justified in using Goedel's theorem as evidence for their position, and that scepticism can only be justified on other grounds -- he's not saying that there is no justification for scepticism. -- Alan Smaill |