Re: Fitch's paradox and OWA
From: Nam Nguyen <namducnguyen_at_shaw.ca>
Date: Thu, 31 Dec 2009 16:40:52 -0700
Message-ID: <aEa%m.273$Mv3.262_at_newsfe05.iad>
>
> Yes, in this context. Since we are finite beings we need to use finite
> systems.
>
>
>
>
> This is the usual first-order initial-algebra definition, and with the
> addition of "succ x = succ y -> x=y" and an induction schema gives
> first-order Peano Arithmetic. First-order logic is indeed formal (i.e.,
> syntactic) in that all inferencing activities consist of finite
> operations on finite strings. But, via Goedel and others, the Peano
> axioms do NOT fully characterise the natural numbers N. N is indeed a
> model (the Standard Model) which satisfies these axioms, but there are
> also *non-standard models* which satisfy these axioms -- these models
> contain infinite elements in addition to the usual naturals.
>
> You can get some of the flavour of non-standard models by considering
> the following non-standard model for just succ, where every element has
> a unique successor and predecessor:
>
> 0, 1, 2, ... ..., w-2, w-1, w, w+1, w+2, ...
>
>
> So, we can readily produce purely formal systems that are satisfied by
> N, but all of them (as far as I know) are also satisfied by other,
> non-standard, models. Try as we might, those pesky infinite
> non-standard integers keep cropping up. That is the sense in which I
> mean that we apparently can not formally fully characterise N.
>
> (Note that we similarly cannot formally define "finite", so the dodge of
> saying something like "the naturals are defined by the Peano axioms plus
> the restriction that everything is finite" can not be expressed purely
> formally.)
>
>
>
> I am not an expert in that field, but I believe that almost all of real
> analysis can be reconstructed using just computable numbers, e.g. the
> work of Bishop.
>
>
> They are true or false in any *particular* model. Since we apparently
> cannot formally pin down arithmetic to have just one particular model
> (the Standard one) then there will always be some arithmetic statements,
> the undecidable ones, which are true in some models and false in others.
Date: Thu, 31 Dec 2009 16:40:52 -0700
Message-ID: <aEa%m.273$Mv3.262_at_newsfe05.iad>
Barb Knox wrote:
> In article
> <a3f061ed-3838-4be9-b73a-836141dc640f_at_u7g2000yqm.googlegroups.com>,
> Marshall <marshall.spight_at_gmail.com> wrote:
>
>> On Dec 30, 8:16 pm, Barb Knox <s..._at_sig.below> wrote: >>> Marshall <marshall.spi..._at_gmail.com> wrote: >>> >>> By the nature of the construction of predicate logic, every arithmetic >>> formula must be either true or false in the standard model of the >>> natural numbers. >>> >>> But, we have no satisfactory way to fully characterise that standard >>> model! We all think we know what the natural numbers are, but Goedel >>> showed that there is no first-order way to define them, and I don't know >>> of *any* purely formal (i.e., syntactic) way to do do. >> I was more under the impression that Goedel showed there >> was no complete finite theory of them, rather than no >> way to define them. Are you saying those are equivalent?
>
> Yes, in this context. Since we are finite beings we need to use finite
> systems.
>
>
>>> (The usual ways >>> to define them are not fully syntactic, but rely on "the full semantics" >>> of 2nd-order logic, or "a standard model" of set theory, both of which >>> are more complicated than just relying on "the Standard Model" of >>> arithmetic in the first place.) >> Here's a possible definition: >> >> nat := 0 | succ nat >> >> x + 0 = x >> x + succ y = succ x+y >> >> x * 0 = 0 >> x * succ y = x + (x * y) >> >> Is there some way this definition is not fully syntactic? >> It uses no quantifying over predicates, so it can't be >> using second order logic.
>
>> It certainly seems to me that the above is fully syntactic, >> and is a complete definition of basic arithmetic. Are >> there statements that are true of this definition that >> can't be captured by any finite theory? Sure there >> are, but that has nothing to do with whether it's >> a proper syntactic definition. To say it's not a syntactic >> definition, you have to point out something about >> it that's not syntactic, or not correct as a model >> of the naturals.
>
> This is the usual first-order initial-algebra definition, and with the
> addition of "succ x = succ y -> x=y" and an induction schema gives
> first-order Peano Arithmetic. First-order logic is indeed formal (i.e.,
> syntactic) in that all inferencing activities consist of finite
> operations on finite strings. But, via Goedel and others, the Peano
> axioms do NOT fully characterise the natural numbers N. N is indeed a
> model (the Standard Model) which satisfies these axioms, but there are
> also *non-standard models* which satisfy these axioms -- these models
> contain infinite elements in addition to the usual naturals.
>
> You can get some of the flavour of non-standard models by considering
> the following non-standard model for just succ, where every element has
> a unique successor and predecessor:
>
> 0, 1, 2, ... ..., w-2, w-1, w, w+1, w+2, ...
>
>
> So, we can readily produce purely formal systems that are satisfied by
> N, but all of them (as far as I know) are also satisfied by other,
> non-standard, models. Try as we might, those pesky infinite
> non-standard integers keep cropping up. That is the sense in which I
> mean that we apparently can not formally fully characterise N.
>
> (Note that we similarly cannot formally define "finite", so the dodge of
> saying something like "the naturals are defined by the Peano axioms plus
> the restriction that everything is finite" can not be expressed purely
> formally.)
>
>
>>>> If it's actually the case (that every statement of basic arithmetic >>>> is either true or false) then it's not a shortcoming to say so. >>>> On the contrary, that would be a virtue. >>> Speaking philosophically (since I'm posting from sci.philoisophy.tech), >>> entities which in some sense exist but are thoroughly inaccessible seem >>> to be of little value. This applies to the truth values of any >>> statements which can never be known to be true or false. >> While I have sympathy for that position, I don't think it's >> tenable in the long run. Or anyway, it's not tenable to go >> from "of little value" to suggesting that we should, say, >> not attend to the real numbers because of the existence >> of uncomputable numbers,
>
> I am not an expert in that field, but I believe that almost all of real
> analysis can be reconstructed using just computable numbers, e.g. the
> work of Bishop.
>
>> or suggest that statements >> that are undecidable one way or the other are somehow >> neither true nor false. What they are is undecidable.
>
> They are true or false in any *particular* model. Since we apparently
> cannot formally pin down arithmetic to have just one particular model
> (the Standard one) then there will always be some arithmetic statements,
> the undecidable ones, which are true in some models and false in others.
Agree. The question - and the heart of my argument - is whether or not there exists a formula F such that it's impossible to know/assert a truth value in the collection K of _all_ arithmetic models: K = {the standard one, the non-standard ones}? I've argued that there exist such statements.
> Thus it is unreasonable to say that an undecidable statement is simply
> "true" or "false" -- we need to specify a particular model, almost
> always the Standard one, which we can not fully characterise formally.
>
> This doesn't prevent doing interesting number theory, but it is at least
> somewhat bothersome from a foundational perspective.
Arguably, FOL isn't just for number theories and so there's always a possibility the existences of such formulas might shed some light about FOL systems that we've largely ignored: e.g. systems that have infinite number of logical symbols, some of which might represent isomorphic - but different - operations. Received on Fri Jan 01 2010 - 00:40:52 CET