# Re: Fitch's paradox and OWA

From: Nam Nguyen <namducnguyen_at_shaw.ca>
Date: Thu, 31 Dec 2009 16:40:52 -0700

Barb Knox wrote:
> In article
> 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

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