Re: Fitch's paradox and OWA
From: Nam Nguyen <namducnguyen_at_shaw.ca>
Date: Thu, 31 Dec 2009 17:57:17 -0700
Message-ID: <PLb%m.284$Mv3.28_at_newsfe05.iad>
>
> "Any effectively generated theory capable of expressing
> elementary arithmetic cannot be both consistent and complete.
> In particular, for any consistent, effectively generated formal
> theory that proves certain basic arithmetic truths, there is an
> arithmetical statement that is true, but not provable in the theory."
>
> So there cannot be a complete finite theory of basic arithmetic.
>
>
>
> Well we agree on one thing. That's unusual.
>
>
> It's easy to extend this with <.
>
>
>
> The goal was to provide a syntactic definition of the
> naturals, which I did. The goal was not to provide
> a FOL model. Nonetheless it's pretty easy to
> get there from here. For example:
>
> {((x, y), z) | x+y=z}
>
>
>
> You keep assuming that the mere fact that a sentence is
> undecidable means that it has some definite truth value
> that is not one of {true, false}. Apparently you just take
> this as a given. I, however, regard it as a false statement.
Date: Thu, 31 Dec 2009 17:57:17 -0700
Message-ID: <PLb%m.284$Mv3.28_at_newsfe05.iad>
Marshall wrote:
> On Dec 31, 12:29 pm, Nam Nguyen <namducngu..._at_shaw.ca> wrote:
>> Marshall 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. >> Godel didn't show any of the 2 you've mentioned.
>
> "Any effectively generated theory capable of expressing
> elementary arithmetic cannot be both consistent and complete.
> In particular, for any consistent, effectively generated formal
> theory that proves certain basic arithmetic truths, there is an
> arithmetical statement that is true, but not provable in the theory."
>
> So there cannot be a complete finite theory of basic arithmetic.
>
>
>>> Are you saying those are equivalent? >> If I'm the one answering this question then "No": defining a model of a formal >> system is not the same as demonstrating anything about a formal system that's >> supposed to be about the model. Naturally.
>
> Well we agree on one thing. That's unusual.
>
>>> It certainly seems to me that the above is fully syntactic, >>> and is a complete definition of basic arithmetic. >> That's *not* the canonical knowledge of arithmetic: what happens to the usual >> syntactical symbol '<', in your "complete definition"?
>
> It's easy to extend this with <.
>
>
>>> 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. >> Setting aside the missing "<", what you've defined up there is >> *in no way* conforming with the _FOL definition of a model_ which >> the naturals is supposed to be collectively. For example, what's >> the set of 2-tuples that would correspond to your '+'?
>
> The goal was to provide a syntactic definition of the
> naturals, which I did. The goal was not to provide
> a FOL model. Nonetheless it's pretty easy to
> get there from here. For example:
>
> {((x, y), z) | x+y=z}
>
>
>>>>> 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, or suggest that statements >>> that are undecidable one way or the other are somehow >>> neither true nor false. What they are is undecidable. >> First order undecidable formulas are in a different class than those >> that aren't model-able, aren't truth assigned-able. >> >> I asked you before: >> >> "(1) There are infinite counter examples of GC. >> >> Tell me what you'd even suspect as a road-map to assign true or >> false to (1)?"
>
> You keep assuming that the mere fact that a sentence is
> undecidable means that it has some definite truth value
> that is not one of {true, false}. Apparently you just take
> this as a given. I, however, regard it as a false statement.
I'm not assuming anything in asking you the question, Marshall. If a simple question that you, I, or anyone could either know or don't know the answer.
If I'm to answer the question I'd say I don't know of any possible road-map. If you you think (1) is false, as you seem to have so, present your road-map, reasons based on the _accepted definitions_ of FOL models etc...to back it up
Don't just evade the question and hope that people would understand your argument! Received on Fri Jan 01 2010 - 01:57:17 CET