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On Dec 31, 7:10=A0am, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) wrote: I don't see a rule saying f |- []f. Where did you see that? He didn't say that there was an explicitly stated rule of that form. He said that in step 8 of the derivation, they use a rule that was explicitly stated as If |- f then |- []f but they use it *as if* the rule was f |- []f |

to assume K(p & ~K(p)), and show that that leads to

a contradiction. That's a proof of ~K(p & ~K(p)).

So we have |- ~K(p & ~K(p)). Then we can apply the

rule "If |- f, then |- [] f" to conclude

[]~K(p & ~K(p))

--

Daryl McCullough

Ithaca, NY

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On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote: Marshall <marshall.spi... (AT) gmail (DOT) 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. |

Quote:

Are you saying those are equivalent? |

system is not the same as demonstrating anything about a formal system that's

supposed to be about the model. Naturally.

Quote:

(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? |

of FOL, being syntactical is being part of a FOL language/formula which

this part doesn't seem to be].

Quote:

It certainly seems to me that the above is fully syntactic, and is a complete definition of basic arithmetic. |

syntactical symbol '<', in your "complete definition"?

Quote:

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. |

*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 '+'?

Quote:

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. |

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)?"

Now if you let (1') be defined as:

(1') df= (1) /\ A1 /\ A2 /\ ... A9

where A1 - A9 are Q's axioms (a la Shoenfield). Tell us, Marshall, what models or

what kinds of models that you think you could assign 'true' or 'false' to

(1')? If you really can't - which I don't think you can - then don't you at

least think of the possibility that there are arithmetic statements that can't

be true or false?

Why is it that a statement has to be true or false while _there's no way_ to

assign a truth value to it any way? Other than we might have grown up accustomed

to it, what kind of reasoning is that?

Ok I might sound a bit rhetorical here. But can you technically answer my question

about (1')?

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Nam Nguyen says... Daryl McCullough wrote: Nam Nguyen says... Daryl McCullough wrote: By the way, I haven't thought about it a huge amount, but I don't have any problems with the paradox, because I don't accept the premise: Every true proposition is potentially knowable. It seems to me that sufficiently complex true propositions may never be known. But how can we know it's true in the first place, when its being true can't be known? I didn't say that we can *know* it is true. That's my point---something can be true without anyone knowing that it is true. It might be true, for example, that there is an even number of grains of sand in the world, but we may never find that out. Is e^pi rational? We may never find out. Don't want to beat a dead horse so to speak but not knowing a truth because its proof (knowledge) is _finitely_ larger than what one can possibly know is *not* the same as not knowing a truth value because the statement is not *genuinely* truth-assigned-able. The "sand in the world" being an even number example above is of the 1st kind: not the 2nd kind. That was my point. |

on the ground that we can't assign a truth value to them; i.e., on the ground

what we've _intuitively perceived_ as the "natural numbers" is _not adequate_ for

us to say they are true or false?

Quote:

we will never know that they are true. There can also be statements that are true, but which we have no way of ever knowing that they are true. For example, I flip a coin, and before I see whether it lands heads up or tails up, it is run over by train, smashing it into a flat, smooth chip of metal. Now, there is no way of ever knowing whether it was heads-up or tails-up. But it is possible that "It was heads-up before it was smashed" is true. Statements can be true even if there is no way to ever know that they are true. |

can _not_ be assigned true or false, in the first place! Do you see that they

aren't of the same kind of statements you've alluded to?

Quote:

-- Daryl McCullough Ithaca, NY |

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On Dec 30, 8:16*pm, Barb Knox <s... (AT) sig (DOT) below> wrote: *Marshall <marshall.spi... (AT) gmail (DOT) 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? |

systems.

Quote:

*(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. |

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.)

Quote:

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, |

analysis can be reconstructed using just computable numbers, e.g. the

work of Bishop.

Quote:

or suggest that statements that are undecidable one way or the other are somehow neither true nor false. What they are is undecidable. |

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.

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.

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Marshall says... On Dec 31, 7:10=A0am, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) wrote: I don't see a rule saying f |- []f. Where did you see that? He didn't say that there was an explicitly stated rule of that form. He said that in step 8 of the derivation, they use a rule that was explicitly stated as * *If |- f then |- []f but they use it *as if* the rule was * *f |- []f No, I don't think they did that. What they did was to assume K(p & ~K(p)), and show that that leads to a contradiction. That's a proof of ~K(p & ~K(p)). So we have |- ~K(p & ~K(p)). Then we can apply the rule "If |- f, then |- [] f" to conclude []~K(p & ~K(p)) |

the assumption being K(p & ~K(p)).

However what I was referring to was specifically

how they get from step 7 to step 8 within that

RAA proof. Your response does not seem to

address that particular issue.

Are your comfortable with how step 8 is

obtained from step 7 via Rule C as described

on this page?

http://plato.stanford.edu/entries/fitch-paradox/

It's entirely possible that I misunderstand

Jan Hidder's point, or rule C, or something

else entirely, however I would like to at

least feel that we were discussing the same

step in the proof.

Marshall

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*Marshall <marshall.spi... (AT) gmail (DOT) com> wrote: On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote: Marshall <marshall.spi... (AT) gmail (DOT) 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. |

don't see how that point leads to saying that a theory is

the same thing as a definition. That is rather tantamount to

saying that theories are all there are, and that's just not

true. There are things such as computational models,

for examples. It seems entirely appropriate to me to

use a computational model as the definition of something,

which is why I gave a computational model of the naturals

as a definition.

Perhaps worse, if it's not possible to have a definition of

anything, then I don't see how you can have any

theories, either. Theory of what? If you have no

definition, I don't see how you can even claim to

have an object under discussion.

Quote:

(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 of all, I claim "succ x = succ y -> x=y" is necessarily

the case via the definition of =.

Secondly, I claim we don't need to explicitly add any

induction schema, because induction on the naturals

in this case is merely a special case of structural

induction, which is itself merely a special case of

case analysis on the constructors for nat, and case

analysis is always available, as it were.

These are perhaps just quibbles.

Quote:

*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. |

I don't see how it's supposed to be something that conforms

to the definition I gave.

For one thing, addition on the naturals is supposed to be total.

What is the result of "2 + w" under my definition of +? It does

not terminate, because you have introduced elements with

infinite descending deconstruction. That my addition operator

is total over (nat, nat) is provable; if there is some value

for which it is not total that value must therefor not

belong to nat.

For another thing, my definition doesn't have any "w" in

it, so you don't get to insert them in to the process.

We are supposed to be being syntactical here; recall

that you wanted to keep out second order logic and

set theory, so no "w".

Perhaps most importantly, I defined "nat" as those

things that are constructed via one of the two

specified constructors. Your w-elements are not

so constructed, so they cannot meet the definition

I gave.

I have noticed in the past that logicians and set

theorists don't necessarily buy the idea that

the universe consists only of those objects that

can be constructed using explicitly defined

construction rules. I am rather inclined to say

"tough," but perhaps I'll get better results if

I just say that's fine, but anything that isn't so

constructed isn't a natural, by definition.

Quote:

(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.) |

this is perhaps just a quibble.

Quote:

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. |

same as "all". For example, the order relation on computable

numbers is not itself computable, sadly. Also isn't it the

case that the least-upper-bound property is lost if we

limit ourselves to computables?

Regardless, the bigger issue, it seems to me, is

that any such system is going be be distinctly more

complex than the reals, and that complexity has a

nontrivial cost.

Quote:

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. * |

know if they are true or false. It might require an infinite amount

of computation to decide. Or just more than we will ever have.

Quote:

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. |

narrow it down to one of two possibilities, even if we can't narrow

it any further.

Quote:

This doesn't prevent doing interesting number theory, but it is at least somewhat bothersome from a foundational perspective. |

Marshall

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In article a3f061ed-3838-4be9-b73a-836141dc640f...oglegroups.com>, Marshall <marshall.spight (AT) gmail (DOT) com> wrote: On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote: Marshall <marshall.spi... (AT) gmail (DOT) 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. |

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.

Quote:

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. |

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.

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Marshall wrote: On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote: *Marshall <marshall.spi... (AT) gmail (DOT) 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. |

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.

Quote:

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. |

Quote:

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"? |

Quote:

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 '+'? |

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}

Quote:

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)?" |

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.

Quote:

Now if you let (1') be defined as: (1') df= (1) /\ A1 /\ A2 /\ ... A9 where A1 - A9 are Q's axioms (a la Shoenfield). Tell us, Marshall, what models or what kinds of models that you think you could assign 'true' or 'false' to (1')? If you really can't - which I don't think you can - then don't you at least think of the possibility that there are arithmetic statements that can't be true or false? |

way. I suppose if someone were to supply some convincing

argument as to why there must be some third possibility,

I would at least consider it.

However, I have yet to hear any convincing argument

in favor of there being a third possibility. The mere fact

of a decision being hard, even infinitely hard, does not

suggest to me the existence of some third truth value

for a sentence to have.

Quote:

Why is it that a statement has to be true or false while _there's no way_to assign a truth value to it any way? Other than we might have grown up accustomed to it, what kind of reasoning is that? Ok I might sound a bit rhetorical here. But can you technically answer myquestion about (1')? |

are talking about necessitate that a statement is either

true or false. The definition does not admit to the existence

of any third possibility. That some statements are undecidable

does not alter the definition of the terms the statements

were made with; the definitions remain as they were.

Thus every statement must have one of the two truth

values, by definition.

Now, if you want to make some new system to evaluate

statements in, that could certainly be defined with more

than the usual two possibilities. But that wouldn't be the

usual basic arithmetic; it'd be something new.

Although I don't consider reasoning by analogy to the

real world to be a great technique, it is at least suggestive

that there are real-world statements that we can

narrow down to few possibilities but cannot narrow

down to one. For example, Mr. McCullough's coin-and-

railroad story. We could even further say we were

close enough to see the coin landed definitely on

one side, but we weren't close enough to say

which side it was.

Marshall

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Barb Knox wrote: 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. |

are truth values other than true or false?

Marshall

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On Dec 31, 1:08 pm, Barb Knox <Barb... (AT) LivingHistory (DOT) co.uk> wrote: Marshall <marshall.spi... (AT) gmail (DOT) com> wrote: On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote: Marshall <marshall.spi... (AT) gmail (DOT) 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. I have no disagreement with the point about finiteness, but I don't see how that point leads to saying that a theory is the same thing as a definition. That is rather tantamount to saying that theories are all there are, and that's just not true. There are things such as computational models, for examples. It seems entirely appropriate to me to use a computational model as the definition of something, which is why I gave a computational model of the naturals as a definition. |

systems in general and constructing a _specific_ model _candidate_. In defining

the naturals, say, from computational model ... or whatever, you're just

defining what the naturals be. It's still your onerous to prove/demonstrate

this definition of the naturals would meet the definition of a model for,

say Q, PA, .... So far, have you or any human beings successfully demonstrated

so, without being circular? Of course not.