'When Are Relations Neither True Nor False? [Logic]

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From: Daryl McCullough on 2 Apr 2010 06:24

Newberry says...
>
>On Apr 1, 5:59=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:

>> How do I know that Peano arithmetic is consistent? I know it the way I
>> know any mathematical theorem I have personally proved.
>
>You proved PA consistent?

It's easy to prove in ZF.

--
Daryl McCullough
Ithaca, NY

From: Nam Nguyen on 2 Apr 2010 10:00

Daryl McCullough wrote:
> Newberry says...
>> On Apr 1, 5:59=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
>>> How do I know that Peano arithmetic is consistent? I know it the way I
>>> know any mathematical theorem I have personally proved.
>> You proved PA consistent?
>
> It's easy to prove in ZF.

Is ZF _syntactically_ consistent?

From: Nam Nguyen on 2 Apr 2010 10:10

Daryl McCullough wrote:
> Newberry says...
>
>> What I was getting at is how we know that the system is consistent.
>
> In the case of PA, it's because we know that everything it says
> about the natural numbers is true.

What is the natural numbers collectively? One in which "There are
infinitely many examples of GC" is true? Or one in which "There are
infinitely many counter examples of GC" is true? Are you sure you
know what you're talking about, in talking about the "natural numbers"?

>
> Basically, the axioms of PA consist of:
> 1. Axioms that recursively define plus and times in terms of successor.
> 2. Axioms that say that zero is the smallest natural natural number,
> and that successor is 1-1.

Those are non-induction axioms and there are only finitely number
of them and syntactically they don't mean anything, much less anything
about "recursively".

> 3. The induction axioms, which basically say that every natural
> number is obtained from zero by repeatedly applying the successor
> function.

Again a FOL formula (or schema) doesn't have intrinsic meaning, or they
could mean _anything_!

From: Jesse F. Hughes on 2 Apr 2010 13:55

Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> Daryl McCullough wrote:
>> Newberry says...
>>
>>> What I was getting at is how we know that the system is consistent.
>>
>> In the case of PA, it's because we know that everything it says
>> about the natural numbers is true.
>
> What is the natural numbers collectively? One in which "There are
> infinitely many examples of GC" is true? Or one in which "There are
> infinitely many counter examples of GC" is true? Are you sure you
> know what you're talking about, in talking about the "natural
> numbers"?

You do not have to know everything about N in order to know what N is.

I don't know whether the moon has hundreds of craters 100 yards across
or thousands. Or dozens. Yet, I'm sure I know what the moon is. And
I also know certain things about the moon.

--
Jesse F. Hughes
"You may not realize it but THOUSANDS of people read my posts.
You are putting your stupidity on wide display."
-- James S. Harris knows about wide displays of stupidity.

From: Nam Nguyen on 2 Apr 2010 14:24

Jesse F. Hughes wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Daryl McCullough wrote:
>>> Newberry says...
>>>
>>>> What I was getting at is how we know that the system is consistent.
>>> In the case of PA, it's because we know that everything it says
>>> about the natural numbers is true.
>> What is the natural numbers collectively? One in which "There are
>> infinitely many examples of GC" is true? Or one in which "There are
>> infinitely many counter examples of GC" is true? Are you sure you
>> know what you're talking about, in talking about the "natural
>> numbers"?
>
> You do not have to know everything about N in order to know what N is.
>
> I don't know whether the moon has hundreds of craters 100 yards across
> or thousands. Or dozens. Yet, I'm sure I know what the moon is. And
> I also know certain things about the moon.
>

Right. We'd have an _incomplete_ knowledge about the moon, in this example.
That would enable us to know, say, there's at least 1 crater on its surface.
But that wouldn't allow us to know truth of one simple quantification
predicate statement "All the moon craters are at least 100 yard across".

The issue here is not "what N is" per se. One could easily _define_ N as
a model of some modulo arithmetic. The issue is how we can know the
_syntactical consistency_ of PA, using strictly definition of syntactical
consistency.

Just defining "things" that we don't completely know does not _prove_
in meta level that PA is syntactically consistent.

We shouldn't invent something that isn't compatible with rules of inference,
and then "prove" things that the rules themselves can *not* prove.