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

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From: MoeBlee on 30 Mar 2010 18:11

On Mar 30, 2:42 pm, Transfer Principle <lwal...(a)lausd.net> wrote:

> Multiplication of ordinals is defined using
> transfinite recursion -- which requires the Replacement Schema!

See your next paragraph:

> At this point, one might point out that we only need to define the
> operations on the _finite_ ordinals, and so we ought to use finite,
> rather than _trans_finite, recursion. But finite recursion -- though
> it
> apparently avoids Replacement Schema -- requires the Axiom of
> _Infinity_ in its proof.

(1) I must have overlooked that you're not allowing the axiom of
infinity. (2) Of course, per ordinary set theory, without the axiom of
infinity, it is not possible to have a function whose domain is the
set of ordered pairs of finite ordinals. But still, if I am not
mistaken, even without the axiom of infinity, we can define a 2-place
operation symbol "+" that yields the ordinary arithmetic sum of two
finite ordinals (and yields "junk" 0 otherwise). This is done by using
such techniques as the ancestral, as I recall. It's mentioned in
various texts in set theory.

> "1 is the multiplicative identity" and all that. But MoeBlee and
> Ullrich
> were explicitly working in Z(-Regularity),

I mentioned Ullrich in connection with ".999..=1"; I didn't say
anything about Ullrich in connection with multiplicative identity.
Also, in the proof of ".999...=1", I don't recall whether I mentioned
the explicit set theoretic axioms, but, in any case, yes, no axioms
not in Z-regularity were used.

> I'm not sure whether MoeBlee and Ullrich wish to avoid Replacement
> in the same way that a finitist/"crank" wishes to avoid Infinity.

No, I don't think replacement is to be avoided in the manner that some
people think infinity ought to be avoided.

> It
> may
> be that MoeBlee and Ullrich are open to using Replacement when it
> is absolutely inevitable, but as long as they can avoid the schema --
> and one can construct R without it -- they'll do so.

I have no idea what Ullrich thinks about this, but the above is okay
for me. Sure, if I can prove something without replacement then I
prefer to do it that way. But it's not much of a philosophical issue
for me (I have no philosophical quibbles with replacement); rather
it's more an admiration of the relative economy and power of Z-
regularity.

MoeBlee

From: MoeBlee on 30 Mar 2010 18:17

On Mar 30, 2:57 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Mar 26, 10:04 am, MoeBlee <jazzm...(a)hotmail.com> wrote:
>
> > On Mar 26, 12:42 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> > > as powerful
> > This cries for a definition or explanation of what is meant by 'as
> > powerful'. In mathematical logic we have various notions such as
> > 'intepretability' and 'conservative extension'. But some of your
> > comments seem not to use 'as powerful' in such technical senses
> > (otherwise some of your questions in this regard would be non-
> > starters).
>
> Marshall Spight was the first to use the phrase in this thread. I only
> use the phrase in reponse to Spight's post.
>
> Here is a quote from the post from earlier in this thread, back on the
> 2nd of March, at 4:11PM Greenwich time:
>
> "One thing that the cranks and crankophiles never understand
> is that the systems they come up with add a lot of complexity
> while actually removing functionality or utility. To do so
> merely to avoid some counterintuitive but harmless
> property (such as vacuous truth, in Newberry's case) is
> a huge waste of time.
> Less powerful; more work to use: that's a crank theory for you."
>
> So Spight criticizes so-called "crank" theories as less powerful than
> some other theory -- presumably the standard theory (ZFC). My
> goal, therefore, is to find a theory that's _as_ powerful as ZFC, so
> that Spight would have less reason to criticize it.

Whether you're adopting his terminology or not, for such remarks as
the previous one to have much meaning, you'd need to say what YOU mean
(or what you think Spight might mean, or SOMETHING) by 'as powerful' .
For if you mean it in it's most natural sense in mathematical logic
(i.e., the sense of 'as strong'), then your project is a non-starter,
since any theory at least as strong as ZFC is not one that would be
accepted by someone who eschews ZFC (let alone the axiom of infinity
itself), since such a theory INCLUDES all the theorems of ZFC.

MoeBlee

From: Nam Nguyen on 30 Mar 2010 23:52

Tim Golden BandTech.com wrote:
> On Mar 27, 11:25 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> Tim Golden BandTech.com wrote:
>>> On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>>> Nam Nguyen wrote:
>>>>> Alan Smaill wrote:
>>>>>> Nam Nguyen <namducngu...(a)shaw.ca> writes:
>>>>>>>> Seriously, if you could demonstrate a truly absolute abstract truth
>>>>>>>> in mathematical reasoning, I'd leave the forum never coming back.
>>>>>>> If you can't (general "you") then I'm sorry: my duty to the Zen council,
>>>>>>> so to speak, is to see to it that "absolute" truths such as G(PA) is a
>>>>>>> thing of the past, if not of oblivion.
>>>>>> one day you will realise that your duty to the Zen council
>>>>>> is to overcome your feeling of duty to what is purely subjective ...
>>>>> I'm sure your belief in the "absolute" truth of G(PA) is subjective, which
>>>>> you'd need to overcome - someday. Each of us (including Godel) coming to
>>>>> mathematics and reasoning has our own subjective "baggage".
>>>>> Is it FOL, or FOL=, that you've alluded to? For example.
>>>> Note how much this physical reality has influenced and shaped our
>>>> mathematics and mathematical reasonings. Euclidean postulates had their
>>>> root in our once perception of space. From P(a) we infer Ex[P(x)]
>>>> wouldn't be an inference if the our physical reality didn't support
>>>> such at least in some way. And uncertainty in physics is a form
>>>> relativity.
>>>> The point is relativity runs deep in reality and you're not fighting
>>>> with a lone person: you're fighting against your own limitation!
>>>> Any rate, enough talk. Do you have even a single absolute truth you
>>>> could show me so that I'd realize I've been wrong all along? Let's
>>>> begin with the natural numbers: which formula in the language of
>>>> arithmetic could _you_ demonstrate as absolutely true?
>>> There is a fairly straightforward construction that can yield both
>>> boolean logic and continuous higher forms, and even a lower form that
>>> I will call universal.
>>> Constrain the real numbers to those values whose magnitude is unity.
>>> We see two options
>>> +1, -1 .
>> It's relative as to how many real numbers one could "constrain". So
>> "constraint" is a relative notion, not an absolute one.
>>
>> In any rate, in all the below (including the URL) I still couldn't
>> see an absolute truth. Could you state such truth here?
>
> By accepting the generalization of sign the existence of dimension
> follows directly.
> That is the most absolute truth that I've come up with.

So what would happen if one doesn't accept the "the generalization
of sign"? Would we get a relative truth, or an absolute falsehood?

From: Newberry on 31 Mar 2010 00:21

On Mar 30, 3:30 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Mar 29, 10:17=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> >> How are we to apply your ideas about vacuity, meaningfulness, truth,
> >> proof, what not, in context of the following mathematical observation:
> >> for any consistent theory T extending Robinson arithmetic, either
> >> directly or through an interpretation, in which statements of the form
> >> "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" can
> >> be expressed, there are infinitely many Diophantine equations
> >> D(x1, ...,xn) = 0 that have no solutions but for which
> >> "the Diophantine equation D(x1, ..., xn) = 0 has no solutions"
> >> is not provable in T.
> >> On an ordinary understanding, a statement of the form "the Diophantine
> >> equation D(x1, ..., xn) = 0 has no solutions" is true just in case
> >> D(x1 ..., xn) = 0 has no solutions. According to your account some
> >> such statements are neither true nor false
>
> >No.
>
> What do you mean, no? You are proposing to equate truth and provability.
> Some sentences of the form "the Diophantine equation D(x1, ..., xn) = 0
> has no solutions" are neither provable nor refutable.

In which theory?

It follows from
> your equating of truth and provability that they are neither true nor
> false.
>
> The reasoning that there are statements that are true, but unprovable
> goes like this:
>
> 1. Let D(x1,...,xn) be a polynomial in the variables x1, ..., xn.
>
> 2. If m1, ..., mn are n integers such that D(m1,...,mn) = 0,
> then the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0"
> is provable. We can easily prove this by plugging in m1, ..., mn and
> checking to see if the result is 0.
>
> 3. If the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0"
> is not provable, then there are no integers m1, ..., mn such that
> D(m1, ..., mn) = 0. This follows immediately from 2.
>
> 4. Note that if Phi is the formula
> "There exists x1, ..., xn such that D(x1,...,xn) = 0",
> then 3. has the form: "If Phi is not provable, then ~Phi".
> In other words, if Phi is not provable, then the negation of Phi
> holds.
>
> 5. Therefore, if Phi is neither provable nor refutable, then
> the negation of Phi holds. So if Phi is neither provable nor
> refutable, then Phi is false. ("Phi is false" means the same
> thing as "The negation of Phi holds").
>
> 6. Therefore, if Phi is neither provable nor refutable, then
> there is a statement, Phi, that is false, but not provably false.
> There is another statement, ~Phi that is true, but not provable.
>
> 7. Therefore, if there is a statement Phi (of the appropriate
> form) that is neither provable nor refutable, then provability
> and truth are not the same.
>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -

From: Newberry on 31 Mar 2010 00:26

On Mar 30, 6:07 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > On Mar 28, 9:01 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> >> Newberry <newberr...(a)gmail.com> writes:
> >> > On Mar 28, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> >> >> Newberry <newberr...(a)gmail.com> writes:
>
> >> >> > But I can. In a system with gaps Tarski's theorem does not apply. We
> >> >> > can then simply equate truth with provability.
>
> >> >> Your second sentence does not follow. You have to show that you have
> >> >> a logic in which provability turns out to be equivalent to truth.
> >> >> Tarski's theorem may not preclude this possibility, but it doesn't
> >> >> follow that you can then "simply equate truth with provability."
>
> >> > Did I say it follows? I meant that it is possible. In classical logic
> >> > withuot gaps it is impossible. Why did you not interpret what I said
> >> > this way?
>
> >> "We can then simply equate truth with provability."
>
> > It does automatically folow but we can nevertheless do that.
>
> You have to *show* that this can be done in your system.

And to a reasonable degree I have shown it. And I do NOT mean by
pointing out that Tarski does not apply.

>
> And, indeed, the word "equate" is still misleading, since it suggests
> that define true to mean "provable". That can certainly be done. I
> can say that, hereafter, when I say that a statement of PA is true, I
> mean that there is a proof of P in PA. Of course, such semantic play
> is unsatisfactory.
>
> --
> "After years of arguing I realize that your intellects are too limited
> to fully grasp my work. [...] Still, no matter how child-like your
> minds are, [...] since you have language, [...] there's a chance that
> I'll be able to find something that your minds can handle." --JSH- Hide quoted text -
>
> - Show quoted text -