Nice to see you treat it this way.

On Mon, Mar 10, 2014 at 6:01 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:

>
> On 09 Mar 2014, at 21:46, LizR wrote:
>
> On 10 March 2014 02:15, Edgar L. Owen <edgaro...@att.net> wrote:
>
>> Russell,
>>
>> Yes, but that is crazy because it assumes all theories are equally valid
>> with which I disagree. Science selects theories based on which best explain
>> the observable universe.
>>
>
> This is true. David Deutsch argues for this view convincingly in "The
> Fabric of Reality". (Russell and Brent are not disputing this view as a
> practical approach, I think, they are just pointing out that there are
> metaphysical assumptions built into it....unless they correct me on this.)
>
>
>> Therefore it is reasonable to assume that theories DO reflect actual
>> reality. They are not just made up by humans willy nilly....
>>
>
> Not willy nilly, certainly. However the assumption that they reflect an
> actual reality is only an assumption, partly because it's impossible to
> prove and partly because, in any case, all theories are open to revision.
> (This is why people keep asking you for some testable predictions of p-time
> and Bruno for testable predictions of comp, for example.)
>
>
> And that is why I keep answering that there are there. The *observable* is
> given by Z1*, or qZ1* (a whole sort of quantum "mathematics), so it is
> enough to compare the propositions of Z1* with the quantum logics of the
> physicist, to test comp+Theatetetus, and to abandon it or improve it.
>
> And then what I say, is not a proposition of a theory, but something
> derived from an hypothesis, already made by many if not most scientists,
> unfortunately made often in the materialist context, which leads to
> eliminativism of the person, when correct.
>
> Hmm... Ad we are not yet close to Z1*, especially that the main shortcut
> was in the post that you did not comment. I have also teach this to some
> students here, and it heps me to understand that this is not so simple ...
>
> Yet, I will just reprint it below, to not lost it (!), and called it "the
> real things", an obligatory passage which unfortunately, when done with all
> details is very long, as we have to explain to "a very dumb machine", the
> not so simple functioning of that "very dumb machine".
>
> I slightly correct it, I think it will be an evolving post. Don't comment
> it, as long as it looks "chinese".
>
>
>
> ======================  T H E   R E A L   T H I N G
> =================================
> And don't worry, at some point I will have to re-explained all this, to
> what some people might take as a very dumb machine, which indeed believes
> only few axioms of elementary arithmetic.
> That will be the real thing.  Some modal logics will impose themselves
> there, including the one corresponding to alternating consistent
> extensions, whose measure should provide the physical laws.
>
> The theory of everything, here, is  classical first order logic + the
> following formula:
>
> 0 ≠ s(x)
> s(x) = s(y) -> x = y
> x+0 = x
> x+s(y) = s(x+y)
> x*0=0
> x*s(y)=(x*y)+x
>
> or
>
> 0 ≠ (x + 1)
> ((x + 1) = (y + 1))  -> x = y
> x + 0 = x
> x + (y + 1) = (x + y) + 1
> x * 0 = 0
> x * (y + 1) = (x * y) + x
>
> The hard task will consist in defining an "observer" using only the theory
> above. It will be defined to be a sound extension believer of the axioms
> above, + some amount of induction axioms, of the type:
>
> (F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in the
> arithmetical language (with "0, s, +, *).
>
> We have to explain to a dumb machine, which understands only 0, s(0),
> s(s(0)), ... and can only add and multiply, but yet can reason in classical
> logic, the very functioning of such a dumb machine.
>
> There is no miracle. To define the variables, we can use the letter x, y,
> ..., it works well for many human people, but the dumb machine understands
> only 0, s(0), s(s(0)), so we will have to decide to say something like let
> the variable be defined by 0, s(s(0)), s(s(s(s(0)))), that is, the even
> number, so we will defined in arithmetic, the variable by the even numbers.
>
> Variable(x) <-> even(x) <-> Ey(2*y = x)
>
> And about "&", "->" "t", and even what about "(", and ")" ?
>

These parts are difficult to relate to people. To have you distill it here
like this, is nice.


>
> Well, again, there is no magic, you have to chose particular odd numbers
> (to not confuse them from variable) to represent them.
> That is both logic and polite.
>
> And then, how about finite sequences of symbols like "0≠s(x)"?
>
> They too must be defined in terms of number relations, and in this case a
> simple way, if we allow ourselves the use of exponentiation, is given by
> the uniqueness of prime decomposition. If g(0), g(≠), ... represents the
> particular odd number symbol for "0", "≠", etc. then you can represent
> "0≠s(x)" by
> 2^g(0)*3^g(≠)*5^g(s)*7^g(()*11^g(x)*13^g()).
>
> Or better: 3^g(0)*5^g(≠)*7^g(s)*11^g(()*13^g(x)*17^g()). To avoid the
> confusion with the variable, we start from the prime number 3, if not we
> would get an even number which represents already a variable.
>
> Then the theory itself can be defined or represented, as a number, being a
> finite sequences of the number corresponding to the axioms above.
>

Gödel was a cool, prime example of a person.


>
> We will have to defined in arithmetic what we mean by a valid proof. A
> proof is itself a finite (or infinite) sequences of application of
> inference rules, making proof "easy" to check (and hard to find in many
> domains).  So we can define in arithmetic a predicate b(x, y) true when y
> is a proof (in the dumb number language) of x.
>
> Then provable(x) can be defined by EyB(x, y). It is a Turing complete
> sigma_1 arithmetical predicate, a Löbian once it  get few induction axioms.
>
> That "provable(x)", or "believable(x)", or "assertable(x)" by the modest
> believer in the axiom above, is an arithmetical modality.
> Solovay theorem shows that a modal logic G characterize completely and
> soundly (and the logicians' sense) the logic of that provability.
> G characterized actually what the machine (theory, believer) can prove
> about this, and Solovay second theorem provides a logic G* which get
> completely and soundly the truth about the machine. Amazingly enough.  The
> main axiom of G is the Löb formula []([]p->p) -> []p. We will talk about it
> later.
>
> That provability is the "[]" of the modal logic G. It is 3p
> self-reference. 1p self-reference(s) will be obtained by weakening or
> strengthening of the "[]" in G. The 8 hypostases, and infinitely many
> others, are consequences of incompleteness, some of them should gives the
> core physical observable (mainly the []p & <>t nuances), on the sigma_1
> sentences.
>
> Modal logic can be used to describe quantum logic, (by results from
> Goldblatt and others), and on the sigma_1 sentences (on the UD, if you
> want), the "observable" (modal nuance) obeys the right modal laws needed
> for an arithmetical quantization, which should normally defined the core
> invariant laws obeyed by the measure "one" on the consistent extensions.
>
>
You loose me on the last paragraph; too dense for now. But even if I don't
comment too much instead of clowning around these days, I applaud everybody
working towards and thinking of archiving these posts, in some searchable
sense. Because the book and your papers are one thing, but to see you focus
and clarify common confusion that people have is also useful and it would
be sad if it was only NSA that keeps all these exchanges and contributions,
metadata or not, lol! PGC


>
> Bruno
>
>
>
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> http://iridia.ulb.ac.be/~marchal/
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