Hi Brian,

Le 10-avr.-08, à 04:35, Brian Tenneson a écrit :

>
> Hi Bruno,
>
> It's not a new idea, no. However, I find the classical logic
> restriction to make set theories with a universal set as unnatural
> (e.g., some automatically sacrifice choice) as one that uses FL might
> seem to others.


Although I feel certainly being a platonist with respect to the  
arithmetical reality and machines, I am not sure about sets. The notion  
of set is far too rich.




> I mainly want to know if Russel type paradoxes are
> completely universal,

It is hard for me to figure out what you mean here. In both NF, or in  
computability theory you can give meaning to V is-in V. In Quine New  
Foundation you have model with V-is-in V, but nobody knows if NF is  
consistent. In computability V is-in V (for example the universal  
dovetailer does run the universal dovetailer), and this does not lead  
to paradoxes or contradictions but only to infinities or non stopping  
machines.




> which would be interesting, in all
> generalizations of classical logic or even logics without excluded
> middle.  W/o Em, I think no paradoxes are even paradoxes, though.
> (Boring w/o Em?)

?


>
> Not as interesting when one changes logic, maybe to some.  But to me,
> changing the logic, it's like non Euclidean geometry.  Why assume (not
> you, but the mathematical masses) that non-classical logic need be
> uninteresting?


I wrote in some paper that the beauty of classical logic is that it  
forces us to see the beauty of non-classical logics.
Classical logic is the simpler and common way to describe what is a  
non-classical logic. Like common sense is the best tool to go beyond  
common sense.
You will not find a fuzzy book on fuzzy logic, with fuzzy theorems and  
fuzzy price.



>  It seems to have been quite opposed until recently
> (ie, basically less than a century).

Even that could be a matter of debate, but ok: formal non classical  
logic is a recent development. Yet there is a sense to say that very  
old mathematics and engineering were primarily intuitionistic.  
Classical logic could be attributed to the greeks. The main power of  
classical logic is that it allows the possibility of having partial  
information, and allowing ignorance. It makes possible the theological  
reasoning.
I guess you know Garden's proof of the existence of irrational numbers  
x y such that x^y is rational. It illustrates how simple and powerful  
and tolerant classical logic can be. But Garden's proof can be replaced  
by a constructive proof. Yet this one is hard and can be communicated  
only to expert in number theory. Then, in theoretical computer science,  
and even more in theoretical artificial intelligence, most proofs are  
*necessarily* not constructive. Those vast landscape are threw away, or  
made less accessible when you weaken the logic.
Also, most weakened logics can get some sense by having epistemical  
interpretations in classical logic. It is one of the main use of  
(classical) modal logic.
In my (humble) opinion: believing that a non-classical logic can be  
"fundamental" or absolute is the same as Berkeley, Wittgenstein,  or  
Brouwer's type of mistake: mainly to confuse the unknown reality with  
one of its many mode of apprehension. Like in "esse est percipi".



>
>
> I have been mainly working on different approaches recently.  Now I'm
> trying to stay in classical logic and challenge your valid objections
> to Tegmark-like mathematizations of physics, based on Russell-style
> lines of thought.

OK.



> The idea is that a non-well founded set theory,
> where a set could easily be an element of itself, is all I really need
> to resolve (seemingly?) the objections.

With the comp assumption we do not need more that the fact that some  
number n belongs to the domain of the nth partial functions (n is-in  
W_n, or f_n(n) converges). Now I have no conceptual objection against  
non-well founded set theories.
Hmmm .... except that I don't really believe in sets, ...

It is ok if you like sets. I am personally problem driven, and try to  
avoid discussion on theories. They are like people who discuss days  
after days about which programming language is the better and who never  
programs, or like people who discuss about which car they like the most  
and never drive.
A theory is just a source of light to see in the dark 'reality', and  
should not be confused with the "reality" itself. Except in  
metamathematics were the (formal or mechanical) theories are the object  
under study. (yes theories and machine also belong to "reality"). In  
that case, they are all interesting a priori. For me, axiomatic set  
theories are just good examples of very strong lobian machine, capable  
of knowing the whole of the "theology" of much simpler lobian machine  
like Peano arithmetic. (Lobian machine are, roughly speaking, universal  
machine capable of knowing that they are universal; they obey Lob's  
theorem (a generalisation of Godel's theorem). Such machine knows that  
thay are terribly ignorant: Lob's theorem is a self-modesty result.

I love machine for their unbounded imagination ...

Best,

Bruno



>
>
> This post at "project virgle" might be interesting as some of Rose's
> ideas are invoked:
> http://groups.google.com/group/virgle/browse_thread/thread/ 
> 76f99835bbfc9b30/4c617765852ced87? 
> hl=en&lnk=gst&q=tenneson#4c617765852ced87
>
> On Mar 27, 3:57 am, Bruno Marchal <[EMAIL PROTECTED]> wrote:
>> Hi Brian,
>>
>> Your idea of a universal set, in case it works, would indeed meet one
>> of the objection I often raised against Tegmark-like approaches,  
>> mainly
>> that the whole of mathematical reality cannot be defined as a
>> mathematical object. Of course this is debatable, and a case can been
>> made that such a universal set can exist (see the Forster reference
>> below).
>> Nevertheless I have no clues why do you want such an universal set to
>> be fuzzy, except perhaps by the analogy which can exist between the
>> empirical multiverse and some sort of fuzzy physical universe. A
>> problem with fuzzy set is that there are many approaches, and they do
>> not seem to converge on some standard apprehension. Perhaps you know
>> better. Have you written a longer text?
>> Now, once you assume the computationalist hypothesis in the cognitive
>> science (NOT in the physical science!) and once you are aware of the
>> mind-body problem (or the first person/third person relationship
>> problem) then you will be confronted with my other objections to
>> Tegmark, mainly the fact that the mind-body problem is still somehow
>> put under the rug. I suggest you read my texts (url below, or see the
>> Archive of this list) for appreciating that a universal structure
>> definitely cannot exist. Like in Plotinus or Cantor the big whole
>> cannot be made first order citizen.
>> Of  course with comp (actually with only Church's Thesis) we do have
>> some "universal structure" like the universal *machine* or the
>> universal dovetailer, and those are embedded in the structure they
>> deploy. That is why comp works. But of course a universal machine does
>> not describe a universal set in your sense.
>>
>> For the existence of a universal set in the context of Quine New
>> Foundation set theory (NF) I suggest you consult the book by
>>
>> T. E. FORSTER, Set Theory with a Universal Set. Oxford Science
>> Publications, 1992. Oxford.
>>
>> Brunohttp://iridia.ulb.ac.be/~marchal/
>>
>> Le 23-mars-08, à 05:46, Brian Tenneson a écrit :
>>
>>
>>
>>> I would tend to think that most mathematicians and even more
>>> physicists and even more engineers and even more laymen would say  
>>> that
>>> 'just' is a huge, huge understatement.
>>
>>> However, from the perspective of Non-Classical logic (be it
>>> paraconsistent or fuzzy), that sentence was perfectly formulated, in
>>> my humble opinion, and that article was not written with all forms of
>>> non-classical logic in mind.
>>
>>> What I need to show is that the answer is different or the same in  
>>> all
>>> MV-Algebras.  My guess is looking at just [0,1], as proofs done in
>>> [0,1] can sometimes be carried over to all MV-algebras using Chang's
>>> theorems, mentioned above, which connect just [0,1] to all of these
>>> types of fuzzy logics, would be a big step towards settling my
>>> investigation for all MV-algebras.
>>
>>> In other words, I want to investigate Russell's "paradox" for as many
>>> types of logic that already have been developed, to determine how
>>> "true" Russell's "paradox" is for any logic that is not binary logic.
>>> I don't know, it could be false in +all+ logics that could be
>>> reasonably called logics or, more interestingly to me, true in some
>>> but not all.  Then, in that event, the investigation would be to find
>>> out in which logics Russell's Theorem (ie, no universal set exists in
>>> that logic-set-theory combo) is true and in which is false.  Then I'd
>>> like to know why Russell's Theorem is true sometimes and why not
>>> sometimes.  Or why it's always true.  Why being the main question for
>>> me.  I think the physicist would mainly be interested in whether any
>>> universal (fuzzy) sets can consistently exist, and the logician more
>>> interested in why it exists.  However, why it exists is, I think,  
>>> also
>>> interesting to the philosopher in that it is like asking "why does  
>>> the
>>> universe exist" assuming the MUH and that any universal sets can
>>> consistently exist.
>>
>>> On Mar 22, 9:30 pm, <[EMAIL PROTECTED]> wrote:
>>>>> Does 'any theory' in the following quote include theories that
>>>>> involve
>>>>> logics with every MV-algebra as their truth set and every set of
>>>>> syntactical axioms or is this just any theory using binary logic?
>>
>>>> my guess is: just any theory using binary logic.
> >
>
http://iridia.ulb.ac.be/~marchal/


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