Consciousness is information?

[Torgny Tholerus]

Quentin Anciaux skrev:

> 2009/6/13 Torgny Tholerus <torgny@...>:
>  
>> What do you think about the following deduction?  Is it legal or illegal?
>> -------------------
>> Define the set A of all sets as:
>>
>> For all x holds that x belongs to A if and only if x is a set.
>>
>> This is an general rule saying that for some particular symbol-string x
>> you can always tell if x belongs to A or not.  Most humans who think
>> about mathematics can understand this rule-based definition.  This rule
>> holds for all and every object, without exceptions.
>>
>> So this rule also holds for A itself.  We can always substitute A for
>> x.  Then we will get:
>>
>> A belongs to A if and only if A is a set.
>>
>> And we know that A is a set.  So from this we can deduce:
>>
>> A beongs to A.
>> -------------------
>> Quentin, what do you think?  Is this deduction legal or illegal?
>>    
>
> It depends if you allow a set to be part of itselft or not.
>
> If you accept, that a set can be part of itself, it makes your
> deduction legal regarding the rules.

OK, if we accept that a set can be part of itself, what do you think
about the following deduction? Is it legal or illegal?

-------------------
Define the set B of all sets that do not belong to itself as:

For all x holds that x belongs to B if and only if x does not belong to x.

This is an general rule saying that for some particular symbol-string x
you can always tell if x belongs to B or not.  Most humans who think
about mathematics can understand this rule-based definition.  This rule
holds for all and every object, without exceptions.

So this rule also holds for B itself.  We can always substitute B for
x.  Then we will get:

B belongs to B if and only if B does not belong to B.
-------------------
Quentin, what do you think?  Is this deduction legal or illegal?


--
Torgny Tholerus

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[Quentin Anciaux-2]

Well it is illegal regarding the rules meaning with these rules set B
does not exist as defined.


2009/6/13 Torgny Tholerus <torgny@...>:

>
> Quentin Anciaux skrev:
>> 2009/6/13 Torgny Tholerus <torgny@...>:
>>
>>> What do you think about the following deduction?  Is it legal or illegal?
>>> -------------------
>>> Define the set A of all sets as:
>>>
>>> For all x holds that x belongs to A if and only if x is a set.
>>>
>>> This is an general rule saying that for some particular symbol-string x
>>> you can always tell if x belongs to A or not.  Most humans who think
>>> about mathematics can understand this rule-based definition.  This rule
>>> holds for all and every object, without exceptions.
>>>
>>> So this rule also holds for A itself.  We can always substitute A for
>>> x.  Then we will get:
>>>
>>> A belongs to A if and only if A is a set.
>>>
>>> And we know that A is a set.  So from this we can deduce:
>>>
>>> A beongs to A.
>>> -------------------
>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>
>>
>> It depends if you allow a set to be part of itselft or not.
>>
>> If you accept, that a set can be part of itself, it makes your
>> deduction legal regarding the rules.
>
> OK, if we accept that a set can be part of itself, what do you think
> about the following deduction? Is it legal or illegal?
>
> -------------------
> Define the set B of all sets that do not belong to itself as:
>
> For all x holds that x belongs to B if and only if x does not belong to x.
>
> This is an general rule saying that for some particular symbol-string x
> you can always tell if x belongs to B or not.  Most humans who think
> about mathematics can understand this rule-based definition.  This rule
> holds for all and every object, without exceptions.
>
> So this rule also holds for B itself.  We can always substitute B for
> x.  Then we will get:
>
> B belongs to B if and only if B does not belong to B.
> -------------------
> Quentin, what do you think?  Is this deduction legal or illegal?
>
>
> --
> Torgny Tholerus
>
> >
>

--
All those moments will be lost in time, like tears in rain.

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[Jesse Mazer]

Some parts of this message have been removed. Learn more about Nabble's security policy.

> Date: Sat, 13 Jun 2009 11:05:22 +0200
> From: torgny@...
> To: everything-list@...
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Jesse Mazer skrev:
> >
> > > Date: Fri, 12 Jun 2009 18:40:14 +0200
> > > From: torgny@...
> > > To: everything-list@...
> > > Subject: Re: The seven step-Mathematical preliminaries
> > >
> > > It is, as I said above, for me and all other humans to understand what
> > > you are talking about. It is also for to be able to decide what
> > > deductions or conclusions or proofs that are legal or illegal.
> >
> > Well, most humans who think about mathematics can understand
> > rule-based definitions like "0 is a whole number, and N is a whole
> > number if it's equal to some other whole number plus one"--you seem to
> > be the lone exception.
> >
> > As for being "able to decide what deductions or conclusions or proofs
> > that are legal or illegal", how exactly would writing out all the
> > members of the "universe" solve that? For example, I actually write
> > out all the numbers from 0 to 1,038,712 and say that they are members
> > of the "universe" I want to talk about. But if I write out some axioms
> > used to prove various propositions about these numbers, they are still
> > going to be in the form of general *rules* with abstract variables
> > like x and y (where these variables stand for arbitrary numbers in the
> > set), no? Or do you also insist that instead of writing axioms and
> > making deductions, we also spell out in advance every proposition that
> > shall be deemed true? In that case there is no room at all for
> > mathematicians to make "deductions" or write "proofs", all of math
> > would just consist of looking at the pre-established list of true
> > propositions and checking if the proposition in question is on there.
>
> What do you think about the following deduction? Is it legal or illegal?
> -------------------
> Define the set A of all sets as:
>
> For all x holds that x belongs to A if and only if x is a set.

It's well known that if you allow sets to contain themselves, and allow arbitrary rules for what a given set can contain, then you can get contradictions like Russell's paradox (the set of all sets which do not contain themselves). But what relevance does this have to arithmetic? Are you afraid the basic Peano axioms might lead to two propositions which can be derived in finite time from the axioms but which are mutually contradictory? If so I don't see how allowing only a finite collection of numbers actually helps--like I said in an earlier post, the number of propositions that can be proved about a finite set of numbers can still be infinite. I suppose it might be possible to make it finite by disallowing propositions which are created merely by connecting other propositions with the AND or OR logical operators, but it's still the case that if your largest whole number BIGGEST is supposed to be at least as large as some numbers humans have already conceived--say, as large as 10^100--then there is no way we could actually write out all possible propositions about these numbers that follow from some Peano-like axiom system to check manually that no two propositions contradicted each other (do you want to try to calculate 10^100 + A and A + 10^100 for every possible value of A smaller than 10^100 to verify explicitly that they are equal in every case?) So, it seems that unless you want to make your universe of numbers *very* small, you have to rely on some sort of mental model of arithmetic to be confident that you won't get contradictions from the axioms you start from, just like how people are confident in the non-contradictoriness of the Peano axioms based on their mental model of counting discrete objects like marbles (see my comments in the last paragraph of the post at http://www.mail-archive.com/everything-list@.../msg16564.html ).

Jesse

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[david.nyman]

On Apr 24, 4:39 pm, Bruno Marchal <marc...@...> wrote:

> Any content of consciousness can be an illusion. Consciousness itself
> cannot, because without consciousness there is no more illusion at all.

- just catching up with the thread, but I feel compelled to comment
that this is beautifully and clearly put.  Why does this insight
escape so many whose grasp of logic in other respects seems quite
adequate?  The word 'illusion' is often brandished in a scarily
'eliminative' way, but those who do so seem quite
'unconscious' (ironically) that the subtle knife they wield for this
excision is precisely that which they seek to excise!

David

> On 24 Apr 2009, at 06:14, Kelly wrote:
>
>
>
> > On Apr 22, 12:24 pm, Bruno Marchal <marc...@...> wrote:
> >>> So for that to be a plausible scenario we have to
> >>> say that a person at a particular instant in time can be fully
> >>> described by some set of data.
>
> >> Not fully. I agree with Brent that you need an interpreter to make
> >> that person manifest herself in front of you. A bit like a CD, you
> >> will need a player to get the music.
>
> > It seems to me that consciousness is the self-interpretion of
> > information.  David Chalmers has a good line:  "Experience is
> > information from the inside; physics is information from the outside."
>
> First person experience and third person experiment. Glad to hear  
> Chalmers accept this at last.
> In UDA, inside/outside are perfectly well defined in a pure third  
> person way: inside (first person) = memories annihilated and  
> reconstructed in classical teleportation, outside = the view outside  
> the teleporter. In AUDA I use the old classical definition by Plato in  
> the Theaetetus.
>
>
>
> > I still don't see what an interpreter adds, except to satisfy the
> > intuition that something is "happening" that "produces"
> > consciousness.  Which I think is an attempt to reintroduce "time".
>
> I don't think so. The only "time" needed is the discrete order on the  
> natural numbers. An interpreter is needed to play the role of the  
> person who gives some content to the information handled through his  
> local "brain".   (For this I need also addition and multiplication).
>
>
>
> > But I don't see any advantage of this view over the idea that
> > conscious states just "exist" as a type of platonic form (as Brent
> > mentioned earlier).
>
> The advantage is that we have the tools to derive physics in a way  
> which is enough precise for testing the comp hypothesis. Physics has  
> became a branch of computer's psychology or theology.
>
> > At any given instant that I'm awake, I'm
> > conscious of SOMETHING.
>
> To predict something, the difficulty is to relate that consciousness  
> to its computational histories. Physics is given by a measure of  
> probability on those comp histories.
>
> > And I'm conscious of it by virtue of my
> > mental state at that instant.  In the materialist view, my mental
> > state is just the state of the particles of my brain at that
> > instant.
>
> Which cannot be maintained with the comp hyp. Your consciousness is an  
> abstract type related to all computations going through your current  
> state.
>
>
>
> > But I say that what this really means is that my mental state is just
> > the information represented by the particles of my brain at that
> > instant.  And that if you transfer that information to a computer and
> > run a simulation that updates that information appropriately, my
> > consciousness will continue in that computer simulation, regardless of
> > the hardware (digital computer, mechanical computer, massively
> > parallel or single processor, etc) or algorithmic details of that
> > computer simulation.
>
> OK. But it is a very special form of information. Consciousness is  
> really the qualia associated to your belief in some reality. It is a  
> bet on self-consistency: it speed up your reaction time relatively to  
> your most probable histories.
>
>
>
> > But, what is information?  I think it has nothing to do with physical
> > storage or instantiation.  I think it has an existence seperate from
> > that.  A platonic existence.  And since the information that
> > represents my brain exists platonically, then the information for
> > every possible brain (including variations of my brain) should also
> > exist platonically.
>
> You make the same error than those who confuse a universal dovetailer  
> with a counting algorithm or the babel library. The sequence:
>
> 0, 1, 2, 3, 4, ... , or 0 1 10 11 100 101 110 111 go through all  
> description of all information, but it lacks the infinitely subtle  
> redundancy contained in the space of all computations (the universal  
> dovetaling). You work in N, succ, you lack addition and  
> multiplication, needed for having a notion of interpreter or universal  
> machine, the key entity capable of giving content to its information  
> structure. This is needed to have a coherent internal interpretation  
> of computerland.
>
>
>
> >> Conscious experience is more the content, or the interpretation of
> >> that information, made by a person or by a universal machine.
> >> If the doctor makes a copy of your brain, and then codes it into a  
> >> bit
> >> string, and then put the bit string in the fridge, in our probable
> >> history, well in that case you will not survive, in our local  
> >> probable
> >> history.
>
> > Given the platonic nature of information, this isn't really a
> > concern.  In Platonia, you always have a "next moment".  In fact, you
> > experience all possible "next moments".  The "no cul-de-sac" rule
> > applies I think.
>
> By definition, indeed, once we want to quantify the first person  
> indeterminacy.
> "next moment" makes sense only relatively to (universal) machine. It  
> is the "next step" relatively to some computation and thus universal  
> machine interpreting that "machine".
> The cul-de-sac/no-cul-de-sac depends on the points of view adopted by  
> the machine itself.
>
>
>
> >> If you say yes to a doctor for a digital brain, you will ask for a
> >> brain which functions relatively to our probable computational
> >> history. No?
>
> > I won't worry about it too much, as there is no doctor, only my
> > perceptions of a doctor.  Every possible outcome of the "brain
> > replacement operation" that I can perceive, I will perceive.
>
> Not in the relative way. You have to explain why you see apples  
> falling from a tree, and not any arbitrary information-theoretical data.
>
>
>
> > Including outcomes that don't make any sense.
>
> You have to explain why they are *rare*. If not your theory does not  
> explain why you put water on the gas and not in the fridge when you  
> want a cup of coffee.
>
>
>
> > Additionally, every possible outcome of the operation that the doctor
> > can percieve, he will perceive.  Including outcomes that don't make
> > any sense.
>
> > So it seems to me that a lot of your effort goes into explaining why
> > we don't see strange "white rabbit" universes.
>
> Indeed.
>
> > Thus the talk of
> > probabilities and measures.  I'm willing to just say that all
> > universes are experienced.
>
> That is absolutely true. But we don't live in the absolute (except  
> perhaps with salvia :). We live in the relative worlds/states. I  
> cannot go to my office by flying through the window. The probability  
> that I end up in an hospital is far greater than arriving in piece to  
> my job place.
>
> > Strange ones, normal ones, good ones, bad
> > ones, ones with unbreakable physical laws, ones with no obvious
> > physical laws at all.  It's all a matter of perception, not a matter
> > of physical realization.
>
> That is true, but we want to explain "the stable appearance of atoms  
> and galaxies", and what happens when we die.
>
>
>
>
>
>
>
> >> Yes there is a world in which you computer will transform itself into
> >> a green flying pig. The "scientific", but really everyday life
> >> question, is, what is the "probability" this will happen to "me" here
> >> and now.
>
> > I'm not sure what it means to ask, "what is the probability that my
> > computer will turn into a green pig".  One of me will observe
> > everything that can be observed in the next instant.  How many things
> > is that?  I'm not sure.  More than 10...ha!  Setting aside physical
> > limits, maybe infinitely many?  Given that I might also get extra
> > sensory capacity in that instant, or extra cognitive capacity, or
> > whatever.
>
> > So, of course all of that sounds somewhat crazy, but that's where you
> > end up when you try to explain consciousness I think.  Any explanation
> > that doesn't involve eliminativism is going to be strange I think.
>
> The comp theory explains why we cannot explain consciousness, nor  
> truth. But we can bet on computational states, then the thought  
> experiments show that physics is derivable from computer science/
> number theory in term of probabilities, and we can compare those  
> probabilities with the one we extract from the long observation of our  
> neighborhoods. Comp is a concrete testable theory, but we have to  
> derive the physics from it to do so. There is a gift because it gives  
> a complete arithmetical interpretation of an earlier type of theory  
> like Plotinus theology, which does not eliminate the person like  
> modern materialism, comp lead to a natural distinction between truth  
> about us, and provable by us.
>
>
>
> > But, if you are willing to say that consciousness is an illusion, then
> > you can just stick with materialism/physicalism and you're fine.
>
> You are right. But consciousness is the only thing I have no doubt  
> about. The *only* undoubtable thing. The fixed point of the cartesian  
> systematic doubting attitude. A theory which eliminate my first  
> person, or my consciousness, although irrefutable by me, is wrong, I  
> hope, I hope it to be wrong for you too. (Why would I send a post on  
> consciousness to a zombie?)
>
> > In
> > that case there's no need to invoke any of this more esoteric stuff
> > like platonism.  Right?
>
> Right. Materialism is a trick based on a lie (consciousness, and thus  
> pain, suffering are illusions) and an illusion (there is matter). This  
> is used to stop fundamental inquiry. It is not a coincidence that  
> authoritative theologies insists on materialism so much. Before  
> Darwinism God created the man, after Darwinism God created the matter.  
> Assuming comp, couple matter-man
>
> Any content of consciousness can be an illusion. Consciousness itself  
> cannot, because without consciousness there is no more illusion at all.
>
> Best,
>
> Bruno
>
> http://iridia.ulb.ac.be/~marchal/

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[Bruno Marchal]

On 12 Jun 2009, at 17:16, A. Wolf wrote:




>
>> We agree then.
>
> Yes, it's my fault for creating a semantics argument.  I'm usually  
> too busy
> to even read the list...every once in a while something pops up and  
> I feel
> obliged to comment even when it's the middle of a conversation.

No problem.


>
>
> I actually have some questions for the list members that are  
> relevant to the
> list content, and this coming week is break.  I may have a chance to  
> post
> them.  They're much more on the philosophical side than the  
> mathematical,
> though.


Don't hesitate,

Bruno

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




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[Bruno Marchal]

On 12 Jun 2009, at 20:31, Jesse Mazer wrote:

>Even for just an arithmetical realist. (All mathematicians are arithmetical realist, much less are mathematical realist. I am not an arithmetical realist).

I assume you meant to write "I am not a mathematical realist"?

Yes.

OK, but this leads to a further question. I remember from Penrose's book that he talked about various levels of oracle machines (hypercomputers)--for example, a first order-oracle machine was like a Turing machine but with an added operation that could decide in one step whether any Turing program halts or not, a second-order oracle machine was like a Turing machine but with operations that could decide whether a Turing machine program *or* a first-order oracle machine program halts, and so forth.

>Hmm... let us say "OK" (but this could be ambiguous). This gives mainly the arithmetical hierarchy when you start from the oracle for the halting problem. There are relativized hierarchy based on any oracle, and then starting from the halting problem in that oracle. The degrees are structured in a very complex way. 

I don't know the meaning of the phrases "arithmetical hierarchy" or "relativized hierarchy", is there any simple way of explaining? In any case, the problem I am mainly concerned with is the set of all propositions that would be considered well-formed-formulas (WFFs) in the context of Peano arithmetic (so it would involve arithmetical symbols like + and x as well as logical symbols from first-order logic, and they'd be ordered in such a way that the symbol-string would express a coherent statement about arithmetic that could be either true or false). Is there some way to come up with a rule that would allow you to judge the true or falsity of *every* member of this set of propositions using some kind of sufficiently powerful hypercomputer (presumably a fairly powerful one like the 'omega-order oracle machine' or beyond), just by checking every possible value for the numbers that could be substituted in for variable symbols? That would allow us to make sense of the distinction between "truths about arithmetic" and "statements about arithmetic provable by some axiomatic system like the Peano axioms" (the issue Brent Meeker was talking about in the post at http://www.mail-archive.com/everything-list@.../msg16562.html ), without having to worry about the "meaning" that we assign to arithmetical symbols like the number "2", or about philosophical questions about where our understanding of that meaning comes from.

I think you are asking something impossible. The notion of elementary aritmetical truth will always be simpler than the notion you need to define the hypercomputer. Just Gödel's incompleteness theorem justifies in an transparent way the separation between truth and provability in such or such formal theory. The arithmetical hierachy is the one I was describing with the sequence of alternation of quantifiers starting from decidable formula. That hierarchy does indeed described a sequence of "hypermachine" (because each level posses a Sigma_i or Pi_i completeness notion, which generalize the notion of universality for machine-with-oracle. I will have some opportunity to describe notion even more powerful. But only (with Church thesis) the Sigma_1 universality has an effective universal counterpart, represented by any computer, or universal language interpreter. 

Instead we'd have a purely formal definition about how to judge the truth-value of WFFs beyond those the Peano axioms can judge,

Remember that  PA+"consistent PA" is uncomputably more powerful than PA. And ZF set theory is much much more powerful than PA, ...

This is something I will have to explain, at least for AUDA, but in all those discussions we have to distinguih the notion of computability (which is universal and does not depend on the choice of formalism), with the notion of provability which is formalism dependent. 

And then we can generalize the notion of computability, and even of provability.

Humans provability is generally considered as much larger than PA arithmetic. 

albeit one that cannot actually be put into practice for arbitrary propositions without actually having such a hypercomputer (but for some specific propositions like the Godel statement for the Peano axioms, I think we can come up with an argument for why the hypercomputer should judge this statement 'true' as long as we believe the Peano axioms are consistent, so in this sense defining arithmetical truth in terms of such a hypercomputer is *conceptually* useful).

The whole incompleteness phenomena bears on all notion of hypercomputations. I prefer to call "hypercomputers" Gods or angels, and both humans and machines can accelerate their work by invoking them. The notion of real numbers are based on such notions. It is an open problem if we really needs those God to do mathematics, but everyone agree they simplified life considerably, like infinities in analysis, etc. Eventually such existence question is related to the relation between first and third points of view.

You can go even past finite-order oracle machines into oracle machines for higher ordinals too...

>This leads to the hyperarithmetical hierarchy and/or analytical hierarchy, where you consider formula with variables for functions or sets. There are many non trivial theorems which relate those notions (and open problems, but I have not follow the recent developments). Imo, the best book on that subject is still the book by Rogers.

Again, I'm only interested here in the type of propositions that would be judged WFFs in the context of the Peano axioms, and I think in this case the variables only refer to particular numbers, right? Or is it possible to write a WFF in this context where the variables refer to "functions or sets"?

Yes, that is second order logic. And the theories of set have been invented for doing higher order logic in a pure first order way. In recursion or computability theory this leads to the analytical hierarchy. Naïve math is of infinite or indeterminate order.

Like you I am an arithmetical realist but not necessarily a realist about arbitrary sets.

I can be, but by distinguish levels. 

I think it may be problematic to talk about sets that are so big that most of their numbers have no finite description, as must be true of any uncountable set. If there is no *rule* which maps countable ordinals to real numbers and which can be described in finite terms by a humanlike being, for example, then is it really meaningful to ask whether or not a completely incomprehensible mapping exists between these two sets? I'm not so sure, and thus I'm not sure whether I believe there is any "real truth" about the Continuum Hypothesis.

OK. It is a very difficult subject. I am a long way from having any definite opinion. Even assuming comp.

which means we should also have a notion of an omega_1^CK-order oracle machine.

>You are quick here! There are more than one way to make this precise.

More than one non-equivalent way to make it precise, so that a number computable by one version of an "omega_1^CK-order oracle machine" might not be computable by another version? Of course even with ordinary Turing machines, there are multiple ways to design the internal state-changing rules for the machine, such that different Turing machines might respond to the same input string differently...still if a given output is computable by one universal Turing machine it should be computable by any other, so they are all "equivalent" in that sense. If we define the "order" of oracle machines in the sense I discussed (so that the third-order oracle machine can decide whether the first- or second-order oracle machines will halt given any input string of the type that could be given to a Turing machine, and so forth), then even if there could be different input-output relations for an oracle machine of a given "order", would the set of outputs "computable" by differently-designed oracle machines of that particular order not be "equivalent" in a similar sense?

There are many hierarchies there, and many results which would be difficult to describe in any short way. I hope I can give later more information. 

The important result is that the "order defined by oracle for the halting problem, the totality problem, and similar can be associated with the Pi-i and Sigma_i hierarchy of alternation of quantifiers? Then all that -hierarchy can itself be relativized on any oracle. Like you can define the Phi_i^A, and their domain W_i^A. The Phi_i^A are the partial functions from N to N that you can compute with a universal machine "in connection" of the oracle A.

The whole of recursion theory can be lifted in such "machines". But their rôle can only be "technical" for a computationalist. Of course they could be of interest fro some A-computationalist. But those have not yet appear, except in the sense of taking our data as an oracle. But this should be avoided, I would say by definition, in any scientific enterprise, it would leads to a confusion between necessary and contingent.

You should study books on unsolvability, like those by Shoenfield, or Rogers, if those questions interest you. 

Bruno

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

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[Bruno Marchal]

Many people believe something like

objectivity = serious, truth, rationality etc.
subjectivity = not serious, childish, unscientific, irrational

when truth is  (if I can say, to be short):

subjectivity = what you cannot doubt, what you know, truth
objectivity = hypothetical, theoretical, but sharable, learnable and  
refutable. You have to doubt the theories, and you have to take them  
seriously so as to clarify them, and doubt them even more,  up to  
their replacement.

Now, that confusion is made greater when you begin to make objective  
(doubtable)  theories on subjectivity (undoubtable).

It is good to keep always in mind that

subjectivity = undoubtable (not improvable)
objectivity = doubtable (improvable)

This can be made more precise in the language and theorems/non-
theorems of the universal machine which introspects herself. This is  
really the AUDA thing .... Plato, Descartes and Popper have grasped  
similar things, imo.

Bruno


On 14 Jun 2009, at 03:40, David Nyman wrote:

>
> On Apr 24, 4:39 pm, Bruno Marchal <marc...@...> wrote:
>
>> Any content of consciousness can be an illusion. Consciousness itself
>> cannot, because without consciousness there is no more illusion at  
>> all.
>
> - just catching up with the thread, but I feel compelled to comment
> that this is beautifully and clearly put.  Why does this insight
> escape so many whose grasp of logic in other respects seems quite
> adequate?  The word 'illusion' is often brandished in a scarily
> 'eliminative' way, but those who do so seem quite
> 'unconscious' (ironically) that the subtle knife they wield for this
> excision is precisely that which they seek to excise!
>
> David
>
>> On 24 Apr 2009, at 06:14, Kelly wrote:
>>
>>
>>
>>> On Apr 22, 12:24 pm, Bruno Marchal <marc...@...> wrote:
>>>>> So for that to be a plausible scenario we have to
>>>>> say that a person at a particular instant in time can be fully
>>>>> described by some set of data.
>>
>>>> Not fully. I agree with Brent that you need an interpreter to make
>>>> that person manifest herself in front of you. A bit like a CD, you
>>>> will need a player to get the music.
>>
>>> It seems to me that consciousness is the self-interpretion of
>>> information.  David Chalmers has a good line:  "Experience is
>>> information from the inside; physics is information from the  
>>> outside."
>>
>> First person experience and third person experiment. Glad to hear
>> Chalmers accept this at last.
>> In UDA, inside/outside are perfectly well defined in a pure third
>> person way: inside (first person) = memories annihilated and
>> reconstructed in classical teleportation, outside = the view outside
>> the teleporter. In AUDA I use the old classical definition by Plato  
>> in
>> the Theaetetus.
>>
>>
>>
>>> I still don't see what an interpreter adds, except to satisfy the
>>> intuition that something is "happening" that "produces"
>>> consciousness.  Which I think is an attempt to reintroduce "time".
>>
>> I don't think so. The only "time" needed is the discrete order on the
>> natural numbers. An interpreter is needed to play the role of the
>> person who gives some content to the information handled through his
>> local "brain".   (For this I need also addition and multiplication).
>>
>>
>>
>>> But I don't see any advantage of this view over the idea that
>>> conscious states just "exist" as a type of platonic form (as Brent
>>> mentioned earlier).
>>
>> The advantage is that we have the tools to derive physics in a way
>> which is enough precise for testing the comp hypothesis. Physics has
>> became a branch of computer's psychology or theology.
>>
>>> At any given instant that I'm awake, I'm
>>> conscious of SOMETHING.
>>
>> To predict something, the difficulty is to relate that consciousness
>> to its computational histories. Physics is given by a measure of
>> probability on those comp histories.
>>
>>> And I'm conscious of it by virtue of my
>>> mental state at that instant.  In the materialist view, my mental
>>> state is just the state of the particles of my brain at that
>>> instant.
>>
>> Which cannot be maintained with the comp hyp. Your consciousness is  
>> an
>> abstract type related to all computations going through your current
>> state.
>>
>>
>>
>>> But I say that what this really means is that my mental state is  
>>> just
>>> the information represented by the particles of my brain at that
>>> instant.  And that if you transfer that information to a computer  
>>> and
>>> run a simulation that updates that information appropriately, my
>>> consciousness will continue in that computer simulation,  
>>> regardless of
>>> the hardware (digital computer, mechanical computer, massively
>>> parallel or single processor, etc) or algorithmic details of that
>>> computer simulation.
>>
>> OK. But it is a very special form of information. Consciousness is
>> really the qualia associated to your belief in some reality. It is a
>> bet on self-consistency: it speed up your reaction time relatively to
>> your most probable histories.
>>
>>
>>
>>> But, what is information?  I think it has nothing to do with  
>>> physical
>>> storage or instantiation.  I think it has an existence seperate from
>>> that.  A platonic existence.  And since the information that
>>> represents my brain exists platonically, then the information for
>>> every possible brain (including variations of my brain) should also
>>> exist platonically.
>>
>> You make the same error than those who confuse a universal dovetailer
>> with a counting algorithm or the babel library. The sequence:
>>
>> 0, 1, 2, 3, 4, ... , or 0 1 10 11 100 101 110 111 go through all
>> description of all information, but it lacks the infinitely subtle
>> redundancy contained in the space of all computations (the universal
>> dovetaling). You work in N, succ, you lack addition and
>> multiplication, needed for having a notion of interpreter or  
>> universal
>> machine, the key entity capable of giving content to its information
>> structure. This is needed to have a coherent internal interpretation
>> of computerland.
>>
>>
>>
>>>> Conscious experience is more the content, or the interpretation of
>>>> that information, made by a person or by a universal machine.
>>>> If the doctor makes a copy of your brain, and then codes it into a
>>>> bit
>>>> string, and then put the bit string in the fridge, in our probable
>>>> history, well in that case you will not survive, in our local
>>>> probable
>>>> history.
>>
>>> Given the platonic nature of information, this isn't really a
>>> concern.  In Platonia, you always have a "next moment".  In fact,  
>>> you
>>> experience all possible "next moments".  The "no cul-de-sac" rule
>>> applies I think.
>>
>> By definition, indeed, once we want to quantify the first person
>> indeterminacy.
>> "next moment" makes sense only relatively to (universal) machine. It
>> is the "next step" relatively to some computation and thus universal
>> machine interpreting that "machine".
>> The cul-de-sac/no-cul-de-sac depends on the points of view adopted by
>> the machine itself.
>>
>>
>>
>>>> If you say yes to a doctor for a digital brain, you will ask for a
>>>> brain which functions relatively to our probable computational
>>>> history. No?
>>
>>> I won't worry about it too much, as there is no doctor, only my
>>> perceptions of a doctor.  Every possible outcome of the "brain
>>> replacement operation" that I can perceive, I will perceive.
>>
>> Not in the relative way. You have to explain why you see apples
>> falling from a tree, and not any arbitrary information-theoretical  
>> data.
>>
>>
>>
>>> Including outcomes that don't make any sense.
>>
>> You have to explain why they are *rare*. If not your theory does not
>> explain why you put water on the gas and not in the fridge when you
>> want a cup of coffee.
>>
>>
>>
>>> Additionally, every possible outcome of the operation that the  
>>> doctor
>>> can percieve, he will perceive.  Including outcomes that don't make
>>> any sense.
>>
>>> So it seems to me that a lot of your effort goes into explaining why
>>> we don't see strange "white rabbit" universes.
>>
>> Indeed.
>>
>>> Thus the talk of
>>> probabilities and measures.  I'm willing to just say that all
>>> universes are experienced.
>>
>> That is absolutely true. But we don't live in the absolute (except
>> perhaps with salvia :). We live in the relative worlds/states. I
>> cannot go to my office by flying through the window. The probability
>> that I end up in an hospital is far greater than arriving in piece to
>> my job place.
>>
>>> Strange ones, normal ones, good ones, bad
>>> ones, ones with unbreakable physical laws, ones with no obvious
>>> physical laws at all.  It's all a matter of perception, not a matter
>>> of physical realization.
>>
>> That is true, but we want to explain "the stable appearance of atoms
>> and galaxies", and what happens when we die.
>>
>>
>>
>>
>>
>>
>>
>>>> Yes there is a world in which you computer will transform itself  
>>>> into
>>>> a green flying pig. The "scientific", but really everyday life
>>>> question, is, what is the "probability" this will happen to "me"  
>>>> here
>>>> and now.
>>
>>> I'm not sure what it means to ask, "what is the probability that my
>>> computer will turn into a green pig".  One of me will observe
>>> everything that can be observed in the next instant.  How many  
>>> things
>>> is that?  I'm not sure.  More than 10...ha!  Setting aside physical
>>> limits, maybe infinitely many?  Given that I might also get extra
>>> sensory capacity in that instant, or extra cognitive capacity, or
>>> whatever.
>>
>>> So, of course all of that sounds somewhat crazy, but that's where  
>>> you
>>> end up when you try to explain consciousness I think.  Any  
>>> explanation
>>> that doesn't involve eliminativism is going to be strange I think.
>>
>> The comp theory explains why we cannot explain consciousness, nor
>> truth. But we can bet on computational states, then the thought
>> experiments show that physics is derivable from computer science/
>> number theory in term of probabilities, and we can compare those
>> probabilities with the one we extract from the long observation of  
>> our
>> neighborhoods. Comp is a concrete testable theory, but we have to
>> derive the physics from it to do so. There is a gift because it gives
>> a complete arithmetical interpretation of an earlier type of theory
>> like Plotinus theology, which does not eliminate the person like
>> modern materialism, comp lead to a natural distinction between truth
>> about us, and provable by us.
>>
>>
>>
>>> But, if you are willing to say that consciousness is an illusion,  
>>> then
>>> you can just stick with materialism/physicalism and you're fine.
>>
>> You are right. But consciousness is the only thing I have no doubt
>> about. The *only* undoubtable thing. The fixed point of the cartesian
>> systematic doubting attitude. A theory which eliminate my first
>> person, or my consciousness, although irrefutable by me, is wrong, I
>> hope, I hope it to be wrong for you too. (Why would I send a post on
>> consciousness to a zombie?)
>>
>>> In
>>> that case there's no need to invoke any of this more esoteric stuff
>>> like platonism.  Right?
>>
>> Right. Materialism is a trick based on a lie (consciousness, and thus
>> pain, suffering are illusions) and an illusion (there is matter).  
>> This
>> is used to stop fundamental inquiry. It is not a coincidence that
>> authoritative theologies insists on materialism so much. Before
>> Darwinism God created the man, after Darwinism God created the  
>> matter.
>> Assuming comp, couple matter-man
>>
>> Any content of consciousness can be an illusion. Consciousness itself
>> cannot, because without consciousness there is no more illusion at  
>> all.
>>
>> Best,
>>
>> Bruno
>>
>> http://iridia.ulb.ac.be/~marchal/
> >

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




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[Torgny Tholerus]

Quentin Anciaux skrev:
> Well it is illegal regarding the rules meaning with these rules set B
> does not exist as defined.
>  

What is it that makes set A to exist, and set B not to exist?  What is
the (important) differences between the definition of set A and the
definition of set B?  In both cases you are defining a set by giving a
property that all members of the set must fulfill.

Why is the deduction legal for set A, but illegal for set B?  There is
the same type of deduction in both places, you are just making a
substitution for the all quantificator in both cases.

--
Torgny Tholerus

>
> 2009/6/13 Torgny Tholerus <torgny@...>:
>  
>> Quentin Anciaux skrev:
>>    
>>> 2009/6/13 Torgny Tholerus <torgny@...>:
>>>
>>>      
>>>> What do you think about the following deduction?  Is it legal or illegal?
>>>> -------------------
>>>> Define the set A of all sets as:
>>>>
>>>> For all x holds that x belongs to A if and only if x is a set.
>>>>
>>>> This is an general rule saying that for some particular symbol-string x
>>>> you can always tell if x belongs to A or not.  Most humans who think
>>>> about mathematics can understand this rule-based definition.  This rule
>>>> holds for all and every object, without exceptions.
>>>>
>>>> So this rule also holds for A itself.  We can always substitute A for
>>>> x.  Then we will get:
>>>>
>>>> A belongs to A if and only if A is a set.
>>>>
>>>> And we know that A is a set.  So from this we can deduce:
>>>>
>>>> A beongs to A.
>>>> -------------------
>>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>>
>>>>        
>>> It depends if you allow a set to be part of itselft or not.
>>>
>>> If you accept, that a set can be part of itself, it makes your
>>> deduction legal regarding the rules.
>>>      
>> OK, if we accept that a set can be part of itself, what do you think
>> about the following deduction? Is it legal or illegal?
>>
>> -------------------
>> Define the set B of all sets that do not belong to itself as:
>>
>> For all x holds that x belongs to B if and only if x does not belong to x.
>>
>> This is an general rule saying that for some particular symbol-string x
>> you can always tell if x belongs to B or not.  Most humans who think
>> about mathematics can understand this rule-based definition.  This rule
>> holds for all and every object, without exceptions.
>>
>> So this rule also holds for B itself.  We can always substitute B for
>> x.  Then we will get:
>>
>> B belongs to B if and only if B does not belong to B.
>> -------------------
>> Quentin, what do you think?  Is this deduction legal or illegal?
>>
>>
>> --
>> Torgny Tholerus
>>
>>    
>
>
>
>  

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[Quentin Anciaux-2]

2009/6/14 Torgny Tholerus <torgny@...>:

>
> Quentin Anciaux skrev:
>> Well it is illegal regarding the rules meaning with these rules set B
>> does not exist as defined.
>>
>
> What is it that makes set A to exist, and set B not to exist?  What is
> the (important) differences between the definition of set A and the
> definition of set B?  In both cases you are defining a set by giving a
> property that all members of the set must fulfill.

Yes and one fulfil it according to the given rules the other not.

I would add that your "excercise" is inconsistent from the start,
whatever a set is, your argument is contradictory whatever the rules
are.

> Why is the deduction legal for set A, but illegal for set B?  There is
> the same type of deduction in both places, you are just making a
> substitution for the all quantificator in both cases.

That's all the point of puting rules and checking that something is
correct or not according to it. 1+1=3 is false according to PA... that
doesn't mean you couldn't find a rule or mapping that would render
this statement true ***regarding the chosen rules***/

Regards,
Quentin

> --
> Torgny Tholerus
>
>>
>> 2009/6/13 Torgny Tholerus <torgny@...>:
>>
>>> Quentin Anciaux skrev:
>>>
>>>> 2009/6/13 Torgny Tholerus <torgny@...>:
>>>>
>>>>
>>>>> What do you think about the following deduction?  Is it legal or illegal?
>>>>> -------------------
>>>>> Define the set A of all sets as:
>>>>>
>>>>> For all x holds that x belongs to A if and only if x is a set.
>>>>>
>>>>> This is an general rule saying that for some particular symbol-string x
>>>>> you can always tell if x belongs to A or not.  Most humans who think
>>>>> about mathematics can understand this rule-based definition.  This rule
>>>>> holds for all and every object, without exceptions.
>>>>>
>>>>> So this rule also holds for A itself.  We can always substitute A for
>>>>> x.  Then we will get:
>>>>>
>>>>> A belongs to A if and only if A is a set.
>>>>>
>>>>> And we know that A is a set.  So from this we can deduce:
>>>>>
>>>>> A beongs to A.
>>>>> -------------------
>>>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>>>
>>>>>
>>>> It depends if you allow a set to be part of itselft or not.
>>>>
>>>> If you accept, that a set can be part of itself, it makes your
>>>> deduction legal regarding the rules.
>>>>
>>> OK, if we accept that a set can be part of itself, what do you think
>>> about the following deduction? Is it legal or illegal?
>>>
>>> -------------------
>>> Define the set B of all sets that do not belong to itself as:
>>>
>>> For all x holds that x belongs to B if and only if x does not belong to x.
>>>
>>> This is an general rule saying that for some particular symbol-string x
>>> you can always tell if x belongs to B or not.  Most humans who think
>>> about mathematics can understand this rule-based definition.  This rule
>>> holds for all and every object, without exceptions.
>>>
>>> So this rule also holds for B itself.  We can always substitute B for
>>> x.  Then we will get:
>>>
>>> B belongs to B if and only if B does not belong to B.
>>> -------------------
>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>
>>>
>>> --
>>> Torgny Tholerus
>>>
>>>
>>
>>
>>
>>
>
>
> >
>

--
All those moments will be lost in time, like tears in rain.

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[Bruno Marchal]

Torgny,

I agree with Quentin.
You are just showing that the naive notion of set is inconsistent.  
Cantor already knew that, and this is exactly what forced people to  
develop axiomatic theories. So depending on which theory of set you  
will use, you can or cannot have an universal set (a set of all sets).  
In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the  
collection of all sets is not a set. In NF, some have developed  
structure with universal sets, and thus universe containing  
themselves. Abram is interested in such universal sets. And, you can  
interpret the UD, or the Mandelbrot set as (simple) model for such  
type of structure.

Your argument did not show at all that the set of natural numbers  
leads to any trouble. Indeed, finitism can be seen as a move toward  
that set, viewed as an everything, potentially infinite frame (for  
math, or beyond math, like it happens with comp).

The problem of naming (or given a mathematical status) to "all sets"  
is akin to the problem of giving a name to God. As Cantor was  
completely aware of. We are confused on this since we exist. But the  
natural numbers, have never leads to any confusion, despite we cannot  
define them.

You argument against the infinity of natural numbers is not valid. You  
cannot throw out this "little infinite" by pointing on the problem  
that some "terribly big infinite", like the "set" of all sets,  leads  
to trouble. That would be like saying that we have to abandon all  
drugs because the heroin is very dangerous.
It is just non valid.

Normally, later  I will show a series of argument very close to  
Russell paradoxes, and which will yield, in the comp frame,  
interesting constraints on what computations are and are not.

Bruno


On 13 Jun 2009, at 13:26, Torgny Tholerus wrote:

>
> Quentin Anciaux skrev:
>> 2009/6/13 Torgny Tholerus <torgny@...>:
>>
>>> What do you think about the following deduction?  Is it legal or  
>>> illegal?
>>> -------------------
>>> Define the set A of all sets as:
>>>
>>> For all x holds that x belongs to A if and only if x is a set.
>>>
>>> This is an general rule saying that for some particular symbol-
>>> string x
>>> you can always tell if x belongs to A or not.  Most humans who think
>>> about mathematics can understand this rule-based definition.  This  
>>> rule
>>> holds for all and every object, without exceptions.
>>>
>>> So this rule also holds for A itself.  We can always substitute A  
>>> for
>>> x.  Then we will get:
>>>
>>> A belongs to A if and only if A is a set.
>>>
>>> And we know that A is a set.  So from this we can deduce:
>>>
>>> A beongs to A.
>>> -------------------
>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>
>>
>> It depends if you allow a set to be part of itselft or not.
>>
>> If you accept, that a set can be part of itself, it makes your
>> deduction legal regarding the rules.
>
> OK, if we accept that a set can be part of itself, what do you think
> about the following deduction? Is it legal or illegal?
>
> -------------------
> Define the set B of all sets that do not belong to itself as:
>
> For all x holds that x belongs to B if and only if x does not belong  
> to x.
>
> This is an general rule saying that for some particular symbol-
> string x
> you can always tell if x belongs to B or not.  Most humans who think
> about mathematics can understand this rule-based definition.  This  
> rule
> holds for all and every object, without exceptions.
>
> So this rule also holds for B itself.  We can always substitute B for
> x.  Then we will get:
>
> B belongs to B if and only if B does not belong to B.
> -------------------
> Quentin, what do you think?  Is this deduction legal or illegal?
>
>
> --
> Torgny Tholerus
>
> >

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




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[Torgny Tholerus]

Bruno Marchal skrev:
> Torgny,
>
> I agree with Quentin.
> You are just showing that the naive notion of set is inconsistent.  
> Cantor already knew that, and this is exactly what forced people to  
> develop axiomatic theories. So depending on which theory of set you  
> will use, you can or cannot have an universal set (a set of all sets).  
> In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the  
> collection of all sets is not a set.

It is not the naive notion of set that is inconsistent.  It is the naive
*handling* of sets that is inconsistent.

This problem has two possible solutions.  One possible solution is to
deny that it is possible to create the set of all sets.  This solution
is chosen by ZF and VBG.

The second possible solution is to be very careful of the domain of the
All quantificator.  You are not allowed to substitute an object that is
not included in the domain of the quantificator.  It is this second
solution that I have chosen.

What is illegal in the two deductions below, is the substitutions.  
Because the sets A and B do not belong to the domain of the All
quantificator.

You can define "existence" by saying that only that which is incuded in
the domain of the All quantificator exists.  In that case it is correct
to say that the sets A and B do not exist, because they are not included
in the domain.  But I think this is a too restrictive definition of
existence.  It is fully possible to talk about the set of all sets.  But
you must then be *very* careful with what you do with that set.  That
set is a set, but it does not belong to the set of all sets, it does not
belong to itself.  It is also a matter of definition; if you define
"set" as the same as "belonging to the set of all sets", then the set of
all sets is not a set.  This is a matter of taste.  You can choose
whatever you like, but you must be aware of your choice.  But if you
restrict yourself too much, then your life will be poorer...

>  In NF, some have developed  
> structure with universal sets, and thus universe containing  
> themselves. Abram is interested in such universal sets. And, you can  
> interpret the UD, or the Mandelbrot set as (simple) model for such  
> type of structure.
>
> Your argument did not show at all that the set of natural numbers  
> leads to any trouble. Indeed, finitism can be seen as a move toward  
> that set, viewed as an everything, potentially infinite frame (for  
> math, or beyond math, like it happens with comp).
>
> The problem of naming (or given a mathematical status) to "all sets"  
> is akin to the problem of giving a name to God. As Cantor was  
> completely aware of. We are confused on this since we exist. But the  
> natural numbers, have never leads to any confusion, despite we cannot  
> define them.
>  

The "proof" that there is no biggest natural number is illegal, because
you are there doing an illegal deduction, you are there doing an illegal
substitution, just the same as in the deductions below with the sets A
and B.  You are there substituting an object that is not part of the
domain of the All quatificator.

--
Torgny Tholerus

> You argument against the infinity of natural numbers is not valid. You  
> cannot throw out this "little infinite" by pointing on the problem  
> that some "terribly big infinite", like the "set" of all sets,  leads  
> to trouble. That would be like saying that we have to abandon all  
> drugs because the heroin is very dangerous.
> It is just non valid.
>
> Normally, later  I will show a series of argument very close to  
> Russell paradoxes, and which will yield, in the comp frame,  
> interesting constraints on what computations are and are not.
>
> Bruno
>
>
> On 13 Jun 2009, at 13:26, Torgny Tholerus wrote:
>
>  
>> Quentin Anciaux skrev:
>>    
>>> 2009/6/13 Torgny Tholerus <torgny@...>:
>>>
>>>      
>>>> What do you think about the following deduction?  Is it legal or  
>>>> illegal?
>>>> -------------------
>>>> Define the set A of all sets as:
>>>>
>>>> For all x holds that x belongs to A if and only if x is a set.
>>>>
>>>> This is an general rule saying that for some particular symbol-
>>>> string x
>>>> you can always tell if x belongs to A or not.  Most humans who think
>>>> about mathematics can understand this rule-based definition.  This  
>>>> rule
>>>> holds for all and every object, without exceptions.
>>>>
>>>> So this rule also holds for A itself.  We can always substitute A  
>>>> for
>>>> x.  Then we will get:
>>>>
>>>> A belongs to A if and only if A is a set.
>>>>
>>>> And we know that A is a set.  So from this we can deduce:
>>>>
>>>> A beongs to A.
>>>> -------------------
>>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>>
>>>>        
>>> It depends if you allow a set to be part of itselft or not.
>>>
>>> If you accept, that a set can be part of itself, it makes your
>>> deduction legal regarding the rules.
>>>      
>> OK, if we accept that a set can be part of itself, what do you think
>> about the following deduction? Is it legal or illegal?
>>
>> -------------------
>> Define the set B of all sets that do not belong to itself as:
>>
>> For all x holds that x belongs to B if and only if x does not belong  
>> to x.
>>
>> This is an general rule saying that for some particular symbol-
>> string x
>> you can always tell if x belongs to B or not.  Most humans who think
>> about mathematics can understand this rule-based definition.  This  
>> rule
>> holds for all and every object, without exceptions.
>>
>> So this rule also holds for B itself.  We can always substitute B for
>> x.  Then we will get:
>>
>> B belongs to B if and only if B does not belong to B.
>> -------------------
>> Quentin, what do you think?  Is this deduction legal or illegal?
>>    

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[Quentin Anciaux-2]

2009/6/17 Torgny Tholerus <torgny@...>:

>
> Bruno Marchal skrev:
>> Torgny,
>>
>> I agree with Quentin.
>> You are just showing that the naive notion of set is inconsistent.
>> Cantor already knew that, and this is exactly what forced people to
>> develop axiomatic theories. So depending on which theory of set you
>> will use, you can or cannot have an universal set (a set of all sets).
>> In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the
>> collection of all sets is not a set.
>
> It is not the naive notion of set that is inconsistent.  It is the naive
> *handling* of sets that is inconsistent.
>
> This problem has two possible solutions.  One possible solution is to
> deny that it is possible to create the set of all sets.  This solution
> is chosen by ZF and VBG.
>
> The second possible solution is to be very careful of the domain of the
> All quantificator.  You are not allowed to substitute an object that is
> not included in the domain of the quantificator.  It is this second
> solution that I have chosen.
>
> What is illegal in the two deductions below, is the substitutions.
> Because the sets A and B do not belong to the domain of the All
> quantificator.
>
> You can define "existence" by saying that only that which is incuded in
> the domain of the All quantificator exists.  In that case it is correct
> to say that the sets A and B do not exist, because they are not included
> in the domain.  But I think this is a too restrictive definition of
> existence.  It is fully possible to talk about the set of all sets.  But
> you must then be *very* careful with what you do with that set.  That
> set is a set, but it does not belong to the set of all sets, it does not
> belong to itself.  It is also a matter of definition; if you define
> "set" as the same as "belonging to the set of all sets", then the set of
> all sets is not a set.  This is a matter of taste.  You can choose
> whatever you like, but you must be aware of your choice.  But if you
> restrict yourself too much, then your life will be poorer...
>
>>  In NF, some have developed
>> structure with universal sets, and thus universe containing
>> themselves. Abram is interested in such universal sets. And, you can
>> interpret the UD, or the Mandelbrot set as (simple) model for such
>> type of structure.
>>
>> Your argument did not show at all that the set of natural numbers
>> leads to any trouble. Indeed, finitism can be seen as a move toward
>> that set, viewed as an everything, potentially infinite frame (for
>> math, or beyond math, like it happens with comp).
>>
>> The problem of naming (or given a mathematical status) to "all sets"
>> is akin to the problem of giving a name to God. As Cantor was
>> completely aware of. We are confused on this since we exist. But the
>> natural numbers, have never leads to any confusion, despite we cannot
>> define them.
>>
>
> The "proof" that there is no biggest natural number is illegal, because
> you are there doing an illegal deduction, you are there doing an illegal
> substitution, just the same as in the deductions below with the sets A
> and B.  You are there substituting an object that is not part of the
> domain of the All quatificator.

No the proof is based on PA and in PA you do not have an axiom
restricting the successor function and as such it is defined in the
axiom that you don't have an upper bound limit. The proof is *valid*
against the axioms. *You* are doing an illegal deduction by not taking
into accound the rules with wich you work.

Regards,
Quentin

> --
> Torgny Tholerus
>
>> You argument against the infinity of natural numbers is not valid. You
>> cannot throw out this "little infinite" by pointing on the problem
>> that some "terribly big infinite", like the "set" of all sets,  leads
>> to trouble. That would be like saying that we have to abandon all
>> drugs because the heroin is very dangerous.
>> It is just non valid.
>>
>> Normally, later  I will show a series of argument very close to
>> Russell paradoxes, and which will yield, in the comp frame,
>> interesting constraints on what computations are and are not.
>>
>> Bruno
>>
>>
>> On 13 Jun 2009, at 13:26, Torgny Tholerus wrote:
>>
>>
>>> Quentin Anciaux skrev:
>>>
>>>> 2009/6/13 Torgny Tholerus <torgny@...>:
>>>>
>>>>
>>>>> What do you think about the following deduction?  Is it legal or
>>>>> illegal?
>>>>> -------------------
>>>>> Define the set A of all sets as:
>>>>>
>>>>> For all x holds that x belongs to A if and only if x is a set.
>>>>>
>>>>> This is an general rule saying that for some particular symbol-
>>>>> string x
>>>>> you can always tell if x belongs to A or not.  Most humans who think
>>>>> about mathematics can understand this rule-based definition.  This
>>>>> rule
>>>>> holds for all and every object, without exceptions.
>>>>>
>>>>> So this rule also holds for A itself.  We can always substitute A
>>>>> for
>>>>> x.  Then we will get:
>>>>>
>>>>> A belongs to A if and only if A is a set.
>>>>>
>>>>> And we know that A is a set.  So from this we can deduce:
>>>>>
>>>>> A beongs to A.
>>>>> -------------------
>>>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>>>
>>>>>
>>>> It depends if you allow a set to be part of itselft or not.
>>>>
>>>> If you accept, that a set can be part of itself, it makes your
>>>> deduction legal regarding the rules.
>>>>
>>> OK, if we accept that a set can be part of itself, what do you think
>>> about the following deduction? Is it legal or illegal?
>>>
>>> -------------------
>>> Define the set B of all sets that do not belong to itself as:
>>>
>>> For all x holds that x belongs to B if and only if x does not belong
>>> to x.
>>>
>>> This is an general rule saying that for some particular symbol-
>>> string x
>>> you can always tell if x belongs to B or not.  Most humans who think
>>> about mathematics can understand this rule-based definition.  This
>>> rule
>>> holds for all and every object, without exceptions.
>>>
>>> So this rule also holds for B itself.  We can always substitute B for
>>> x.  Then we will get:
>>>
>>> B belongs to B if and only if B does not belong to B.
>>> -------------------
>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>
>
>
> >
>

--
All those moments will be lost in time, like tears in rain.

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