On 09 Jun 2012, at 21:53, Abram Demski wrote:

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Bruno, Wei,I've been reading the book "saving truth from paradox" on and off,and it has convinced me of the importance of the "inside view" wayof doing foundations research as opposed to the "outside view".At first, I simply understood Field to be referring to the languagevs meta-language distinction. He criticises other researchers fortaking the "outside view" of the system they are describing, meaningthat they are describing the theory from a meta-language which mustnecessarily exist outside the theory.

`Since Gödel we know that for "rich" theory we can embed the metatheory`

`in the theory. That is what Gödel's provability predicate does, and`

`what Kleene predicate does for embedding the reasoning on the Turing`

`machines, and the phi_i, in terms of number relations.`

Arithmetic contains its own interpreter(s).

I thought that his complaint was frivolous; of course you need todescribe a theory of truth via a meta-language. That is part of thestructure of the problem. Yes, it makes the entire theory dubious;but without a concrete alternative, the only reply to this is "suchis life!". So I was confused when he refused to take other logiciansliterally (accepting the logic which they put forward as the logicwhich they put forward), and instead claimed that their logiccorresponded to the 1-higher theory (the metalanguage in which theydescribe their theory).At some point, though, the technique "clicked" for me, and Iunderstood that he was saying something very different. For example,the outside view of Kripke's theory of truth says that truth is a'partial' notion, with an extension and an anti-extension, but alsoa 'gap' between the two where it is undefined. (It is a "gap theory".)

I am not sure I understand well.

On the inside view, however, it does not make this kind ofcommitment; it does not claim there is a gap. What the theory saysabout itself makes no commitment about the status of the (would-be)gap sentences; they could well be both true and false. The "outsideview" will insist on giving a semantic status to these, but this ispathological; we cannot develop a theory of truth in this way (weknow that it leads to paradox).Instead, we need to take the inside view seriously, and developtheories from that perspective.This generally means taking the truth predicate as basic, andlooking for deduction rules about it which capture what we want,rather than trying to define its semantics in a set-theoretic orotherwise external way.I don't feel that I have an excellent grasp of this technique,though. So, I'm looking for feedback. Do you have any thoughts oradvice here?

`Better! A theory. Not mine, but the one by the "rich" universal`

`machine itself (that I call Löbian). Basically a machine is Löbian if`

`it is universal (in Church Turing sense) and can prove (in a technical`

`weak sense) that she is universal. Basically it is a universal system`

`+ an induction axiom (or axiom scheme). Examples are Peano Arithmetic,`

`ZF, etc.`

`The machine's inside view is already unameable by the machine, it is a`

`"time" creator, (in some semantics), a kind of intuitionist knower.`

`Yes, it is important to take its view too.`

`All löbian machines are able to distinguish two forms of self-`

`reference: a third person one, and a first person one. And other`

`modalities, notably those needed to extract physics from arithmetic`

`(as UDA enforced).`

`The computationalist hypothesis suggest using computer science and`

`mathematical logic for dealing with the complex aspects of relative`

`self-reference, in apparent simple ideal case. I think.`

Bruno

Wei, Concerning your "undefinability of induction" paradox...In this view, the answer is more or less "there can be no truthpredicate which acts like that"... truth is an "open" notion, muchlike ordinals are an open notion.To some extent, this is an acceptance of the fact that if an alienshowed up claiming to have a box which determined the truth orfalsehood of any statement, we should ascribe this 0 probability; orrather, we won't fully understand the statement (there is no way tosay such a thing; the idea is incoherent). We can ascribe someprobability to much weaker statements concerning the connectionbetween the output of the box and the truth of statements, however.In particular, probability can be ascribed to any partial notion oftruth which can be discussed.This feels like "accepting the problem statement as a statement ofthe solution". The problem is that there is no notion of semanticsfor which allows a system to refer to all its own semantic values.The 'solution' is to say that semantics simply "isn't likethat" (there is no 'completion' of the semantics). If we state theseformally, the problem and the solution are the same statement; itseems like we've made no progress! Again, any comments on thisapproach are appreciated.Best, -- Abram Demski http://lo-tho.blogspot.com/ --You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

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