# Re: The limit of all computations

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On 28 May 2012, at 04:00, Russell Standish wrote:```
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```On Sun, May 27, 2012 at 06:20:29PM +0200, Bruno Marchal wrote:
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On 27 May 2012, at 12:15, Russell Standish wrote:
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```I still don't follow. If I have proved a is true in some world, why
should I infer that it is true in all worlds? What am I missing?
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I realize my previous answer might be too long and miss your
question. Apology if it is the case.

Here is a shorter answer. The idea of proving, is that what is
proved in true in all possible world. If not, a world would exist as
a counter-example, invalidating the argument.
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I certainly missed that. Is that given as an axiom?
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That would be a meta-axiom in a theory defining what is logic. But that does not exist. It is just part of what logic intuitively consists in. Logicians are not interested of truth or interpretation of statements. They are interested in validity. What sentences follow from what sentences, independently of interpretations, and thus true in all possible worlds.
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```It seems like that
would be written p -> []p.
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This means that if p then p is provable. "p -> Bp", if B = provable, is completeness (with the meaning of completeness = its meaning in incompleteness). This is false in non rich theory (by the fact that their are non rich) and false in rich theory, by the fact that rich theory obeys to the incompleteness theorem. So, it is true for rare exception (like the first order theory of real numbers) which is not rich (not sigma_1 complete).
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Take the proposition (a v b) in propositional logic. Take the world {(a t), (b, f)}, i.e. the world with a true, and b false. Let p = (a v b). This provides a counter-example to p -> Bp. p is true in that world (because a v b is true if a is true), yet it is not provable, because it is false in some other world, like the world with both a and b false.
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Or take p = Dt. Dt -> BDt contradicts immediately the second incompleteness theorem which says that Dt -> ~BDt.
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When I say p is true in a world, I can only prove that p is true in
that world.
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I don't think so. If p is true, that does not mean you can prove it, neither in your world, nor in some other world.
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```I am mute on the subject of whether p is true in any other
world (unless I can use an axiom like the above).
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By the logicians notion of proof, if you prove a proposition, it is true in all worlds/model/interpretation.
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In what class of logics would such an axiom be taken to be true.
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All.

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```(Of
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course it is true in classical logic, but there is only one "world" there).
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In classical propositional logic, a world is just anything to which we attach a valuation t, or f, to the atomic proposition, p, q r, ... This makes 2^aleph_zero worlds. A world can be identified with a function from {p, q, r, ...} to {t, f}. In first order logic, worlds can be identified with interpretations, or models. All first order theories have many models. In fact for any cardinal, there is a model having that cardinal. The number of worlds exceeds the cardinals nameable in set theory.
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Bruno

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Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpco...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au
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