# Re: To extract physics, you need only the self-referential invariant?

```Hello Stephen,
```
```
On 23 Feb 2011, at 20:39, Stephen Paul King wrote:

```
```Dear Bruno,

Could you explain this a bit more?

“The ideally correct machine
is to the human what a material point is to the sun. My answer tries
only to help you to understand what I mean by a knowing machine, not
really a knowing human. Human have non-monotonic layers, they can
update beliefs. The logic G and G*, and the six intensional variants
can be seen as the tangential theology, but we are variable machine. G
and G* remains invariant, but get each nanosecond (say) a different
arithmetical interpretations. To extract physics, you need only the
self-referential invariant.”

```
How does one define a self-referential invariant? What might be an example of this?
```

```
G, G*, all the arithmetical points of view (hypostases), are such invariant.
```
```
It is a theorem that all the sound recursively enumerable extensions of PA have the modal logics G and G* describing (completely at the propositional level) their probability and consistency predicates. G describe what they can prove, and G* what is true about them, including what they cannot prove (like their consistency Dt = ~B~f).
```
All this is a consequence of
```
1) the second recursion theorem of Kleene, which you can see as a fixed point theorem. 2) Gödel's diagonalization lemma: which really is the fact that PA (and extensions) proves the Kleene recursion theorem themselves.
```
```
G and G* remain correct for the supermachines (machine with oracles), and are even still correct and complete for many interesting "gods" (non-turing emulable entities). G and G* remain correct but no more complete, when getting very close to "God" (arithmetical truth). I still don't know if they make sense for some truth theory. I doubt so.
```
Bruno

PS I recall (for the new people) that G is the modal logic having

B(p->q) -> (Bp -> Bq)    (Kripke's formula)
B(Bp -> p) -> Bp    (Löb's formula)

```
as axioms, together with classical propositional calculus, and with the usual modus ponens rule, and the necessitation rule (from p deduce Bp) as deductive inference rule.
```
```
G* has as axioms the theorem of G, + the formula Bp -> p, but is NOT close for the necessitation rule.
```
```
See the book by Boolos for more, or the archive (search 'S4Grz', perhaps, or read the second half the sane04 paper).
```
Best

Bruno

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

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