On 03 Jan 2011, at 17:26, Brian Tenneson wrote:

Ah, ok. Well, as your friend checked my proof, what I was/am working on is an effective theory.


You and most mathematician and logicians manage to have checkable proofs, but they usually don't work in a formal setting, so it makes no sense to say that their proofs are recursively enumerable, unless they add (in a footnote) that what they do can be formalised in ZF, or NBG, as it is usually the case. Mathematicians and logicians don't work in formal systems. Logicians works *on* formal system, but their meta-theories are the usual informal high order mathematics (and easily implemented in ZF or NBG, but never actually done so).

Bruno



Bruno Marchal wrote:


On 02 Jan 2011, at 18:01, Brian Tenneson wrote:

Bruno Marchal wrote:


On 02 Jan 2011, at 11:31, silky wrote:

On Sun, Jan 2, 2011 at 8:31 PM, Brian Tenneson <tenn...@gmail.com> wrote:
In the case of a TOE, the model IS reality.

Okay, I won't reply further, this has become irrelevant noise.


I suspect the traditional confusion between "model" in the sense of physicists (where model = a theory, like a toy model), and model in the sense of the logician, where model = the reality studied (like a woman serving as model for a painter, or the mathematical structure (N, +, x) for PA or RA).

Logicians and physicists use the word "model" in the complete opposite sense, and this leads often to complete deaf dialog.

This makes even more problem with computationalism, where an observer accept that some "theories/brains/finite-describable- objects" fits the reality. When you say "yes" to the doctor, it is because you believe that the artificial brain does capture (locally, with respect to your current environment) the real thing (your conscious you). In that case *you* are a fixed point where a model-theory correspond to a model-reality, a bit like in Brouwer fixed point theorem, where a map of a territory is shown to have a point on it matching the real point in the territory, provided the map is not ripped in two disconnected parts, but only transformed continuously. The point is that in some contexts some overlap can exist between a theory and its (or one of its) model, between description and realities, like with the painting of a painting of a pipe (cf Magritte).

Things get confusing also if, like Brian, (but also logicians in some circumstances) people makes a model (a "reality") into a (non effective) theory. This can be justified for some technical reason, when working on super non effective structure, but is really out of topic, imo.

Bruno

What makes a theory effective?

That its proofs are checkable. That its set of theorems is recursively enumerable.




I'm going to be less precise given that my audience has changed in a way I do not know.

We argue in an interdisciplinary field.




Given a couple of assumptions, which are essentially that (1) reality is independent of humans (which will imply that a model (in the logical sense) can be a TOE as defined in this thread) and

I don't see this. I prefer to use "theory" for something finitely presentable (finite or recursively enumerable).




(2) a model every model can be embedded within endows that model with a universality that makes it a candidate for being reality. This is then a brief description of reality, though I couldn't hope to give all the details about reality.

In that case the model (N, +, x) is the "TOE" that your are searching. It is rather a ROE (realm of everything), and the embedding relation is simulation (emulation or partial emulation). My point is that we have no choice in the matter once we assume that brains work like a digital machine at some level of description.



I am also working on the hypothesis that a TOE can be given in an finite/infinite presentation such as found in ZF with axioms and axiom schemata. Question: what is the theory with no assumptions? I know that in logic, the consequent closure of the empty set of statements is the set of tautologies, which is not really what I'd call an effective theory.

The set of (classical tautologies) is effective. But the empty theory as all models, the structure of which depending of the meta- theory. It is the trivial theory satisfied by all structures.




But what about if we remove all assumptions? Sounds like chaos to me. This is connected to all this as I can explain.

In fact, I can prove (1) on the grounds that there is no largest number. It took me a while to find this argument.

"1)" follows from comp, which assumes arithmetical realism (used in "there is no largest number").


Bruno







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