Is there a transcendental-logical or arithmetic system that could account for everything or is this just stating the set of all sets that is also a self-containing set paradox? — Posty McPostface

It's a tree. To pick a familiar example, take set theory ZF. This is Zermelo-Fraenkel set theory, the standard basic set theory used in modern math. The Axiom of Choice is independent of ZF. So we may form the new theory ZFC, ZF with choice. And we may also form the new theory ZF-C, which is ZF plus the negation of choice. These two set theories are each individually consistent, but they are mutually inconsistent.

Then we can keep on adding new axioms to each system to get more and more variants of set theory. You get an ever-expanding tree of theories.

Modern set theorists like to contemplate the entire structure at once. They study the set theoretic multiverse. Joel David Hamkins is one of the world's big time set theorists and he's also a blogger who tries to bring some of this heady stuff to the masses.

If I'm understanding your question, modern set theorists are attempting to figure out what is mathematical truth, even in a sea of incompleteness. After all ‎Gödel himself was a Platonist. I think the point of incompleteness isn't that we can never know truth. It's just that axiomatic systems have their limitations. But even so we can still strive to know the truth.

And of course if we can know the truth but not by means of logical deduction ... what does that mean, exactly? This gets very philosophical.

What are your thoughts about the compactness theorem in logic and entailment of smaller sets by larger sets that obey the compression theorem. Would that point to a never ending set of sets that contains all provable theorems?

What are your thoughts about the compactness theorem in logic — Posty McPostface

I'm no specialist in these things. I was pretty good at set theory but logic made my eyes glaze. I recall the compacness theorem says that if every finite subset of a collection of axioms has a model, then the entire collection has a model.

People usually learn about compactness in real analysis or topology class so this connection between topology and logic often comes as a surprise.

That's everything I know about it. What aspect of the compactness theorem is of interest in the present context?

and entailment of smaller sets by larger sets that obey the compression theorem. — Posty McPostface

Well you've got me there. I had to look it up. The compression theorem is a theorem of computational complexity that says:

The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.

I know enough of the buzzwords to see that they are putting some sort of topology on the set of complexity classes. Yet more topology applied to logic.

If I had to hazard a guess it looks like the set of complexity classes is an open set. It does not contain its boundary. So you can't draw a sharp line and say that on one side it's computable and on the other it's not. It's fuzzy in the exact same way that the open unit disk is fuzzy. The open unit disk is the set of points in the plane whose distance from the origin is strictly less than 1. It doesn't include its boundary.

Is that a reasonable interpretation? If so, what aspect of this did you want to ask about?

Would that point to a never ending set of sets that contains all provable theorems? — Posty McPostface

There is no set of all sets. It's a logical impossibility, if by set you mean something that satisfies the usual mathematical notions of sets. Russell proved that in 1901.

It is legitimate to talk about the class of all sets. A class is a collection that's "too big" to be a set. A class is often something we can form with an unrestricted predicate. Take the predicate S(x) which is true just in case x is a set.

Then V = {x : S(x)} is a class that can not possibly be a set. In that case we call it a proper class.

There is no set of sets. Better to find another metaphor for your intuition.

Is that a reasonable interpretation? If so, what aspect of this did you want to ask about? — fishfry

Yeah, so are emergent phenomena by that understanding mystical phenomena or indeterminate?

Yeah, so are emergent phenomena by that understanding mystical phenomena or indeterminate? — Posty McPostface

You are saying that emergence has some connection to the compression theorem? I'm afraid I don't understand that. Explanation please?

In any event, emergence is a far murkier idea than people generally realize. Hydrogen's not wet and oxygen's not wet but water is wet. Emergence! Call me unimpressed. The whole often has qualities not evident in the parts. Thats the most natural thing in the world.

Here's the exchange I had over at PhysicsForums. Maybe you could see the connection from that exchange.

Here's the exchange I had over at PhysicsForums. Maybe you could see the connection from that exchange. — Posty McPostface

That wasn't much of a thread. In it you said, "I'm probably spouting nonsense." It didn't seem to get any better from there. I'm not being harsh, there's nothing in that thread at all. Nor does the thread mention the compression theorem. You haven't made your point in a way I can understand.

My assumption is that the compression theorem can serve as a mathematical foothold to understanding that things like the Church-Turing-Deutsch principle as impossible to achieve. I've been fixated on that principle and Godel's Incompleteness Theorem for a while now.

My assumption is that the compression theorem can serve as a mathematical foothold to understanding that things like the Church-Turing-Deutsch principle as impossible to achieve. I've been fixated on that principle and Godel's Incompleteness Theorem for a while now. — Posty McPostface

I get that you are interested in this. I just don't follow your reasoning. I don't know enough about the compression theorem to have any opinion at all as to what implications it might have. I should mention that I personally don't believe the universe is computation so that, to the limited extent that I know anything about it at all, I'm not kindly disposed toward CTD. I'm the wrong person to be asking about any of this. My only contribution to this thread was to respond to your speculation that there's a hierarchy of axiomatic theories. In fact it's a tree. But modern set theorists do study the entire structure of axiomatic set theories as the "set-theoretic multiverse." I pointed you to Hamkins. I really can't comment on how this relates to CTD. I just don't know anything about it.

The Wiki article on the compression theorem is less than enlightening to say the least. It offers no clue as to the meaning or application of the theorem.