Thank you Cameron.  A helpful point to notice, I think.  I mention both terms, self-reference and circularity, simply to emphasise what I'm talking about.  The audience I'd like ultimately to have impact on, with these questions, are the physicists.  But in physics, self-reference is not currently a consideration -- I don't think so anyway.  Generally, I'd have no right to expect physicists to follow their meaning.

When I say "Where should we look for self-reference", I mean, where in physical theory can we expect to come across it?  If it's in the Universe, then a correct physical theory should express it.

If the universe should be self-referent in the sense that it simulates itself, we should look everywhere but in some timing (tact) which is completely unknown... (There are Turing Machines that simulates other Turing Machines, such that one step of machine B occurs at [for example] eight steps of machine A, according to some coding, which is also unknown...] If the universe should be self-referent in the sense of some logical circularity, I presume that such situations could be better observed by singularities - like big bangs, black holes, empty zones which are too far away of any galaxy, star births, etc... by global space time behavior in their neighborhood. By space time behavior I mean that one should observe the neighborhood also in the temporal direction, in its evolution, not only the spatial neighborhood.

I guess theself reference Steve is asking about is a self reference similar what we have in Goedel's proof where using the rules of logic one arrives the statement that in essence says "this statement is unprovable".

In my opinion self reference of this type has no relevance in nature.  

We cannot have such a self reference in nature, because as we know nature itself doesn't work by numbering some statements in order to refer them later. 

Thank you Atilla.

This is indeed the kind of self-reference I am asking about.  To convince you then, of my idea, would I have to persuade you that Nature does involve arithmetical processes?

No,  it may involve processes that may be arithmetical like addition or subtraction of natural numbers. In Goedel's proof what we have is met a mathematics not mathematics.  It contains  statements like "the statement with Goedel's number n has no proof ". The nature does not number mathematical statements and it doesn't try to prove a statement.  It just "works" according to mathematical rules.  Self reference like "this statement is unprovable" corresponds to nothing more then only certain neural firing pattern in our brains. The neural processes that creates this firing pattern themselves are without self reference. 

That self referencing statements look weird has been known since ancient Greece.  Thecontribution of  Goedel's 

The contribution of Goedel  is that he has shown that one can end up with this type of self referencing statements using rules of logic if one:...uses recursive notation.like  n! = n (n - 1)!and... Numbers statements in order to refer them

Thank you Atilla,

The question I have asked is one of a series on the subject.  I carefully worded this particular question: "If we accept, as a premise, that logical circularity exists in the Universe".  My reason for those words was to attract answers that would advance discussion on the matter, rather than argue against my premise.

Regards and thanks.

I accept your premise as a necessary complement to the assumption that the universe is infinite. The universe provides no completely provable starting points for thought. We need to jump on the merry-go-round at some arbitrary point even though subsequent logic will always produce circularity. For instance, consider the proper definition of matter: Matter always contains other matter ad infinitum. Consider the proper answer to the question: Where do things come from? Answer: From somewhere else. We observe that there is an endless concatenation of material causes for all effects. The infinite universe, being infinite in both time and space, has no first cause. The passing of the buck has neither a beginning nor an ending. Thus, logical circularity is an inherent characteristic of the infinite universe.

What  can "logical circularity" mean to a physical universe? In your example you use the term "self-referent." Give a physical example. Since we can't observe any boundaries in what we are able to see, it has been proposed that our global geometry could be closed, meaning that if you go far enough in one direction, you return to where you started. I don't think that's what you mean.

It has also been proposed that every black hole has at its center the beginnings of another universe. Now that has possibilities.

Thank you James,

The kind of self-reference I'm talking about is cause and effect in the Universe.  If the Universe is "Everything" and self contained, there is a problem in explaining the Original Effects. We may also question the idea of Physical Laws motivating processes. Some tent to think of Physical Laws standing outside the Universe and motivating what goes on inside.  But then we can ask: Where are these Physical Laws?   And what LAWS motivated existence of The Physical Laws?....... And so on.

I can't see cause and effect as self-referent. If there were true feedback, there could be a circular effect and this effect is utilized to make oscillators. But examining it carefully, it's not true self reference because the effect is always somewhat delayed in time. I would agree that if you think causality is a law then you have a problem. But the current explanation of the initiation of the big bang is quantum in nature which is not causal in the small, only in the large. So it was presumably started by a quantum fluctuation, a probabilistic effect. This is how some are able to claim the universe as something from nothing.

Thank you James.  My ultimate suggestion is that logical self-reference in the Universe is linked to the logic of quantum phenomena that are not explained by classical logic.

in finding it

I wrote previously that self referencing statements like "this statement doesn't have a proof" as we have in Goedel's proof have no relevance in how nature works. In this sense there is no self reference in nature (I don't consider dynamic processes with feedback as self reference. 

However there is a self reference in Copenhagen interpretation of quantum mechanics. The formulation states that the dynamics of the system is determined by the dynamics of the probability wave function(or the dynamics of the operators depending whether one uses Schrodinger or Heisenberg picture) in the time span that doesn't contain a measurement event. The uncertain value of an observable as represented by the probability wave function becomes certain after a measurement of the observable. 

To the question "what is a measurement? " the Copenhagen interpretation doesn't have a definite answer. Since measurement itself is a physical process, the axiom of the theory contains a term that refers to a complex physical process that should be described by the theory. To avoid this self reference one can include the device into the wave function but then there can be no collapse of the system. Or to get a collapse namely a certain outcome for an observable one needs another measuring device measuring now the combined system and this goes on unlimitedly. Thus if one includes all particles in the universe into the wave function there is no measuring device left. 

However of course this is not self reference of the nature itself but only self reference caused by an ill formulation of the theory. The Copenhagen formulation is a practically useful tool to evaluate the results of an experiment statistically but it certainly can not be an ultimate formulation. 

I don't know whether you meant this self reference when you mentioned self reference in context with quantum mechanics