**Quote** (8472Zo @ Feb. 24 2005, 1:44 am) |

I came to this conclsion a) becuase it's true and b) becuase it is the raison d'etre of stable state theory itself. |

The stable state is a paradox in itself. How could you have a stable state universe when entropy drives down a system to thermal equalibrum where no life could exist. also it introducts the dark sky paradox or Olbers' paradox

If the universe is assumed to be infinite, containing an infinite number of uniformly distributed luminous stars

A star is any massive gaseous celestial body in outer space. Stars appear as shining points in the nighttime sky that twinkle because of the effect of the Earth's atmosphere and their distance from us. The Sun is an exception: it is the only star sufficiently close to Earth to appear as a disc and to provide daylight.

then every line of sight should terminate eventually on the surface of a star. The observed brightness of a surface is independent of its distance, and the apparent area of a star diminishes as the square of its distance, and the number of expected stars increases as the square of the distance. Thus, every point in the sky should be as bright as the surface of a star.

It should be noted that for stars to appear "uniformly distributed" in space they must also be uniformly distributed in time, because the further away one looks, the older what one sees is. On an infinite scale, this means the universe must be infinitely old with no dramatic changes in the nature of stars in that time...

And you truth about paradoxes has also been disproven by Godel's Incompleteness Theorem:

The proof begins with Godel defining a simple symbolic system. He has the concept of a variables, the concept of a statement, and the format of a proof as a series of statements, reducing the formula that is being proven back to a postulate by legal manipulations. Godel only need define a system complex enough to do arithmetic for his proof to hold.

Godel then points out that the following statement is a part of the system: a statement P which states "there is no proof of P". If P is true, there is no proof of it. If P is false, there is a proof that P is true, which is a contradiction. Therefore it cannot be determined within the system whether P is true.

As I see it, this is essentially the "Liar's Paradox" generalized for all symbolic systems. For those of you unfamiliar with that phrase, I mean the standard "riddle" of a man walking up to you and saying "I am lying". The same paradox emerges. This is exactly what we should expect, since language itself is a symbolic system.

Godel's proof is designed to emphasize that the statement P is *necessarily* a part of the system, not something arbitrary that someone dreamed up. Godel actually numbers all possible proofs and statements in the system by listing them lexigraphically. After showing the existence of that first "Godel" statement, Godel goes on to prove that there are an infinite number of Godel statements in the system, and that even if these were enumerated very carefully and added to the postulates of the system, more Godel statements would arise. This goes on infinitely, showing that there is no way to get around Godel-format statements: all symbolic systems will contain them.

Your typical frustrated mathematician will now try to say something about Godel statements being irrelevant and not really a part of mathematics, since they don't directly have to do with numbers... justification that might as well turn the mathematician into an engineer. If we are pushing for some kind of "purity of knowledge", Godel's proof is absolutely pertinent.

In addition, some known mathematical phenoma already exhibit the Godel incompleteness property. For instance, in set theory mathematicians define different degrees of infinity based on the number of members of the set of all integers, rational numbers or reals. The first degree of infinity, called (aleph-nought), is the number of integers or the number of rational numbers (these numbers are the same "degree of infinity"). The second degree of infinity is aleph-nought raised to the power aleph-nought. For a long time people were trying to decide whether 'C', the number of real numbers, was the same as the second degree of infinity. Finally it was proven that whether C and 2nd infinity were equivalent came down to the truth or falsehood of a statement that could not be proven from the existing axioms of mathmatics. This statement was absorbed as a new axiom, just as Godel statements would have to be. So there is the first of many Godel-style statements that we'll probably see popping up in mathematics.

Of course, a more familiar example is the parallel-postulate axiom, since it cannot be proven from any other axioms of Euclidean geometry, and in this case the way you define it leads to at least three different self-consistent systems.

In any case, what does it mean that a symbolic system based on deriving truth from axioms is incomplete? Could we make a complete system? The only way I can see to do that would be to include an infinite number of axioms, which deterministicly describe all happenings in the past, present and future. This would only work in a deterministic universe, and it would be difficult to draw a distinction between the data of this 'complete' system and reality itself.

Thinking of the data required is perhaps the right direction to move in: it is the reason the symbolic system is incomplete. The symbolic systems we use to describe the universe are not separate from the universe: they are a part of the universe just as we are a part of the universe. Since we are within the system, our small understandings are 'the system modelling itself' (system meaning reality in this case). Completion of the model can never happen because of the basic self-referential paradox: the model is within the universe, so in effect the universe would have to be larger than itself. Or you can view it iteratively: the model models the universe. The universe includes the model. The model must model itself. The model must model the model of itself.. ad absurdum.

So paradoxes are woven into the fabric of logic itself.