jaekwon

The program or instructions embedded in the PS only construct statements consistent with the axioms of the symbolic system, barring external factors like cosmic rays. I think you are asking why a computer behaves consistently. I assume it does for the sake of argument. If it does not, my original theorem holds. 1. Yes. Show me a complete set of physical laws and I will derive you a contradiction. 2. Sort of. You bring up an interesting question here. Let me get back to you in a bit. 3. Well, yes, at least for every repeatably observed event. And physics does this, for repeatably observed inconsistent events. 4. No, it just means that any mathematical laws of the Universe are inconsistent, or incomplete. It doesn't make sense (for me) to make statements about the (in)consistency of reality, because empirically (my) reality is consistent. 1. Yes and no. Some phenomena are observable but they can't be considered "events" if they have no time component. But yes, I mean all observable "stuff" (minus emphasis on time) which I call "phenomena". 2. Yes, that is the common understanding of law. Otherwise it wouldn't be applicable and it wouldn't have predictive power. 3. Yes and no. Let's leave it at X is consistent with L. To "predict" has wuzzy connotations when it comes to probabilistic laws like QM, and "consistency" is hand-wavy enough.

Thank you. All I want to do is apply existing mathematical insight into physics, and see what emerges. I would appreciate your feedback.

The problem with questions of determinism in QM (which includes elements of non-determinism) is, there exists a "kind of determinism" beyond the inherent non-determinism of QM as long as all real physical phenomena appear to be sufficiently modeled by QM. That is, in repeated experiments it should appear that all experimental results converge towards those as predicted by QM. Then, one can think that QM "determines" the universe. I would argue that no set of theorems consistent with the Universe can have this kind of determinism; that the Universe is non-deterministic under any set of theorems. It follows from an analog of Godel's Incompleteness Theorem; an Incompleteness Theorem of Physics. link to discussion: http://www.scienceforums.net/topic/63883-incompleteness-theorem-of-physics/

In the topic below I try to prove that free will is necessarily possible with Turing machines. http://www.scienceforums.net/topic/63883-incompleteness-theorem-of-physics/ The theorem, which is derived from Godel's Theorem of Incompleteness, states that "there exist physical phenomena that cannot be explained from laws derived from empirical observations". I define free will as your ability to affect those unexplainable, un-model-able physical phenomena.

Your ability to affect physical phenomena consistent with known physical laws derived from empirical observations, but unexplainable from those laws alone. In this mapping, unprovable statements that are true should be consistent with physical phenomena. The mapping is symbolic "truth" to physical "phenomena". Otherwise a "sufficiently strong" computer would not be deterministic when performing computations on "sufficiently strong systems". BTW, even the lack of a physical phenomena (e.g. particles of type X never appears in two places) is a phenomena. Does that address your point? I do. But this assumption is true, otherwise your computer would not exist. Godel himself would not have existed. I agree. This is called "progress" in science. But there will be no end to it so long as computers or minds exist. Yes, that's a nice way of putting it. Or, less palatably: the universe is nondeterministic. I read conflicting opinions on this forum about quantum theory and what it means for determinism, but the universe is not determined by any theory. For me, I hypothesize that free will is intimately tied to all physical phenomena invariant of scale. I won't try to prove this, but it's exciting to think that we may have a crucial role in the workings of the cosmos. Indeed!

Hello all, this is my first post here. It occurred to me that Godel's Incompleteness Theorem may have a physical counterpart. I propose one here: The Incompleteness Theorem of Our Universe: "There exist physical phenomena that can not be explained from laws derived from empirical observations". Sketch of a proof: 0. Assume otherwise, that all physical phenomena can be explained from laws derived from empirical observations. 1. Some physical systems PS like the human brain or a sufficiently strong computer can be instructed to run in accordance to rules (axioms) of a "sufficiently strong (symbolic) system". ** 2. There exists in PS a mapping between (a) the statements in the symbolic system and (b) empirical observability of physical phenomena. 3. From Godel's Incompleteness Theorem, (2), and (0) it follows that there exists physical phenomena of this physical system PS that cannot be explained from its own empirical observations. (3) is a contradiction of (0). QED. ** if you are self aware and capable of formulating statements about yourself, "you have free will" is a Godel sentence. I don't believe we can ever fully understand the laws of this universe with only symbols and equations. There is always room for free will in the universe, if you choose to believe it. - Jae