You weren’t asked to accept it, but assume it for the purposes of discussion. 
If that particular discussion has ended, I feel comfortable discussing its viability.
Okay, sorry, Eris. If you prefer to discuss the soundness of the assumption in a different thread, that’s fine. I didn’t mean to hijack this one, although it seemed to be sort of meandering a bit. But I don’t see where there is a debate if we assume the universe to be a deductive system based on axioms. By definition, GIT applies.
On the other hand, I think a compelling case can be made for my own assumption:
Select any arbitrary coordinate of the universe, using any arbitrary grid. Call the coordinate U. Then, at U, for any energy, E, and for any time, T, [symbol]D[/symbol]E[symbol]D[/symbol]T[symbol]³[/symbol]h/2, where h is Planck’s constant. Since for every U, the relation between the change in energy and the change in time is identically uncertain, the universe is a composite of probabilities. Thus, it is a probability field.
I was rethinking Math Geek’s post, and now I’m wobbly. The Axiom of Choice is a classic undecidable proposition, is it not? And yet it is a possible one (that is, it finds a means of expression; we can write it down, I presume, in the symbols that are allowed). So a Turing machine would never “get” to the Axiom of Choice’s expression, but we would have no problem starting it in such a state; thus, it is possible, but it would never happen running any program designed to create all theorems.
Lib, this is classic. You want to say the universe isn’t an axiomatic system of deduction, and then throw a math equation my way to demonstrate that. 
Secondly, a quantum computer (for example) should still be a Turing machine. Just because it would solve some problems almost instantaneously doesn’t mean it can solve all problems, yeah?
And the universe is a probability field, or it probably is a probability field? Can we phrase your equation in a manner where the existence of energy and time (and normal mathematical operations) are the axioms and state that in any space S there exists etc ? Why not?
Oh, no need for a different thread, I just wanted to stay off a tangent if possible. But, you’re right, this thread has gone all over the place anyway, so what the hell?
Are the two mutually exclusive? I’m sorta-kinda inclined to agree with you, but I think you’re making a big assumption here.
Not mutually exclusive, no. But it is a matter of which contains the other. Deduction is but one of the epistemic systems that the probabilistic universe presents. Fortunately, Eris’s concern is assuaged by that fact.
But Libertarian, how delightful. For let us hypothesize that the universe is an axiomatic deductive system in the sense that one needs an axiomatic deductive system to comepletely describe the universe. This does not rule out your probability field.
So does GIT still hold? In a sense it does, but in a much more complicated sense. Given the probability field, the universe can discretely jump to possible states without intervening states, meaning the universe is not forbidden to reach “true” states, yet since it never has to “get there” it isn’t proving undecidable propositions.
How very perplexing.
But now we reach a point where the behavior is that of adding axioms to the system (reaching true states without “getting there”). Now the question is: are these axioms recursively enumerable? If they are not then, as I understand it, GIT cannot hold.
It’s certainly possible to state an undecidable proposition, but the point of Godel’s theorem is that it isn’t possible to prove it.
And the hypothetical programs we’ve been discussing aren’t stating theorems, they’re also stating proofs of the theorems. While it is possible to start a Turing machine in state which encodes a statement of an undecidable proposition, we could never start it in a state which encodes a proof of such a proposition because no such proof exists. Or, to be needlessly picky, there is no correspondence between states of the machine and proofs such that some state corresponds to a proof of an undecidable proposition.
(Technically, the Axiom of Choice isn’t the kind of proposition you get from Godel’s theorem. I think it’s the same idea…a proposition which can’t be proved or disproved by the other axioms of set theory…but I don’t know that for certain, and it doesn’t follow from Godel’s theorem. Anyone else know?)
AC is independent of the axioms of Zermelo-Fraenkel set theory, in the sense that there’s a model for which the ZF axioms and AC is true, and there’s a model for which the ZF axioms hold and AC is false, I think. Undecidable statements are either true or false. but can’t be demonstrated to be either way.
I’m a little shaky on this, but I think that’s the distinction.
AFAIK all Godel statements are of the self-referential form, “I can’t be proved”, “I am not a theorem” etc. But that is also all that is necessary for proving the “there exist statements that etc”. It was orginally suggested that GIT was a “trick” that didn’t really say anything about math because godel statements weren’t all that deep (so sayeth books on the recent history of math, at any rate). Finding statements that people thought were important to have the same conditions (can’t be proved or disproved) changed people’s perceptions greatly. It isn’t the same “kind” in the sense that no one has constructed a paradox by godel-numbering AoC, but it is the same “kind” in the sense that accepting AoC or rejecting AoC are both consistent (if not done simultaneously in the same system, of course).
So “same” is a matter of perspective, wouldn’t you say?
Well, that’s where I am trying to see what that means for a TM. I am imagining that a proof of a theorem would require a sequence of states. In such a manner that each individual state is a proposition, and the series of propositions leading to a final proposition can be a theorem. That’s why I’m saying what I’m saying. I would appreciate any input on the matter.
I hate to chime in with my usual themes … but I’ll do it anyway!
There can not be an axiomatic deductive basis to the universe because the base axiom would always have to have a basis (as erl has previously called it, “the god postulate”): the axiom, of course, is not deductively provable within the system, so, in practice, axioms are based on induction. Deductions can then follow. Most formal mathematical systems make these inductions right up front and make all that follows true to deductive rules. Other forms of knowledge are generally a constant mix of inductions and deductions. We hold various inductions to different degrees of doubt.
This statement of erl’s is far from trivial. All knowledge of the universe is ultimately based upon a foundation of induction. Our inductions may be “truth”. But we can’t deductively prove it. Do we need GIT to establish this?
And in fact, it must be noted that Godel’s conclusion, if his proof is sound, is itself unprovable and therefore not proved. Some of the most entertaining comments on GIT come from hard atheists who see it as evidence (even proof!) that God doesn’t exist. And yet, as Ramachandran would point out were he here, the implications of GIT cut both ways: if God is Absolute Truth, then it stands to reason that His nature is entirely outside the scope of the Unprovability Theorem.
Since Eris has graciously extended me some license to discuss the broader issue of what the universe is, I’ll proceed.
I’ll entertain that hypothesis as an excluded middle. But, for the record, I don’t think that an ADS completely describes the universe.
As a probability field, everything in the universe is potentially “true” (if true is being used as a synonym for possible, and apparently it is) except where probabilities would be zero. The only state where that would be the case would be the state where electromagnetism breaks down, with neither electricity nor magnetism inducing the other. That would also be a state where the change in time is zero, which would be absurd since zero is neither greater than nor equal to half of Planck’s constant.
Therefore, every state in the universe is possible, and no deductive system is required to achieve any state, since no state can be predicted. This contradicts the excluded middle, meaning that the universe is not an ADS.
GIT doesn’t even hold in its own system.
I’m having a feeling of deja vu on this one.
You still need an ADS to describe probability. Also, am I to understand that the state of the universe at any one time t[sub]1[/sub] has no bearing on the state of the universe at t[sub]2[/sub]?
Umm…no, sorry, you’re wrong on that one. The proof is sound, and the theorem is true: undecidable sentences do exist.
And also, “probability zero” isn’t equivalent to “never happens”, unless you have a finite sample space (in this case, number of possible states for the universe). Consider the implications of that.
The write-up on Godel’s theorem is coming, it’s just that right now I don’t have a desk and I’m typing on the floor, and it sucks. Hold on, everybody.
Ultra
Ad logicam. The truth of the theorem is irrelevant with respect to the validity of the proof. A proposition can be true even if the argument that led to it is false. For example, 16/64 is equal to 1/4 even if I derive the reduction by cancelling the 6s (which is the wrong way to do it).
Furthermore, I didn’t say that the theorem isn’t true; I said (and you even quoted) that the theorem is unproved. And it is unproved because Godel used an axiomatic deductive system to derive his theorem. Proposition XI does two things in one fell swoop: it establishes Godel’s conclusion while simultaneously knocking down the structure from which his conclusion was built.
I look forward to your paper.
Eris
Only to describe it one way. I can describe it other ways. Frankly, I like my sainted mother’s way best: “You never know what’s around the corner.”
Well, it might. You just can’t know for sure.
What???
Godel’s theorem doesn’t state that all axiomatic systems, or proofs about statements in those systems, are somehow invalid. Godel’s theorem doesn’t invalidate anything. Godel’s theorem only says that there exist statements which can’t be proved in certain axiomatic systems.
I have no idea why you think Godel’s theorem itself is somehow one of those statements, because it isn’t. ultrafilter is correct: Godel’s theorem has been proved. The proof is valid.
Math
Here’s what I mean. Let P be a formal system of deduction. Let N be a proposition that is provable in P. If Neg N is also provable in P, then P is inconsisent. But if P is inconsistent, then Neg N is provable (since every wiff is provable in P). The inconsistency of P and the provability of Neg N are logically equivalent, as are the consistency of P and the unprovability of Neg N.
Proposition XI — Wid©[symbol]®/symbol Q(x,p) — may be stated this way: “If P is consistent, its consistency is unprovable within P”. Since we cannot prove that P is consistent (at least within P), we do not know whether ~[Wid©[symbol]®/symbol Q(x,p)] is provable as well. We are left with a “proof” that is valid, yes, but incomplete since we may not say that N contradicts Neg N.
First, a nitpick: the formula Wid©®(x) Q(x,p) is not the conclusion of proposition XI of Godel’s paper; it’s just a step in the argument.
Secondly, there exists a valid proof of Godel’s theorem. Nothing you have said negates that fact.
I do agree with the following two statements:
[list=1]
[li]If N is provable in P, then P is inconsistent if and only if Neg N is provable in P.[/li][li](By Godel’s Theorem) If P is consistent, then the consistency of P cannot be proved in P.[/li][/list=1]
Combining the above statements and applying them to Godel’s theorem results in: “If P is consistent then the non-existence of a proof of the negation of Godel’s theorem cannot be proved in P.”
But quite frankly, so what? Godel’s theorem has been proved. The negation of Godel’s theorem, by extension, has been disproved. You seem to claim that the job’s not done, however, until we can also prove that the negation of Godel’s theorem cannot be proved…but if we accept that reasoning, then we can’t accept any theorem of number theory until we’ve somehow proved the consistency of the whole system. And that’s just raising the bar too high.
We will probably never prove, at least in the mathematical sense of the word, that number theory is consistent. Godel’s theorem certainly says that we will never prove that fact using number theory alone. But let’s face it: either it is consistent or it isn’t. If it is consistent then Godel’s theorem is true and the proof is valid. If it isn’t consistent then every single mathematical proof is suspect, including the proof of Godel’s theorem…and how can we accept the statement of Godel’s theorem if we can’t accept the proof? Your claim that somehow we have to accept the statement but not the proof…that somehow the statement renders its own proof invalid…is just bizarre.
Finally, it’s interesting to note that Godel himself commented on this issue in the very paper where he presented his result (the full paper can be seen here):
Although it’s a little out of context, note that Godel is saying that his theorem refers to a specific form of proof. Certainly it doesn’t apply to all forms of proof, not even all mathematical ones.
Math
I’ve read the paper and already have that link, thanks.
Yes, the assertion I showed is a step. It’s the last step, or essentially the last step. Proposition XI was his most sloppily constructed one, and quite a bit was left to the imagination. Others have filled in most of the blanks since. All Godel did after the step was to translate it to an equivalent implication with a different syntax. “W Imp (17 Gen r)” is the same as “Wid©[symbol]®/symbol Q(x,p)”. I believe that the very comment you quoted from Godel — or rather, the rest of the paragraph — supports my point, and is in fact the comment that made the idea occur to me.
Of course, he never delivered on his promise to generalize XI. I’ve never heard anyone say why. The part you quoted (the part that saved Hilbert’s students from heart attacks) spared systems that have no multiplication (no recursion).
I don’t know why you want to subjectively determine where the bar should be. In my opinion, it should be where it falls. If the universe is not real existence, if there is no consistency in it or it is incomplete, if it is nothing more than a probability field — so what? It doesn’t bother me.
Math
And by the way:
Strawman. When did I say the proof is invalid? In logic, validity and soundness are not the same.