Spiritus Mundi

[TVAA]

Arrogant and unpleasant, certainly. Dishonest, no – if someone doesn’t understand what I mean, how can I deceive them?

ultrafilter is willing to admit that Spiritus is incorrect about at least one thing. Wow. I thought the chorus’ opinion was that he could do no wrong…

[TVAA]

Such a system is inconsistent, Spiritus.

Perhaps you’re thinking of statements like “this statement is false”, which can be considered neither false nor true, yes?

[TVAA]

*Originally posted by Spiritus Mundi *
**ultrafilter, as an example, consider a system which is consistent yet allows for valid proofs of both § and (~P). Which of those statments would you say is necessarily true and which is necessarily false? **

Definition of Consistency

[ultrafilter]

*Originally posted by Spiritus Mundi *
**ultrafilter, as I have been pointing out to TVAA that conclusion holds only in systems that require a binary truth operation to hold for all statemnts. It fails to hold for systems that do not map to an external model for truth, systems that do not define truth as a binary operation, or systems that do not require a defined truth value for all statements.

The argument you just used is the one standardly applied for a predicate calculus, but GIT holds for many systems that are not equivalent to a predicate calculus. **

Fair enough. I’ve never dealt with such systems.

ultrafilter, as an example, consider a system which is consistent yet allows for valid proofs of both § and (~P). Which of those statments would you say is necessarily true and which is necessarily false?

Neither one is necessarily true. However, if we’ve got a binary truth operation, one is true and one is false. In that system, not all provable statements are true.

Anyway, I stand corrected. Spiritus Mundi is correct here. See how easy it is to communicate using precise terminology?

Also, wikipedia’s definition of consistency is not used in modern higher mathematics. A system is consistent iff there is some statement which is not a theorem of that system.

Consider the system with an empty axiom set, and the following rule of inference: For any statement B, we may conclude that ~B & B. Clearly, it’s possible to derive a contradiction here, but the system is consistent.

[ultrafilter]

*Originally posted by ultrafilter *
Also, wikipedia’s definition of consistency is not used in modern higher mathematics. A system is consistent iff there is some statement which is not a theorem of that system.

And because I know someone’s going to ask, here’s a cite. Please note that “true” as used in this page has a very precise meaning.

[TVAA]

Really? How is it possible to derive a contradiction if the system contains no axioms? What does it derive from?

[ultrafilter]

From the empty set. Whenever you have A |–[sub]K[/sub] B, that’s equivalent to |–[sub]K[/sub] A -> B.

A |–[sub]K[/sub] B means “From A and the axioms and rules of inference of K, you can prove B”.

Remember, a logical system is just the combination of a set of rules for making new statements from old and a collection of initial statements. There’s no requirement that the rules preserve any property of statements that you might care to suggest.

[Jabba]

ultrafilter: The link you provided says ( in section 1)
“As usual K is consistent … iff there is no formula S for which both [K entails S] and [K entails not-S].”
(Paraphrase because I can’t get the formal notation on here). This appears to contradict what you said.

[Spiritus_Mundi]

Dishonest, no – if someone doesn’t understand what I mean, how can I deceive them?

Why is TVAA a liar?

Exhibit 1: I have already posted both the correct definition of consistency and noted that the “not (P) and (~P)” version is not rigorous.

Exhibit 2: TVAA himself crowed not too long ago, when he mistakenly thought that I did not understand the correct definition of consistency: "He neglected to pick up on the point that it’s not the case for every type of system that a contradiction leads to all possible statements becoming valid.

Exhibit 3: I reposted the correct definition of consistency on the top of this very page.

Exhibit 4: In response to a system in which both (P) and (~P) follow from axiom, TVAA posts: Such a system is inconsistent, Spiritus.

Exhibit 5: TVAA then links to an incorrect (non-rigorous) definition of consistency.

ultrafilter, at least now you don’t have to feel bad.

Seriously, it is a mistake that is made very often in discussions of GIT, even many very good mathematicians and logicians often use naive truth values as an easy way to illustrate GIT to lay audiences. Like most such treatments, it is good to get the ball rolling but eventually needs to be sacriiced in the name of greater rigor. BTW, I should make sure to clarify that throughout my discussions about truth with TVAA both in the GD thread and here I have been explicitely speaking about the conditions under which one might say that “G is true in system A”.

It is entirely possible that someone might use an external model of truth in which all statements, no matter how paradoxical or unproveable, must be assigned a definitive truth value. Of course, that does not imply any particular truth value for G in system A.

[ultrafilter]

*Originally posted by Jabba *
**ultrafilter: The link you provided says ( in section 1)
“As usual K is consistent … iff there is no formula S for which both [K entails S] and [K entails not-S].”
(Paraphrase because I can’t get the formal notation on here). This appears to contradict what you said. **

The full (translated) quote is “K is consistent if there is some formula S which is not a theorem of K, or equivalently if there is no formula S such that S is a theorem of K and ~S is a theorem of K”. In the first-order predicate calculus, those two conditions are equivalent. In a more general setting, they are not.

[Jabba]

ultrafilter explained:

In the first-order predicate calculus, those two conditions are equivalent. In a more general setting, they are not.

Ah, I see. Thank you for the clarification.

[TVAA]

Okay, I’ve just done some checking and dug up an old textbook, The Language of First-Order Logic. Oh, that brings up memories…

First, Godel’s Incompleteness Theorem does indeed allow us to show that at least one Godel statement for any system is true but unprovable. (I trust Barwise and Etchemendy more than SM.)

Secondly, an invisible expert of mine assures me that while mathematicians have largely abandoned the concept of ‘truth’, logicians have not – Spiritus can’t get out of the GIT’s conclusion by complaining that ‘truth’ isn’t rigorously defined or isn’t commonly used.

Thirdly, I admit that I don’t know precisely what you mean by “omega-consistency”, but it’s not the same as consistency.

The definition I linked to is indeed the definition of ‘consistency’. If you want to discuss another mathematical concept, go right ahead. Your omega-consistency may be used in higher mathematics, but it has nothing to do with the definition I’m using.

Finally, how exactly am I a liar again? Non-rigorous definitions are not incorrect, Spiritus – the one I linked to is equivalent to the statement you made. You’re not challenging the definition – which is my whole point. There are other definitions possible – why aren’t you mentioning them?

Nice try, but no cigar.

[ultrafilter]

Introductory logic textbooks often use the simplified but intuitive definition of consistency, just like calculus textbooks use the simplified but intuitive definition of continuity. Even Mendelson does it! That doesn’t mean that it’s what working mathematicians or logicians actually use. You’re right that logicians still work with ‘truth’, but they mean it in a very precise sense. Worse, they may mean different things by ‘truth’ from one context to the next.

Would you mind giving the precise statement of GIT from Barwise & Etchmenedy? I strongly suspect that they’re assuming that truth is a binary relation–each statement is either true or false. Things may be different if truth values are drawn from {0, …, n - 1} (what does ‘true’ mean here?) or [0, 1].

[symbol]w[/symbol]-consistency is largely irrelevant to this discussion. However, it is relevant because it’s very difficult to define in general without recourse to formal symbolism. In fact, I don’t think it can be done. That pretty much shoots holes in your idea that discussing everything in plain English is good enough.

btw, the definition of consistency you gave is only equivalent to the definition Spiritus gave when you have the standard rules of inference in effect. Note the system I gave, which is inconsistent in the non-rigorous sense, but consistent in the rigorous sense.

[ultrafilter]

Actually, I take that back. [symbol]w[/symbol]-consistency can be very close to defined in ordinary parlance, but you miss a very subtle shade of meaning.

[Spiritus_Mundi]

Okay, I’ve just done some checking and dug up an old textbook, The Language of First-Order Logic. Oh, that brings up memories…

First, Godel’s Incompleteness Theorem does indeed allow us to show that at least one Godel statement for any system is true but unprovable. (I trust Barwise and Etchemendy more than SM.)

I do not have that particular text on my shelf, so I will not venture to guess whether the authors were careless, simplifying for clarity, or providing a weaker version of GIT specific to a first-order predicate calculus (which requires a binary truth value for all statements). I will suggest that you not consider introductory texts on any subject to be the final arbitrator of what is correct in that field.

Secondly, an invisible expert of mine assures me that while mathematicians have largely abandoned the concept of ‘truth’, logicians have not – Spiritus can’t get out of the GIT’s conclusion by complaining that ‘truth’ isn’t rigorously defined or isn’t commonly used.

Secondly, please ask your invisible expert[sup]Tm[/sup] to point out exactly what words of mine he is misreading to claim that either mathematicians or logicians have abandoned the concept of truth. What I have said is that folks who keep current with mathematical logic do not pretend that GIT generically implies a truth value for Godel statements. “Truth” is used quite often in both mathematics and logic, though usually with far more rigor than you have been using it.

Thirdly, I admit that I don’t know precisely what you mean by “omega-consistency”, but it’s not the same as consistency.

Thirdly, it is obvious that you do not know what omega-consistency is. Omega-consistency is exactly the stronger[sup]1[/sup] form of consistency that you have been calling “consistency” and in which Godel’s orignal proof was valid. Consistency is the weaker[sup]1[/sup] requirement whose definition I gave for “consistency” and in which the stronger Godel-Rosser Theorem (now consisdered the standard form of GIT) is valid.
[ul][li]**Consistency: **For some well-formed statement P, § is not a theorem of A.[/li][li]**TVAA-consistency: **If § is a theorem of system A then (~P) is not a theorem of A. This is almost equivalent to omega-consistency and in the converse is often used as a method for showing omega-inconsistency. [/li][li]**omega-inconsistency: **A system is omega-inconsistent when all wffs in a certain infinite series are theorems while the negation of the summarizing quantified statement is a theorem. [/li](Does that work for you, ultrafilter?)[/ul]

I express shock that one of your invisible experts[sup]Tm[/sup] who are so much more knowledgabel about mathematics and logic than any of the stupid and ignorant Dopers contributing to this thread was not able to clarify this rather simple point for you. What, they just stopped by to misinterpret my remarks about truth and GIT but couldn’t be bothered to tell you that you were mis-labeling “omega-consistency” as “consistency”?

The definition I linked to is indeed the definition of ‘consistency’. If you want to discuss another mathematical concept, go right ahead. Your omega-consistency may be used in higher mathematics, but it has nothing to do with the definition I’m using.

Once again, you are wrong.

Once again, you display that particular combination of arrogance and ignorance that makes you the sebacious anal cyst of the SDMB.

Finally, how exactly am I a liar again?

Actually, I find that I have to retract this particular accusation of dishonesty. I assumed that when you posted your smug little challenge about contradictions and consistency that you actually understood the implications of what you were saying. It is clear now that you did not. I suppose you ran across a note somewhere and posted the text while remaining ignorant of how it applies to consistency and Godel’s proof.

So, your previous dishonesties remain unrefuted, but this time it seems you were just being stupid.

Non-rigorous definitions are not incorrect, Spiritus – the one I linked to is equivalent to the statement you made.

[ol][li]Non-rigorous definition is a polite way of saying, “good enough for non-rigorous contexts but wrong when it really matters”.[/li][li]The definition that you linked to is not equivalent to the one I posted. [/ol][/li]

You’re not challenging the definition – which is my whole point. There are other definitions possible – why aren’t you mentioning them?

Mention them? You pathetic illiterate **I posted the damn thing.

TWICE!**

Now, I am going to link to some treatments of GIT that you will not understand. I’ll start out with a couple of easy ones that don’t get everything right but have some points worth reading. then I will post the one that you really should gather all of your invisible experts[sup]Tm[/sup] around to explain to you. Though if the commentary they have provided you thus far is any indication you might want to start seeking a higher class of imaginary playmate.

THIS SITE
has a pretty good summary of GIT, including a breakdown on the three types of incompleteness implied by Godel. It’s treatment of “truth” is too sketchy, but the author does manage to note that:

There is also an important kind of semantic incompleteness in the fact that some of the wffs undecidable in S are true. We must be careful in how we state this, for Gödel himself proved in 1930 that first order predicate logic was (in one sense) “semantically complete”, namely, AA (see Hunter 46.1). We should also remember that the completeness of predicate logic is a very different thing from the completeness of a system of arithmetic. Because systems of arithmetic contain proper axioms, they “intend” to capture more than logically valid wffs; hence “semantic completeness” in the ordinary sense (containing all logically valid wffs of the language as theorems) is less complete than arithmetic systems intend to be.

You might also ask one of your invisible experts[sup]Tm[/sup] to explain the following passage to you, in re: consistency.

A system is omega-inconsistent when all wffs in a certain infinite series are theorems while the negation of the summarizing quantified statement is a theorem. Historical note: Gödel actually proved that if S is omega-consistent, then it is omega-incomplete. The “modern” formulation that if S is simply consistent, then it is omega-incomplete is due to John Barkley Rosser.

THIS ONE
has a pretty good synopsis of how the work of Godel-Rosser-Church-Turing actually does interrelate and what questions this rasies about minds and computers. It is not perfect by any means, but it is pretty good and the author benefits from being both beter informed and far less arrogant thant TVAA. A few passages that might interest the invisible experts[sup]Tm[/sup] are:

In addition, if the formal system has a certain stronger form of consistency, the sentence which asserts its own unprovability, called the Gödel sentence, is also not refutable in the system. Rosser later constructed a more complicated sentence for which simple consistency is sufficient both for its unprovability and for its unrefutability.
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The fact that a particular sentence is neither provable nor disprovable within a system only means that it is logically independent of the axioms: they are not strong enough to either establish or refute it - they don’t say enough about it one way or the other.
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Thus, Gödel’s theorems do not actually establish any disturbing discrepancy between provability and truth. Furthermore, the implication (*) above is an oversimplification: assuming consistency, Gödel’s sentence is not simply true, because it is not always true i.e. not in all interpretations. If it were, it would be provable, by the completeness theorem (also proved by Gödel), as noted in [Penrose 94, p. 116, note 5]: provability is truth in all interpretations).

THIS IS THE SITE THAT IT IS REALLY A SHAME YOU WILL NOT UNDERTSAND.
And, if you ever decide to actually read Godel’s paper then I strongly recommend you print this out and keep it as a guide, if for no other reason than it does a good job of explaining the non-standard terminology that Godel used in his proof. The whole paper is very much worth reading and understanding, but here are a couple of highlights:

The undecidability of some arithmetical propositions within the deductive system S may be classed among the syntactical metamathematical characteristics of the system S (represented by the calculus P), for the reason that this undecidability derives from the undecidability of some formulae within the calculus P which represents S. Deductive systems, unlike calculi, have also semantical metamathematical characteristics; in particular their propositions have or lack the semantical property of being true–what Gödel in his introductory Section I calls being “correct as regards content” (inhaltlich richtig). Connecting the syntactical property of being provable with the semantical property of being true by taking every proposition provable within S (i.e. every axiom and theorem of S) to be true (see [176]) gives an additional kick to the undecidability in S of g–by adding that g is true . . . This metamathematical argument which, combines semantical with syntactical considerations, establishes the truth of an arithmetical proposition which cannot be proved within S.

In his introductory Section 1 Gödel intermingles semantical with syntactical considerations in sketching a proof of the undecidability of g (which is the reason why I have seldom referred to this section in this Introduction). The distinction between what is syntactical and what semantical was not made explicitly until a year or two later (by Tarski, whose work included rigorously establishing unprovability theorems that were semantical); but it is implicit in Gödel’s remark towards the end of Section I that “the exact statement of the proof [of the undecidability of g], which now follows, will have among others the task of substituting for the second of these assumptions [that every provable formula is also correct as regards content] a purely formal and much weaker one” [176]. Gödel’s proof in Section 2 is a purely syntactical proof about a calculus (the formal system P) whose interpretation as a deductive system for arithmetic is, strictly speaking, irrelevant to his argument. It is true that Gödel explains arithmetization as a way of co-ordinating strings in his calculus with natural numbers, and he discusses recursive functions in terms of natural numbers (and I have followed him in speaking of numbers in both these contexts). But whenever he talks about numbers, and thus makes a remark which is prima facie about a deductive system rather than about a calculus, the remark is always a syntactical remark about the deductive system, and is therefore in essence a remark about the calculus which represents the system.

Why dont you and your invisible experts[sup]Tm[/sup] huddle over that for while until you can figure out how it affects the assertion that for any system to which GIT applies some Godel statements must be true.

[sup]1[/sup][sub]Do not be confused by the terminology. In this context, “stronger” means more restrictive, less general. “Weaker” means less restrictive, more general. Thus, Rosser’s proof in the “weaker” context of consistency is a more powerful result than Godel’s proof in the more restrictive context of omega-consistency. One way to think of it is omega-consistency -> consistency, but the reverse implication does not hold.[/sub]

[Spiritus_Mundi]

By the way, it has no relevance ot anything in this thread, but the [url=“http://home.ddc.net/ygg/etext/” Yggdrasil online library (from which the third cite in my post above came) is a great source of electronic texts for the esoteric taste. It’s hardy comprehensive, but it ranges from the Icelandic Sagas to Boole’s Calculus of Logic.

[Spiritus_Mundi]

Dammit! I meant to say “]”

[TVAA]

Gee, that would mean that… it takes a meta-system to demonstrate that those statements are true.

[pauses]

Still, the statements don’t need proof to be true. We’ve already established that. The meta-system does nothing but allow us to prove that the statements must be true relative to the first system. The statement that the statement is true in the first system can only be proved in the second system… but the first statement is still true regardless.

And Godel’s example… yes, it’s always true. There can be a version of that example for any system, since all it has to do is self-reference the system it’s in. And we can always find a meta-system consistent with the first that will allow us to demonstrate the truth of this G.

So… [long pause] What can we deduce from this?

Well, that Spiritus is wrong, for starters.

Yes, I know you posted definitions of consistency. Why haven’t you begun to discuss the alternate definitions? There are systems of reasoning where the version in which GIT applies isn’t used. Feel free to bring them up anytime…

[Spiritus_Mundi]

Gee, that would mean that… it takes a meta-system to demonstrate that those statements are true.

Wrong.

Still, the statements don’t need proof to be true. We’ve already established that. The meta-system does nothing but allow us to prove that the statements must be true relative to the first system. The statement that the statement is true in the first system can only be proved in the second system… but the first statement is still true regardless.

Wrong.

And Godel’s example… yes, it’s always true. There can be a version of that example for any system, since all it has to do is self-reference the system it’s in. And we can always find a meta-system consistent with the first that will allow us to demonstrate the truth of this G.

Wrong.

Well, that Spiritus is wrong, for starters.

Wrong.

Yes, I know you posted definitions of consistency. Why haven’t you begun to discuss the alternate definitions?

Wrong.

There are systems of reasoning where the version in which GIT applies isn’t used. Feel free to bring them up anytime…

Stupid.

Get better invisible experts[sup]Tm[/sup], you ignorant fuck.

You have no excuse for continuing to post such tripe. It is sheer hubris and intellectual inadequacy. That is not an attractive combination. Then again, you long since demonstrated that you are not interested in any honest examination of what GIT means or implies, you just want to spread your lies, bullshit, and ignorance as widely as possible.

[TVAA]

And you continue to do nothing but post ‘wrong’ after arguments.

What a brilliant strategy that is. :rolleyes:

Once again: the truth of a statement is not dependent on whether it is or can be proven. Whether we recognize a statement is true does depend on proof, but that’s not what we’re discussing here.

That is precisely why mathematicians have essentially given up the concept of “truth” in regards to their arguments. Calling a statement true is still meaningful, and it is absolutely vital in logic (as opposed to mere mathematics).

For any sufficiently powerful system S, the statement “this statement cannot be proven in S” (later referred to as G) cannot be proven or disproven. It is nevertheless true relative to the system, as can be demonstrated by thinking about the properties of the system. The statement “G is true” (referred to as G’) is provable in the meta-system (S’).

That does not mean that the truth of G is only dependent on S’. The truth of G’ is dependent on S’. The truth of G is dependent on S – it’s just not able to be proven in S. It is provable (that it is true relative to S) in S’.

You may complain that this explanation doesn’t use specific terminology, but its meaning is correct. Deal with it.