Using an incomplete Bible to formulate a consistent Gospel of Love

[Liberal]

Tyrrell wrote:

But aren’t you begging the question when you assume that the bible is correct when it asserts its own incompleteness? It may simply be wrong about its own completeness, and therefore (if arithmetic is constructible from the bible) inconsistent.

Well, yes, if I were a literalist, that would be a problem.

I probably should have put a winkie in there somewhere since it is ironic that a literalist would have to accept, from the scripture quoted, that the Bible is incomplete.

However, it is incomplete anyway, and its statement just happens to be true. It is reasonable to assume that Jesus did not transmorgrify from an infant to age 12 and then from that age to adulthood.

The Bible is incomplete by prima facie evidence, and therefore its incompleteness is axiomatic. Offering the scripture is simply for the purpose of amplifying that fact in the minds of those who believe its every word is inerrant.

It is a statement that simply has to be true. The irony is that a literalist must acknowledge that the Bible is incomplete because of John 21:25.

It’s unclear to me what you mean by “recursive axioms”, but predicate logic contains axioms I’d call recursive, like “If A and B are deducible wffs, then A & B is a deducible wff.”

Godel himself explains what he means by recursive axioms:

We now introduce a parenthetic consideration having no immediate connection with the formal system P, and first put forward the following definition: A number-theoretic function f(x1,x2,…xn) is said to be recursively defined by the number-theoretic functions y(x1,x2,…xn-1) and m(x1,x2,…xn+1), if for all x2, … xn, k the following hold:

f(0,x2,…xn) = y(x2,…xn)
f(k+1,x2,…xn) = m(k,f(k,x2,…xn),x2,…xn). (2)

A number-theoretic function f is called recursive, if there exists a finite series of number-theoretic functions f1, f2, … fn which ends in f and has the property that every function fk of the series is either recursively defined by two of the earlier ones, or is derived from any of the earlier ones by substitution, or, finally, is a constant or the successor function x+1. The length of the shortest series of fi, which belongs to a recursive function f, is termed its degree. A relation R(x1,x2,…xn) among natural numbers is called recursive, if there exists a recursive function f(x1,x2,…xn) such that for all x1, x2, … xn

R(x1,x2,…xn) º [f(x1,x2,…xn) = 0].

He then lists a number of examples and corollaries. (I’m not going to insert the symbol tags because it would be too tedious. You can follow my link to the original paper.)

Braithwaite explains in his introduction to Godel’s paper that what Godel called recursive axioms were the initial recursive formulae in a proof-schema, specifically those that hold as he indicated above. The introduction is useful because it is written to help translate Godel’s rather unique vocabulary and syntax into the more ordinary language of philosophical logicians.

Since we’re all being extremely precise at this point, we should also note that it is first-order predicate logic, and not all predicate logic, that is both complete and consistent. (There are no successors, but only inferences.)

We should also note that, even in an inconsistent or incomplete deductive system, not every proposition is unprovable, but only those where…

If c be a given recursive, consistent class of formulae, then the propositional formula which states that c is consistent is not c-provable; in particular, the consistency of P is unprovable in P, it being assumed that P is consistent (if not, of course, every statement is provable).

…which is Proposition XI of the theorem.

[Liberal]

Diogenes wrote:

Isn’t it contradictory to say that Jesus is the source of all authority because the Bible, in its authority, says so? In fact, isn’t this almost a paraphrase of Godel’s fork in and of itself? If the Bible states that it cannot be relied upon as authoritative, then that statement itself cannot be relied on as authoritative. I submit that the quotation from Matthew can be paraphrased as follows:

“Nothing in this book may be relied upon as authoritative”

If this statement can be relied upon, then it can’t be relied upon. Am I crazy or is this a real paradox? (logic makes my head hurt)

The point is that the literalist is dead in the water either way.

If the Bible (formulated as a deductive system with recursive axioms — let’s just assume this from now one) is complete, then provable propositions may be formulated only from inference by cherry picking the propositions that are not formed by successor arithmetic. Godel wrote: “We take it to be recognized that the functions x+y (addition) and x.y (multiplication) are recursive.”

On the other hand, if it is consistent, then provable propositions may be formulated by both inference and succession.

So, if the Bible is complete then Jesus’ authority may be taken as axiomatic from the statement itself. But if it is consistent, then Jesus’ authority may be inferred by deductive successions (or syllogisms) like this: God is the authority; Jesus is God; therefore, Jesus is the authority.

[Apos]

—I probably should have put a winkie in there somewhere since it is ironic that a literalist would have to accept, from the scripture quoted, that the Bible is incomplete.—

Can you elaborate on how that passage in John suggests to you that the Bible is incomplete in the sense that is relevant to Godel?

That is, if you describe a deductive system in the Bible, should the fact that it (perhaps being a collection that both describes that system AND also includes some other litterally true events) doesn’t describe everything Jesus did mean that the deductive system found in the Bible is incomplete?

Couldn’t the story be incomplete (in that it doesn’t describe everything that it could) while the deductive system described therein is complete? A book on Godel might, for instance, discuss an example of a complete (but inconsisent) system, but not fully describe what Godel had for breakfast on his eighteenth birthday.

Or are you saying that the Bible, if litterally true, consitutes a description of a deductive system in its entirety, every word in it necessary for the system? I’m not following the argument here, perhaps because I don’t have a strong background in how people go about translating the Bible into a deductive system.

[Tyrrell_McAllister]

Originally posted by Libertarian
I probably should have put a winkie in there somewhere since it is ironic that a literalist would have to accept, from the scripture quoted, that the Bible is incomplete.

However, it is incomplete anyway, and its statement just happens to be true. It is reasonable to assume that Jesus did not transmorgrify from an infant to age 12 and then from that age to adulthood.

But the bible is complete if it is inconsistent. As you mentioned, inconsistency implies completeness. As you probably know, this is a trivial result, and doesn’t require Gödel’s proof to see. An axiom system that entails a contradiction can be used to prove any wff whatsoever[sup]*[/sup]. So, if the bible is an axiom system, and if it’s inconsistent, that is, if it contains even one contradiction anywhere, then it can be used to deduce both that Jesus did transmorgrify from an infant to age 12, and that he did not.

But if the bible is consistent, then, I agree, it is clearly incomplete. I’m surprised that anyone would seriously claim otherwise. Does anyone really believe, e.g., that it can be deduced from scripture what color shirt Jesus was wearing on his 21st birthday?

Braithwaite explains in his introduction to Godel’s paper that what Godel called recursive axioms were the initial recursive formulae in a proof-schema…

I searched for the phrase “recursive axiom” in both the papers of Gödel and Braithwaite, and was unable to find it. I’m wondering if you are confusing the notion of “recursive functions” and “recursively enumerable axioms”. Recursive functions are just a particular kind of function, but there is no such thing as a recursively enumerable axiom. “Recursively enumerable” does not descibe a particular kind of axiom. Rather it describes, roughly speaking, how many axioms there are, and what kind of order they’re presented in.

Since we’re all being extremely precise at this point, we should also note that it is first-order predicate logic, and not all predicate logic, that is both complete and consistent.

An important clarification. Thank you.

We should also note that, even in an inconsistent or incomplete deductive system, not every proposition is unprovable…which is Proposition XI of the theorem.

True, but, again, we don’t need Gödel, or anything so sophisticated, to see this. If the system is inconsistent, then every proposition is provable (see below); and even if it is incomplete, at least one proposition is provable (e.g. the conjugation of one of the axioms with itself)

---

[sub][sup]*[/sup] Proof: Suppose that both P and ~P are deducable. Then, using conjunctive addition, we can deduce P & ~P. Let an arbitrary wff Q be given. We prove Q.

Since we have P & ~P, it follows, using disjunctive addition, that we may deduce (P & ~P) v Q. Whence, by the definition of implication, ~(P & ~P) ==> Q. De Morgan’s and Double Negation then yield (~P v P) ==> Q. Now, by the Law of the Excluded Middle, we have ~P v P. Therefore, Q follows by Modus Ponens.[/sub]

[ultrafilter]

Just FYI, a consistent system is not one in which no contradiction is provable. It’s one in which there is some sentence which is not a theorem of the system. Maybe only one. Anyway, there’s no more general relationship between consistency and completeness that I know of other than that laid out by Gödel.

“recursive axiom” here is a bit of a misnomer. If a theory K has a recursive axiom set, there is some recursive function f(x) such that f(x) = 0 if x is the Gödel number of an axiom of K, and 1 otherwise. Clearly, every system with a finite axiom set has a recursive axiom set.

Additionally, the system with no axioms and no rules of inference has no theorems. It is consistent, but I’m not sure as to its completeness.

[ultrafilter]

It’s not complete. I quote from Mendelson: “A theory K is said to be complete if for every closed well-formed formula B of K, either there is a proof in K of B, or there is a proof in K of ~B.”

So the empty system is consistent and incomplete. But–here’s the interesting part–it does not meet the conditions for Gödel’s theorem, and so we have to resort to other methods for the proof.

[Tyrrell_McAllister]

Just FYI, a consistent system is not one in which no contradiction is provable. It’s one in which there is some sentence which is not a theorem of the system.

Either condition works as a definition: they are trivially equivalent. In my previous post I showed that given an axiom system S,
[ul]Some contradiction is a theorem of S[/ul]implies
[ul]Every sentence is a theorem of S.[/ul]Proving the converse is even simpler. If every sentence is provable is S, then, in particular, some sentence of the form P & ~P is provable in S.

[Liberal]

Apos wrote:

Can you elaborate on how that passage in John suggests to you that the Bible is incomplete in the sense that is relevant to Godel?

That’s an important question.

The point of the New Testament was not to provide an exhaustive history or documentary, but a morality. Your asking the question provides me the opportunity to clarify that. What is important is not deducing what sort of sandals Jesus wore, but rather deducing what sort of message Jesus brought.

What is written in John generally, as in all the Gospels, is an account of the teachings of Jesus — what He said and what He did as they pertain to the Good News (the Gospel).

If He said and did many other things that could not be contained in the book, then we are missing certain premises, inferences, or conclusions with respect to His moral teachings. And these are what matter.

My assertion is that enough is available to round out the teachings into a deductive system. To do this, some new inferences must be drawn and some old ones must be discarded or reworded.

The heart of the effort is finding the essential axiom. What is the central thrust of His message upon which all else is built? I assert that the essential axiom is that God is Love.

[Liberal]

Tyrrell wrote:

But the bible is complete if it is inconsistent. As you mentioned, inconsistency implies completeness. As you probably know, this is a trivial result, and doesn’t require Gödel’s proof to see. An axiom system that entails a contradiction can be used to prove any wff whatsoever.

Sure. A contradiction proves everything, and everything proves a tautology. But here, I’m talking about a more general relation.

The point of raising Godel’s undecidable propositions was to point out to literalists that if they insist that the Bible is complete, then it cannot be used to prove its own consistency. And if it is inconsistent, then we must discard the inferences we find within it that are contradictions.

But if the bible is consistent, then, I agree, it is clearly incomplete. I’m surprised that anyone would seriously claim otherwise. Does anyone really believe, e.g., that it can be deduced from scripture what color shirt Jesus was wearing on his 21st birthday?

What is of concern is morality.

I searched for the phrase “recursive axiom” in both the papers of Gödel and Braithwaite, and was unable to find it.

You have to do basically what I’m trying to do here with the literalists and put two and two together.

Search for “what he calls axioms”, and you will find that that is what he calls the initial formulae. He uses the term formula to mean WiFF. And he holds that any formula with succession is recursive. As I explained before, Godel’s syntax is unorthodox, and Braithwaite’s purpose is to help contextual his paper with more ordinary language. You can’t successfully approach his paper by applying common interpretations of his vocabulary.

[ultrafilter]

*Originally posted by Tyrrell McAllister *
**Either condition works as a definition: they are trivially equivalent. In my previous post I showed that given an axiom system S,
[ul]Some contradiction is a theorem of S[/ul]implies
[ul]Every sentence is a theorem of S.[/ul]Proving the converse is even simpler. If every sentence is provable is S, then, in particular, some sentence of the form P & ~P is provable in S. **

Only given the current rules of inference. In some systems, a contradiction doesn’t imply everything. For instance, suppose that our axiom is A, and our rule of inference is that B -> B & ~B. You can derive a contradiction, but you can’t conclude A v ~A.

Anyway, that’s more or less tangential. IIRC, Gödel’s theorem will work for systems with rules of inference other than the ones we use.

[Liberal]

Well, since this has gone the way of a Nicene discussion of logical structures I am curious as to why Godel never produced his promised follow-up in which he was to show the proof that Prop XI applies to non-Peano arithmetic. I wonder whether he just didn’t find the time, thought better of it, or what.

[ultrafilter]

I’m not sure that anyone knows. I think he just got involved in other directions.

As far as whether the research was done…you’d be best off looking at Chaitin and those who have followed him.

[Apos]

—The point of the New Testament was not to provide an exhaustive history or documentary, but a morality.—

That’s one take, but is it be the only one literalists would accept?

—If He said and did many other things that could not be contained in the book, then we are missing certain premises, inferences, or conclusions with respect to His moral teachings. And these are what matter.—

Is that certain though? I think it could still be said that we may be missing certain premises, inferences, and conclusions. But for all we know, that passage could simply mean that Jesus cured more ailments, fed more people, consorted more with the unclean: in which case they don’t necessarily involve extra clues that we are missing to his Gospel, simply illustrations of the same principles already revealed. That would be akin to restating a deductive system in different words, not revealing new aspects of it unavailable from the initial description. That seems to be one possible interpretation.

To put things in a much more general sense, a literalist could respond that the Bible is 100% true, AND contains a complete (not Godel meaning) account of all you need to know about Jesus’ moral teachings in the same way that a text on logic can completely describe a logical system. The system that is described may itself be incomplete (in the Godel sense) but that’s not a problem with the Bible, it’s a problem with the system (and any system that wishes to be both complete and consistent).

I’m not sure a literalist would argue that Jesus’ moral system has to be logically complete: if anything it seems that they might just get confused by your meaning and think you are saying that the Bible doesn’t contain a full account of the system (which is not the same thing as discussing the inherent completeness vs. consistency of the system itself). For them, it might be one thing to speak of the Bible, and another to speak of a system described within it.

The literalist position I’m thinking of would be that the teachings found in the Bible are all consistent, that the Biblical account is a full description of the system, and maybe concede that the system itself is incomplete in the Godel sense. This would seem to do away with what you described as the irony of their position, revealing that you and they meant different things when discussing the Bible as being “incomplete.”

More importantly, I still think it key that the Bible as a whole is not simply just an account of the moral system. If you, outside the Bible, can look at the moral system and make comments that you perhaps think are outside of the system that help prove things that one system cannot do for itself internally, then why couldn’t the words of someone making similar comments simply be included in the Bible? Jesus himself could have both described a moral system as well as discussed it, because Jesus is not a system, but a thinking being. To wit:

—My assertion is that enough is available to round out the teachings into a deductive system. To do this, some new inferences must be drawn and some old ones must be discarded or reworded.—

But why couldn’t this process have already been done in the text?

For instance, if, prior to the Bible being compiled, you made these comments (or Jesus made them) about the moral system, and then they were compiled into the Bible, wouldn’t that have removed the problem? What if that was exactly what did happen, and the text contains both the moral system and the discussion of the necessary insights external to it regarding the essential axiom you describe?

[Liberal]

Apos wrote:

That’s one take, but is it be the only one literalists would accept?

Who can say? So far, one has not come in.

To put things in a much more general sense, a literalist could respond that the Bible is 100% true, AND contains a complete (not Godel meaning) account of all you need to know about Jesus’ moral teachings in the same way that a text on logic can completely describe a logical system.

Well, then. They’d be in a worse pickle yet. Jesus never directly said that He was God, although He said and did things that strongly imply that He is God. There’s also that Trinity thing. No mention of it either. If they are to take inferences about His divinity and triune nature, they will have to form conclusions that are not found in scripture.

I’m not sure a literalist would argue that Jesus’ moral system has to be logically complete: if anything it seems that they might just get confused by your meaning and think you are saying that the Bible doesn’t contain a full account of the system (which is not the same thing as discussing the inherent completeness vs. consistency of the system itself). For them, it might be one thing to speak of the Bible, and another to speak of a system described within it.

Then they would agree with me, and we can begin formulating a consistent Gospel of Love.

The literalist position I’m thinking of would be that the teachings found in the Bible are all consistent, that the Biblical account is a full description of the system, and maybe concede that the system itself is incomplete in the Godel sense. This would seem to do away with what you described as the irony of their position, revealing that you and they meant different things when discussing the Bible as being “incomplete.”

But if it is incomplete, then they must draw new inferences, which they don’t like doing typically, unless someone else has already done it for them. Like Nicene priests, for example.

More importantly, I still think it key that the Bible as a whole is not simply just an account of the moral system. If you, outside the Bible, can look at the moral system and make comments that you perhaps think are outside of the system that help prove things that one system cannot do for itself internally, then why couldn’t the words of someone making similar comments simply be included in the Bible? Jesus himself could have both described a moral system as well as discussed it, because Jesus is not a system, but a thinking being.

Well, He discussed it quite a lot, and He lived it quite remarkably. All of His acts of healing, miracles, and such were themselves moral parables that He followed up by discussion or making some point or other.

But why couldn’t this process have already been done in the text?

Oh, I think that it has (if I understand the sense you mean).

For instance, if, prior to the Bible being compiled, you made these comments (or Jesus made them) about the moral system, and then they were compiled into the Bible, wouldn’t that have removed the problem? What if that was exactly what did happen, and the text contains both the moral system and the discussion of the necessary insights external to it regarding the essential axiom you describe?

I think that it has everything needed to formulate a consistent Gospel of Love. But what the literalists will not like is that some of it must be discarded because it contradicts the aforementioned formulation.

[Apos]

—If they are to take inferences about His divinity and triune nature, they will have to form conclusions that are not found in scripture.—

Well, of course they disagree with you on this (whether such things can be found, or inferences drawn), which is probably outside the realm of this debate.

—Then they would agree with me, and we can begin formulating a consistent Gospel of Love.—
—Oh, I think that it has (if I understand the sense you mean).—

Then what’s your problem with their position then? They might think that Jesus and other parts of the text make clear what is the proper system, you disagree (or agree, but disagree about what?).

—But what the literalists will not like is that some of it must be discarded because it contradicts the aforementioned formulation.—

Right, but not because of problems with all the aspects of the moral system described being inconsistent: because while the particular system described isn’t logically complete (it can’t be without being inconsistent), they might think other parts of the text already establish the needed external axioms, and they may or may not disagree with you on what those are (and whether the text describes them).

What about those who don’t think that moral teachings of Jesus are necessarily a logically derived moral system per se in the sense that it has anything to do with Godel, but simply an expression of God’s values and the demand that they be embraced?

[Triskadecamus]

And the love God has for you is a consequence of His nature, not yours.

OK, I know it has nothing to do with logic, I just wanted to bring it up again.

Tris

[Liberal]

Apos wrote:

Well, of course they disagree with you on this (whether such things can be found, or inferences drawn), which is probably outside the realm of this debate.

No, they wouldn’t disagree. Try as they might, they cannot find a Strongs number for “trinity” nor a statement by Jesus saying, “I am God”. They must draw inferences that Jesus is divine and triune and they do.

Then what’s your problem with their position then? They might think that Jesus and other parts of the text make clear what is the proper system, you disagree (or agree, but disagree about what?).

I thought I made plain in the OP [checking… yes, I did] that all sorts of consistent Gospels can be drawn from scripture — a Gospel of Hate, a Gospel of Law, a Gospel of Love…

Here, I am interested in a Gospel of Love because I believe that it derives naturally from the most essential premise (which I’ve already given you).

The problem (again) is that the literalist must discard scriptures that would support a Gospel of Hate or a Gospel of Law in order to formulate a Gospel of Love, and that is something that, for obvious reasons, literalists don’t like doing.

Right, but not because of problems with all the aspects of the moral system described being inconsistent: because while the particular system described isn’t logically complete (it can’t be without being inconsistent), they might think other parts of the text already establish the needed external axioms, and they may or may not disagree with you on what those are (and whether the text describes them).

I’m not sure you get the point. Let me give you an example.

Jesus taught that mere faith in (allegiance to) Him is a guarantee of eternal life in His kingdom. Paul, however, said that faith without commensurate works is dead. (As a matter of fact, Paul contradicted himself on that matter, but let’s leave that aside.)

Now, you might seek to rationalize the two as compatible, and in fact, the literalists do exactly that. They have to because they believe that the Bible is complete and consistent.

What about those who don’t think that moral teachings of Jesus are necessarily a logically derived moral system per se in the sense that it has anything to do with Godel, but simply an expression of God’s values and the demand that they be embraced?

Well, I would imagine they would argue with me here.

[Liberal]

*Originally posted by Triskadecamus *
**And the love God has for you is a consequence of His nature, not yours.

OK, I know it has nothing to do with logic, I just wanted to bring it up again.

Tris **

What’s not logical about that? If God is Love, then it follows that loving you is in His nature.

[lekatt]

Ho, Libertarian, would help you if I could.

I understood perfectly what you said and what the intention was, it is a beautiful piece of writing about God’s love. I believe you covered all the bases very nicely.

Don’t even know who Godel was, or why what he said is important, just ignorant, I guess. The statements you made were consistent to me.

Thomas Jefferson made up his own Bible, it was very small. “Only the things that will hold up in a court of law,” Jefferson said, “are in it.”

In my opinion, the old testament, and the writings of Paul that are contradictory to the teachings of Jesus should be removed. Not much would be left, but a lot less controversary.

Since none of this is likely to happen, some more progressive churches are teaching from other books about love. Unity Churches will teach from any book that promotes kindness, compassion, and love. They also use “The Course In Miracles.” This book is taught in the most churches that teach things other than the Bible. TCIM has been translated into all the major languages of the world, and has sold millions of copies. The author of the book is Jesus.

Love
Leroy

[Apos]

—Now, you might seek to rationalize the two as compatible, and in fact, the literalists do exactly that. They have to because they believe that the Bible is complete and consistent.—

But again, they don’t mean by complete that every word in the Bible is the explicit logical expression of key moral system, so I don’t see that they are facing the problem you seem to think they are.
Some of it could be commentary on the system, some of it is history that sheds light on man’s consistent unworthiness, and so on. The moral system (in their mind) could be logically consistent. It could be incomplete as a matter of logic, and I don’t see that they would care considering that they think the rest of the text expresses. If all you are asserting is that there are contradictions that need to be resolved to form a consistent Gospel, then of course they would argue otherwise (and I’m certainly on your side in thinking that the theology does not seem consistent).

Let me be clear: I’m not attacking the idea of a Gospel of Love, I think it’s an inspiring project for someone interested in the Bible. But I just don’t know that literalists would have to concede what you seem to think they must.