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Gödel's Incompleteness Theorem 


The Name of God




I cannot pretend to do an exhaustive treatment of this material here.  

In my mind's eye this is about a 400 page book, to do it justice.

So, I can either wait until that mega-project is complete, which will be years (if ever), or I can point you in a direction, and see how it goes.

These, then, are basically my notes.  

    This writing remains incomplete, and reflects the latest in my ongoing search for the Infinite.

        That's a very important clue.

You can do the research for any terms you might not understand, or ignore the whole thing.

    I'm still writing less. 

        But if you are going to get anything at all out of this, you are going to have to think a good deal more.



In 1931, the Czech-born mathematician Kurt Gödel developed one of the most important mathematical proofs of the last 100 years.  It has echoed throughout many fields, from math to science to philosophy, computer design, artificial intelligence, even linguistics and psychology.

As is my eternal blessing & curse, I see it spiritually.



There's a Psalm (look it up yourself) that says the Earth shows Yhwh's handiwork.  

Just like all art says something about the artist, so too does everything in the manifest universe say something about its Creator.

For years I've been able to pick-up a book on mathematics and read it like it is scripture.  

    I wish all humans had this ability.  

        I wish we lived in a world that did more to encourage such things.

But I digress.



This is very important.  

It is built into the core of our religious life.  

"As above, so below."  Human morals are based on eternal truths that reside "in Heaven", beyond the physical world.  

"As below, so above."  

What we do here matters.  

What we see here does, indeed, reflect the higher, eternal world beyond.

Otherwise, parables would be meaningless.



What Gödel proved is that no system is complete.  Ever.

More specifically, Any system that is sufficiently powerful to be able to state true propositions must be able to create statements that can neither be proved nor disproved within that system.

Or, yet a bit more mathematical, All consistent axiomatic formulations of number theory include undecidable propositions.

And making a bigger, more powerful system does not help.  Because that bigger, more powerful system will also contain unprovable statements.

On and on.  Forever.



This is not to say that these "undecidable propositions" are untrue.  They may, in fact, BE true.

But we can't prove that.

Leading to the blessed observation that 

    Truth is more important than Proof.



But don't be deceived.

    Gödel's Theorem has been PROVED.  

        Absolutely, guaranteed.



We have a mathematical proof that 

    truth is more important that proof.

We have a mathematical proof that 

    some truths cannot be proven.



Can God make a square circle?

    Think about this.

        It is very important.



We know that a circle is the collection of all points within a plane equidistant from a center.

We know a square also lies within a plane, and is the enclosed space formed by 4 equal length line segments, joined at 90 degree angles.

We know that it is physically, utterly, theoretically and eternally impossible for a square to be a circle.  We have proven this absolutely.



So, again, could God make a square circle?

If "Yes", then we lose all connection between ourselves and the Divine.  

    Logic and truth are arbitrary, and ultimately meaningless.  

        We cannot be made in Yhwh's Image, but are simply little ants for Him to use & abuse however he sees fit.

            Nothing we can ever learn or know can possibly have any lasting value.

The entire universe holds together only by The Word of God.  Who is Truth.

Miracles are only the application of higher truths than we are currently aware of; they can never be violations.


No.  Yhwh cannot make a square circle.

Which means that Gödel's Incompleteness Theorem says something real.

Something Eternal.

About God.



Now about this odd notion of "Proof."

What, exactly, is it?

It is a faith in logic, that things connect and make sense.

We take a small collection of ideas, called axioms, that we consider to be essentially self-evident.

"If 2 lines are everywhere an equal distance apart, then they never meet."  That kind of thing.

So we take the axioms and combine them with logic, building on them.  If "A" is true and "B" is true, then we can logically conclude that "C" is true.  In other words, on the strength of "A" and "B" and logic, we can now prove "C".

"All living people breathe."  (Axiom)

"Paul is a living person."  (Observation)

"Paul is breathing." (Proof, required logically from the observation & the axiom.)



Let us assume, then, that Yhwh's universe is the most complete logical system.

    Must it contain unprovable propositions?

        If so, then there are things God cannot prove.



If Yhwh's Mind contains all logic and thought,

    Must He know unprovable truths?

        His Mind is insufficiently powerful to "prove" Himself.



If human logic says something real about Yhwh's Mind,

    Gödel's Incompleteness Theorem proves that there are things Yhwh cannot prove.

Which means we need to reevaluate our childish notions of "Omniscience."



Is the Universe an Unprovable Proposition?

    Are you?



Let the initials "GS" stand for a Gödel Statement, one that cannot be proved or disproved within a particular system.

Let's say that "God's Mind contains unprovable propositions" cannot be proven or disproven from our perspective.  It is GS#1.

Is GS#1 a GS from Yhwh's Perspective?

If the universe is whole, (we are, after all, told that we have the Mind of Christ), then Gödel's Theorem absolutely proves that GS#1 is a GS from Yhwh's Perspective.


Before the Beginning, God said, "There are things I know, that I cannot prove.  And I can prove this."

    God, too, stares at the Naked Reality of Existence Itself, knowing it to be True, but unable to prove it.

        Yet another thing the Children have in common with their Father.



Perhaps, then, the Tree of Life is an ever-expanding, ever-opening flower, forever growing in depth, breadth and complexity, wherein Yhwh is eternally crafting greater structures by which to decide all undecidable propositions.



It has been said of old

    That the very first moment

        of "existence before there was anything else"

            was the "moment" when God first contemplated His Own Eternal Existence.

Staring at the Face of His own Raw Existence.

Truth, beyond proof.

Eternally growing, on a quest to prove the utterly unprovable,

    "I Will Be what I Will to Be."



One of the most amazing axioms in all of mathematics is The Axiom of Choice.

It basically says that many mathematical propositions can only be proved through making choices.

This is real.

Think about it.

A mathematical axiom, central to our human logic, establishing the primacy of Choice.

If you don't believe me, here's The Axiom of Choice's home page.  Sponsored by Vanderbilt University.  http://math.vanderbilt.edu/~schectex/ccc/choice.html

Choice is Will.

Will is Choice.

I Choose to Be what I Choose to Be.



A mathematical axiom establishing the fundamental nature of Choice.  Also known as Will.

And a proof that proves that proof is a limited, endless spiral.

All of this is a part of your heritage.

Contained, hidden, mystically, within the seeds of The Burning Bush: "I Will Be what I Will Be."

Perhaps meant as a clue, like Clark's Monolith in "2001", to inspire us and point us on our way.

I'm utterly convinced that we'd be a lot farther along if we hadn't taken Yhwh's Name to be vain.



The Incompleteness Theorem,

    The Eternal Quest for the Unprovable,

        By the Infinite "I".

The Axiom of Choice,

    The Primacy of Will, Forever growing and expanding.


"I" + "WILL" = All Existence.

Which is The Name of God.


                                            -- Ahyh, July 2001


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