# Gödel’s Incompleteness Theorem: The Universe, Mathematics and God

80 years ago, Kurt Gödel toppled empires of mathematical philosophy with his famous Incompleteness Theorems.

I’m every bit as interested in science, philosophy and engineering as I am in business. Gödel’s theorem has profound implications for every branch of knowledge.

The materialist view prevails in secular circles. Materialism states that the laws of physics and the universe we know are all that is. It sees the universe as a giant machine. It assumes that everything we experience is purely the result of blind cause and effect. It scoffs at the idea that there is any such thing as God or metaphysics.

This view was epitomized by “Logical Positivism” which was espoused by a group known as “The Vienna Circle” in Austria, led by Ludwig Wittgenstein. Logical Positivism says that anything that cannot be experimentally verified or mathematically proven is invalid.

The Logical Positivists were confident that very soon, all the loose ends of mathematics would be nailed down by a single unifying theory. The world would finally fully embrace reason and logic and leave the failures of religion behind.

Kurt Gödel was a member of the Vienna Circle and in 1931 proved that a single unifying theory was impossible. He proved that the goal of the Logical Positivists was unachievable. This was a devastating blow.

Godel’s Incompleteness Theorem says that any system that is complex enough to express mathematics cannot prove, by itself, that everything it says is true. It will always rely on something outside the system that you have to assume is true but cannot prove.

You can then step outside the system and complete your proof, but in order to do that you will now have to invoke something else from the outside. So you keep expanding ever outward, invoking still more things that you cannot prove.

This was very disturbing to mathematicians, because mathematicians hate uncertainty.

Many people have raised the question of whether Gödel’s incompleteness theorem applies to the universe itself. If the universe is mathematical, then yes in fact it does.

Stated in Formal Language:

**Gödel’s theorem says:** “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”

**The Church-Turing thesis says **that a physical system can express elementary arithmetic just as a human can, and that the arithmetic of a Turing Machine (computer) is not provable within the system and is likewise subject to incompleteness.

**Any physical system subjected to measurement is capable of expressing elementary arithmetic. **(In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.)

**Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.**

Syllogism:

**1. All non-trivial computational systems are incomplete**

**2. The universe is a non-trivial computational system**

**3. Therefore the universe is incomplete**

Some time ago I posted an article about this: http://www.perrymarshall.com/articles/religion/godels-incompleteness-theorem/ and I was greatly interested in seeing if anyone would be able to poke a hole in my argument. (The article is a much more thorough explanation of Gödel than I am giving you here.)

Nearly everyone agrees that math is incomplete. The idea that the universe is also incomplete apparently makes some people very uncomfortable. If the universe cannot explain itself then there has to be some kind of higher power at work.

The debate essentially comes down to this:

- If the universe is illogical and inconsistent then it is possible for it to be complete.
- If the universe is logical and consistent then it is incomplete.
- If the universe is incomplete, then it depends on something on the outside.

In other words, if the laws of mathematics and logic apply to the universe, then the universe has to have a metaphysical source. Atheism can only be true if the universe is irrational.

(By the way, my experience from conversing with literally thousands of atheists via email and on my various blogs is this: When you get down to the core emotional center of *why* they don’t believe in God, it’s often because they feel deep down that the universe *is *irrational. They’re immensely disappointed that the world is full of evil and suffering. Because of this, they reject the idea of God.)

**You cannot prove that the universe is mathematical. But belief that the universe is mathematical is the #1 assumption of modern science. **Toss out that assumption and the whole philosophical framework of western civilization crumbles.

In the history of science, you will find that belief in a God who created an orderly mathematical universe was one of the foundations of scientific discovery.

If you visit the world’s largest atheist website, Infidels, on the home page you will find the following statement:

*“Naturalism is the hypothesis that the natural world is a closed system, which means that nothing that is not part of the natural world affects it.”*

If you know Gödel’s theorem, you know that all logical systems must rely on something outside the system. So according to Gödel’s Incompleteness theorem, the Infidels cannot be correct. If the universe is logical, it has an outside cause.

**Thus atheism violates the laws of reason and logic.**

The Incompleteness of the universe isn’t formal proof that God exists. But… it IS proof that in order to construct a rational, scientific model of the universe, belief in God is not just 100% logical… it’s necessary.

Practically speaking, all knowledge we have about anything is incomplete. There are always some things you’re certain of, some things you’re somewhat sure of, and some things you cannot prove at all. Human knowledge is always enlarging the circle of what is known, but every question that we answer just provokes more questions. In real human experience, the quest to enlarge that circle never stops.

And I would submit to you that this is the essence of faith, as actually practiced by thinking, reasoning people.

Many people assume that religious faith is some mystical imaginary idea that is embraced purely on the basis of emotion or intuition. That it has nothing to do with facts, reason or logic.

This is completely untrue – at least in in the Judeo-Christian tradition. No one is asking you to believe without evidence or rational reason. Belief in God, in Jesus, and even the afterlife is based on historical events, logical propositions, and reasonable arguments.

Science itself originated from theology. Science assumed then, and assumes now, that the universe is rational. That the universe operates according to fixed, discoverable laws. Even science itself is a very practical outworking of *faith* in the reliability and consistency of the natural order.

The practice of faith is in many ways living out a hypothesis: That if you follow the teachings and embrace the Spirit, you will have an excellent opportunity to experience success in your work and your family. And that you will be rewarded in your search for meaning and pursuit of the deepest questions.

Perry Marshall

P.S.: If this intrigues you, make sure you read my more extended article, **Gödel’s Incompleteness Theorem: The #1 Mathematical Discovery of the 20th Century**

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Your argument sounds reasonable. However, it is internally flawed because once you introduce a deity that can somehow know all (omniscient as per your definition/belief), the theorem then must be applied to this God as well. If you leave God out of the bounds of Godel’s theorem, then you do not accept the theorem. According to Godel’s Incompleteness Theorem, God can never fully comprehend himself, thus rendering the idea of an omnipotent, omniscient being scientifically impossible.

Essentially, there are only a few options. 1, that the universe contains everything (is infinite), in which case this argument is debatable at best; 2, that the universe is entirely finite, in which case what is it expanding into, what is outside of it and why does it not have bounds? All are important questions with multiple hypotheses. The problem you face there, is that no matter what is outside the universe, there is no reason Godel’s theorem would not apply. Infinity is a concept, but even if “God” is infinite, he cannot understand himself as things cannot explain themselves or their existence (as per Godel).

So either the God that is outside of the universe is not omniscient (and by implication omnipotent), or the God is not outside the universe at all, or there is something outside of God (other universes, multiverse theory etc). In any case, the argument you’re putting forth doesn’t really accomplish much. It IS an interesting logical experiment, though.

Of course, you did kind of gloss over the fact that mathematicians believed (some still do) for bulk of time since Godel proved his theorems that they were useless logical tricks and not mathematical ones at all. Only in 1977 did mathematicians even find a single “interesting” statement inexplicable in PA which brings the grand total up to two, I believe.

Many mathematicians still feel the theorems are useless in that they only raise questions in terms of themselves (the questions would never arise naturally as most important mathematical questions do and as in pure math).

God cannot comprehend Himself if God is complex. God can comprehend Himself if God is simple and indivisible.

Christian theology teaches that God exists in three expressions: Father (essence), Son (expression = word), Holy Spirit (understanding). Hebrews 1:3, speaking of Jesus: “And He is the radiance of His glory and the exact representation of His nature.” If you examine the logic of what you just said, it becomes evident why self-understanding and self-expression are necessary for God to be logical.

Christian theology also understands that there is no division or contradiction between these three expressions of God. God is community. This is why God IS love. Love is intrinsic to God, and not just something God is capable of.

Love can only exist when there is a plurality. Unity can only exist when there is agreement. The trinity is a complete unity within plurality of expression.

If God is love and love is perfect agreement between a plurality of expressions, then a perfect love is consistent with total self understanding.

You are right, no matter what is outside the universe, there is no reason Godel’s theorem would not apply.

Your statement “either the God that is outside of the universe is not omniscient (and by implication omnipotent)” is a non-sequitur, based on what I just said above. God is infinite and indivisible and therefore omniscient.

A fundamental flaw in your argument, which is insurmountable as far as I can see, is that Godel’s incompleteness theorems only apply to what are known as ‘computably enumerable’ axiom systems i.e. those systems whose axioms can in principle be listed out by a computer (even if this would take an infinitely long time). This is what the phrase ‘effectively generated’ means in your statement of Godel’s theorem.

Even if we accept that the universe is an axiomatic mathematical system, there is no reason whatsoever why the universe’s axioms should be ‘computably enumerable’. It is easy to contruct extensions of Peano arithmetic (one example is called ‘true arithmetic’) whose axioms are not computably enumerable, and Godel’s incompleteness theorems simply do not apply to axiom systems like these.

Before you can apply Godel’s theorems to the universe (assuming that the universe is an axiomatic mathematical system), you first have to show that the universe’s axioms are computably enumerable. There is no reason whatsoever why they should be, and therefore there is no reason whatsoever to presume that Godel’s theorems apply to the universe.

I cannot prove that the universe is computationally enumerable, I can only point out that the entire enteprise of science and mathematics implicitly presumes that it is. See the Church-Turing Thesis above. If you wish to reject science and math instead of accepting God as axiomatic, you are free to do so.

Your statement that science and math implicitly presume computable enumerability of the universe is completely false. It is not necessary for the universe to be computably enumerable in order to do science and math. To give a simple example, the set of all real numbers is an uncountably infinite set and therefore not computably enumerable. Despite this, we use real numbers constantly in science, math and everyday life. There is no need whatsoever for the universe to be computably enumerable in order to be able to do science and math in it. Science and math do not at all presume that the universe is computably enumerable.

You also seem to have a misunderstanding of the Church-Turing Thesis. The Church-Turing Thesis is a statement about the algorithmic computability of functions (it is an unproved hypothesis that all such functions are recursive). There are non-recursive functions (one famous one is called the Halting Problem) which are not algorithmically computable. Again, this has nothing whatsoever to do with the computable enumerability of the universe.

In order to apply Godel’s theorems to an axiomatic mathematical system you must first show that the axioms of that system are computably enumerable. As I said in my previous message and repeated above, there is absolutely no reason to think this is true for the universe (even if we accept that the universe is an axiomatic mathematical system – if you could prove that you would win the Nobel Prize in physics and you’d be more famous than Einstein). We could still do science and math perfectly well even if the axioms of the universe were not computably enumerable.

I am afraid your whole case is based on assumptions which are just as huge as the assumption “god exists”. You are essentially trying to prove one huge assumption (“god exists”) by implicitly making other huge assumptions (“the universe is an axiomatic mathematical system with computably enumerable axioms”) which are just as huge. There is no reason whatsoever why the latter should be true.

Your whole case is fundamentally flawed I’m afraid.

There is no infinite set of real numbers in the universe, because the universe is finite. To the best of our actual knowledge,

everythingin the universe is finite. To the best of our knowledge, everything in the universe is also countable, measurable, weighable, quantifiable. Science always assumes this to be true.A possible exception to this might be consciousness, in which case consciousness has a metaphysical component. Which brings us to the age-old “mind/body problem” and the question of free will, which I think we can set aside for the moment.

Outside of that, everything in the hard sciences is quantifiable and computable.

I reiterate that science and applied mathematics do presume computable enumerability of the universe. It’s not provable, though I certainly would love the fame and fortune that would come with proving it.

If you assume that it is not enumerable then there is nowhere left for science to go.

I’ll try again, being fully explicit this time, because with all due respect I think you have some confusions which prevent you from grasping what I said earlier. I will leave no room for doubt now (I hope). For Godel’s theorems to be applicable to the universe you have to assume two main things, which I will show need not be true at all:

(a). That the universe is in reality an axiomatic mathematical system i.e. that everything in the universe we perceive as being real can be deduced mathematically from a set of axioms.

There is no reason whatsoever why this should be true. In science and applied math what we are doing is using OUR OWN axiomatic mathematical systems to approximate certain features of the universe (e.g. by means of differential equations). Just because we are able to use our own axiomatic systems to do this, it does not mean that the whole universe itself must therefore be an axiomatic system! The latter does not follow logically from the former. It is a logical non sequitur.

So the first insurmountable weakness of your case is that there is no need whatsoever for the universe as a whole to be an axiomatic mathematical system. Science and math do NOT presume this. In doing science and math we are just using OUR OWN axiomatic systems to approximate certain aspects of the universe’s behaviour. That does not require the universe itself to be an axiomatic system.

And note again that the Church-Turing Thesis has NOTHING to do with this. All the Church-Turing thesis says is that algorithmically computable functions are recursive. There are many non-recursive functions which are not algorithmically computable. This has nothing whatsoever to do with the question of whether the universe itself is an axiomatic mathematical system. The two things are completely unrelated.

(b). Even if (a) were true (which I have just shown you need not be the case at all), you also have to assume that the axioms of the universe form a computably enumerable set in order to apply Godel’s theorems. What this means is that, in principle, a computer could list out all the universe’s axioms fully, even if this required the computer to carry on listing forever. In principle, even if it took forever, the computer could list out all the universe’s axioms. This is what it means to be computably enumerable, and this is what is required to apply Godel’s theorems.

Again, there is absolutely no need for this to be true for the axioms of the universe. There are many simple things that cannot be listed out fully by a computer even if it carried on forever e.g. the set of all real numbers. The set of all real numbers is said to be ‘uncountably infinite’ because no matter how hard you try to create a list of all real numbers, you will always find that you have missed out real numbers from the list. A computer could not do it any better than we could, even if it carried on forever trying to produce the list. The set of all real numbers is ‘un-listable’. This is what it means to be non-computably enumerable.

Now, there is no need whatsoever for the axioms of the universe to be computably enumerable. Even if we perceive everything in the universe as finite (which again is an open question in science – it is by no means certain), this does not mean that the universe’s AXIOMS have to be computably enumerable. The axioms could be an uncountably infinite set, and still give us a universe in which we perceive reality as being finite. You simply cannot assume from the perceived finiteness of the universe that the universe’s AXIOMS themselves are computably enumerable. That is again a logical non sequitur.

And again, science and math do not in any way presume that the AXIOMS of a mathematical universe are computably enumerable. All we need in order to do science and math is that the universe can be approximated by OUR OWN axiomatic systems. The fact that we can approximate certain features of the universe using our own axioms in no way implies that the axioms of the universe must themselves be computably enumerable.

In summary, there is absolutely no need whatsoever for assumptions (a) and (b) to be true. We can do science and math perfectly well without them, because all we need in order to do science and math is that OUR OWN axiomatic systems allow us to approximate some features of the universe e.g. using differential equations. The fact that we can do this in no way implies that the universe itself must satisfy the assumptions of Godel’s theorem.

I repeat again, your whole case is fundamentally flawed because you are assuming that (a) and (b) are true when there is no need for this to be the case. We do NOT need (a) and (b) to be true in order to use OUR OWN axiomaic systems in science and math to approximate some features of the universe. You DO need (a) and (b) in order to apply Godel’s theorems to the universe however.

Christian,

Thank you for the cordial debate. I see from a Google search that you have credentials in mathematics and it’s great to have this discussion with a qualified individual.

You said,

This need not be true: (a). That the universe is in reality an axiomatic mathematical system i.e. that everything in the universe we perceive as being real can be deduced mathematically from a set of axioms.I am not saying that the universe ontologically IS an axiomatic mathematical system. I am saying that it strictly obeys mathematical laws and the laws of logic. Mathematical models of physical systems are isomorphic with those systems.

We have two mutually exclusive choices:

1. The universe does not obey mathematical laws.

2. The universe obeys mathematical laws.

If you choose (1) then the foundations of science go out the window.

If you choose (2) then all mathematical truths also apply to the universe. If 1+1=2 in math then 1+1 also equals 2 when you’re counting rocks. It makes no difference. Integration, multiplication, division are all the same with numbers as they are with physical objects, magnetic fields, energy, etc.

If we discover a new property of algebra tomorrow, we can be sure that it works in applied mathematics just as well as the algebra we already knew.

Syllogism:

1. All mathematical truths apply to physical systems

2. Incompleteness is a mathematical truth

3. Therefore incompleteness applies to physical systems

Statement 1 above is an axiom, it’s not provable. But it’s arguably the #1 axiom in all of science. It’s simply a statement that the universe is logical.

Now to your second statement:

(b). Even if (a) were true (which I have just shown you need not be the case at all), you also have to assume that the axioms of the universe form a computably enumerable set in order to apply Godel’s theorems.

Again we have two mutually exclusive choices:

1. The axioms of the universe are not computably enumerable

2. The axioms of the universe are computably enumerable

If you choose #1, that means the universe relies on either an infinite number of axioms, or else a set axioms which are illogical.

If the universe relies on illogical axioms then once again you’ve kicked the stool out from under science.

If the universe relies on an infinite number of axioms then that’s infinite regress. “Turtles all the way down.”

So our choices are:

1. The universe relies on an infinite set of axioms

2. The universe relies on a finite set of axioms.

#1 violates parsimony. So #2 is the only workable assumption. In mathematics you can make an infinite number of statements from a finite set of axioms but you can’t do anything with an infinite number of axioms.

The assumption that (a) the universe obeys mathematical laws and (b) it relies on a finite set of axioms is the only scenario that honors the principles of both science and math.

Perry

consider an opposite notion

can a state of nothing exist

can it be called a state

can it exist

I just lost the game

I just won the game.

Entertaining OP and thread. I’ve just started looking at the topic of Godel’s proof and read how much it has been misused. The OP is a classic example. Thank you

Perry, I see you still do not understand Godel’s Incompleteness Theorem. You’ve misapplied it several times in your essay. Here are the essential feature of the theorem:

In simple terms, “complete” means that all the truths of the logic system can be reached from its axioms or starting statements. Incomplete means there will always be some result that cannot be reached from its axioms. Incompleteness does NOT arise in just any logic system but only to formal systems that are:

1. finitely specified

2. large enough to include full arithmetic i.e. Peano arithmetic

3. consistent

Only if these conditions are present can it be said the system is incomplete. Smaller systems like Euclidean and non-Euclidean Geometry are complete; just as is Presburger arithmetic that does not include the multiplication, x, operation.

Godel did not apply his theorem to theology. It was a theorem in logistics and has been misapplied many times …

You quoted: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete …” (This is poor wording. A system that is incomplete has to be consistent. It is one of the three requirements).

If a computer uses algorithms that are large enough to use full arithmetic, are finite specified and consistent then the algorithms will be incomplete in the Godel sense (not the computer).

The universe is also not subject to Godel’s theorem as it is not a mathematical or logic system. In simple terms the universe is not math. People use math in order to understand elements within the universe and the math we use can either be complete or incomplete in the Godel sense.

You have repeatedly mixed up and misapplied concepts trying to prove god exists. You are spreading and perhaps marketing false ideas/conclusions.

The universe is finite.

The known laws of physics can be expressed with Peano arithmetic.

Any mathematical model of the universe is incomplete.

Therefore if the universe is consistent (logical) it therefore must also be incomplete.

A computer can be modeled by an algorithm or emulator and the algorithm or emulator will be incomplete. Therefore the computer is incomplete as well. This is self-evident: No computer can account for its own existence without reference to something outside of it.

The fundamental premise of science is that the universe obeys mathematical laws. You are welcome to reject that premise and reject science itself, in which case you can maintain the position that incompleteness does not apply to the universe.

The above has been discussed at length in prior blog comments.

Perry,

The laws of the universe are expressed with algebra alone and/or blended with non Euclidean geometry as per Einstein’s General Theory of Relativity. The later is a blend of complete and incomplete mathematics. However this is besides the main points.

Some theorists think the universe is finite but others think it could be infinite in some sense. In any case you are confusing Godel’s concept of incomplete with the common usage of the term. Godel’s usage only goes to MATHEMATICAL systems that have the three qualities that I showed in my first post. The universe is not math and we use both complete and incomplete MATHEMATICS to model phenomena within it.

A computer is not conscious therefore is not aware of itself nor can it appeal to anything outside itself. You are really mixed up when you misapply the Incompleteness Theorem here. The theorem is descriptive of mathematical systems that have the three qualities. You can use an algorithm that is complete in a computer, such as one that does geometry (the later is complete in the Godel sense).

Things in the universe obey natural law that we model using mathematics. Our better models are seen as useful approximations. Still the universe is not MATH! So again you are misapplying Godel in claiming the universe is “incomplete.” The universe is complete, how could it not be? Again Godel’s Incompleteness Theorem only goes to mathematical or logic systems.

God in some sense may exist but it has nothing to do with Godel’s Incompleteness Theorem. If you want to use Godel in a discussion about god’s existence then use his Ontological Proof. At least that goes to theology. Perry wake up!

The universe is not ontologically math.

But the universe is mathematical. It is logical. All science implicitly presumes this to be the case.

If this is not the case then science itself is a failure.

All the theorems in a high school geometry book are not conscious but they still appeal to axioms outside themselves, like Euclid’s 5 postulates. A computer likewise need not be conscious to rely on systems outside itself.

The computer is not self existent. The computer can only do what it does because someone or something outside of itself built and programmed it.

Your statement “The universe is complete, how could it not be?” is telling. You need to think about what you have just said here. Are you really trying to tell me the universe is self-existent? Did it give birth to itself? The matter created the big bang from which the matter came? How does that work? How is that not circular logic?

Godel says if a system is consistent, then it HAS to be incomplete. Science assumes the universe is consistent. If so, it cannot be complete.

Perry you are really confused and are confusing your readers. Godel’s Incompleteness Theorem ONLY applies to MATHEMATICAL systems that have the three features that I spoke to in earlier comments. Let’s go over them in more depth and then hopefully you will see the light. If a mathematical system is incomplete in the Godel sense it must:

1. have finitely specified set of axioms. This means you must be able to list them. (If you can’t then you cannot say the system you are referring to is incomplete. However, realize this is only about the axioms and not about all of the subsequent true statements that are products of the said system.

2. the system must be large enough to include full arithmetic i.e Peano Arithmetic. This means the said system must include all the axioms and symbols including the operators used in the said arithmetic.

Geometry in all of its forms and Presburger arithmetic do not and hence are complete mathematical systems. Godel showed as part of his doctoral thesis that Presburger arithmetic is complete.

Perry take notes and delete your mistakes on your websites about geometry and the silly talk about things inside and outside of circles. It has nothing to do with the Incompleteness Theorem and it makes you look like a fool to those who know better.

3. the system must be consistent or logically follow its axioms and operators.

To make a Godel analysis the subject has to be a mathematical system and you must be able to list its axioms. It cannot be applied to the universe as a whole. Wake up Perry!

1. The physics profession presumes that there is indeed a finite specified set of axioms and you can find them in any physics book. F=MA for example. We have not discovered all of the laws but all the known laws can be contained by a fairly small book.

2. All the laws of physics can be expressed in Peano arithmetic.

3. So far as we know, the universe is consistent.

Mathematical truths also apply to the physics. If 1+1=2 in math then 1+1 also equals 2 when you’re counting rocks. It makes no difference. Integration, multiplication, division are all the same with numbers as they are with physical objects, magnetic fields, energy, etc.

If we discover a new property of algebra tomorrow, we can be sure that it works in applied mathematics just as well as the algebra we already knew.

Syllogism:

1. All mathematical truths apply to physical systems*

2. Incompleteness is a mathematical truth

3. Therefore incompleteness applies to physical systems

*There are some mathematical truths where no application in physics is yet known. But there is nothing in physics that is currently known to violate any mathematical law.

Perry

You have repeatedly been saying and/or implying that geometry is incomplete and that is wrong. You wrote: “All the theorems in a high school geometry book are not conscious but they still appeal to axioms outside themselves, like Euclid’s 5 postulates …” The problem is that geometry is complete in the Godel sense. The axioms of a mathematical system are part of the system.

The axioms of geometry are not proven. You cannot prove that a line can be extended infinitely in both directions. In real physical space, infinite distance does not even exist. The mere possibility must be hypothesized. Euclid’s postulates are taken to be self evident.

Look in particular at the 5th postulate, which because of its complexity, many hoped could be provable from something simpler.

See http://en.wikipedia.org/wiki/Euclid%27s_fifth_postulate#History

From http://www.timstakland.com/2013/06/05/how-rhetoric-shapes-science/ “Finally, in the first quarter of the nineteenth century, a Hungarian and a Russian…Bolyai and Lobachevski…established irrefutably that a proof of Euclid’s fifth postulate is impossible.”

Perry that’s not the Godel issue of incompleteness. It requires that a formal system must be:(1) the axioms are finitely specified and this means list-able,(2) is large enough to include full arithmetic (3) the system is consistent.

Geometry does not meet condition (2)and therefore it is complete in the Godel sense.

The ultimate provability of the parallel line axiom is a separate issue. This means any attacks on it were not Godel’s issue. In any case people can show that the parallel line postulate does hold in any test that has been done. It is a given in astrophysics that it does hold true in a flat universe.

His two Incomplete Theorems break down to the three features. See “New Theories of Everything”, John Barrow, Oxford University Press, 2007. pp. 51 – 61.

Again, you and others are are mixing up issues. You cannot use Godel in the same breath. I have tried to get you to see that … I hope this time it pierces your biases.

Euclidean Geometry is not a formal system (as discussed at http://math.stackexchange.com/questions/90393/why-euclidean-geometry-cannot-be-proved-incomplete-by-godels-incompleteness-the) in the Godel sense, so you are right.

In keeping with the point of my article, though, Euclidean geometry still rests on unprovable axioms; quite obviously so.

Perry

Schopenhauer argued the parallel line postulate is evident by perception even though it was not a logical consequence of the other Euclidean axioms. Forms of non-Euclidean geometry do not contain or use the postulate and are complete in the Godel sense and are not subject to the parallel line criticism. Forms of arithmetic that do not contain all the axioms and operators of Peano Arithmetic are also complete. The bottom line is all of mathematics is NOT incomplete like you claim in your articles.

Complete forms of mathematics as well as incomplete ones can and are being used to understand elements in the universe. Your syllogisms that treat the universe as a mathematical system are incorrect. Humans create or discover mathematical systems and use them to calculate and understand elements in the universe. However the universe isn’t math.

While we are speaking about proofs, no one has proven the universe is finite either. Yet you do not have any problem in claiming it is so.

In any case I am glad you are starting to understand the Godel’s Incompleteness Theorem and how it is distinct from the criticism of the Euclidean parallel line postulate. Unfortunately your articles imply that everything you suggest in your logic chains and your conclusions stem from Godel’s Incompleteness Theorem. It is not so.

Ken,

Every logical system above a certain degree of sophistication is incomplete. Even some systems that do not qualify as formal systems in the Godel sense still rest on axioms you cannot prove but have to assume, which is my thesis.

You are free to reject the notion that the universe performs computation. But in doing so you are rejecting the most fundamental premise of science. If on the other hand we accept that the universe when subjected to measurement adds, subtracts, multiplies, divides, integrates etc. then it is necessarily incomplete.

If you insist that all kinds of algebraic logic and logical operations apply to the universe but also insist incompleteness is somehow an exception, you will have to justify that.

I am FAR from the first person to point this out, by the way. Stephen Hawking for example – http://www.hawking.org.uk/godel-and-the-end-of-physics.html; physicists Stanley Jaki and Freeman Dyson, to name three.

::: it’s done.

“God” is proven.

>> Formalization, Mechanization and Automation of Gödel’s Proof of God’s Existence

Christoph Benzmüller, Bruno Woltzenlogel Paleo

(Submitted on 21 Aug 2013 (v1), last revised 25 Aug 2013 (this version, v3))

Goedel’s ontological proof has been analysed for the first-time with an unprecedent degree of detail and formality with the help of higher-order theorem provers. The following has been done (and in this order): A detailed natural deduction proof. A formalization of the axioms, definitions and theorems in the TPTP THF syntax. Automatic verification of the consistency of the axioms and definitions with Nitpick. Automatic demonstration of the theorems with the provers LEO-II and Satallax. A step-by-step formalization using the Coq proof assistant. A formalization using the Isabelle proof assistant, where the theorems (and some additional lemmata) have been automated with Sledgehammer and Metis. <<

http://arxiv.org/abs/1308.4526

Excellent!

Perry you and others are assuming what humans do directly compares with what elements of the universe does. Firstly we do not have any formalism of the collective i.e. uni-verse.

Secondly all we can do is to tackle some select elements within the universe with our mathematical models. We can get good approximations in process and this is what keeps scientific inquiry alive. And so NO … I do not reject science. I understand it!

One mistake you and others make, is to confuse what we do in science with ALL of the universe. That is the unwarranted giant step that you and some others make. This is part of the reasons why your syllogisms that go to the universe are nonsense. Not to again mention your misuse of Godel’s Incompleteness Theorem as the false starting point in some of your essays!

You ignore and do not comment on the specifics of these criticisms and simply go on to cite some more nonsense. However this is common game playing on internet discussions. I get that. My guess is that you will probably not edit your internet presentations, even though, by now, you know you should.

To fool others is common but it takes a little more to fool oneself. When this is done it makes fooling others easier as it makes you look … well fill in the blanks … !

Are you suggesting that “human mathematics” is some kind of subjective phenomenon that has no independent objective reality?

Are you assuming that the mathematical operations which humans do cannot be compared to machines? I refer you to the Church-Turing thesis.

You are free to assume that science and math only apply to certain parts of the universe. In that case you are postulating that the universe is inconsistent. In which case it can be complete as you claim.

If that is your position, don’t tell me you are still practicing science, because science has no framework for modeling behavior that defies logic and violates mathematical laws.

The amount of mathematical ignorance in this article is, quite frankly, impressive. The author has severely raped and abused mathematics (to say it otherwise would not do it justice), and has the gall to ask “why, what’s wrong with that?”. Perry, would it be ok to do that with another abstract concept and in terms you would understand, because you most certainly do not understand mathematical terms you abuse? I understand this sounds harsh, but this IS what you did, even though you don’t understand it. In the sincerest hope that your goal is search for knowledge and not proving a point by any means, and that the mistakes are made in ignorance and not malice, I recommend the following penance for transgressions against mathematics:

1. You are forbidden from using the label “infinity” until you have a good working knowledge of the concept. You will use boundless, limitless, endless, depending on the context. Hopefully, no more propositions such as “I bound something bound-less” will be made. You can of course always imagine something outside of boundless (ie by adding another dimension, a 2d plane in 3d space), but that process is end-less. Studying proofs (ie 0.9999…=1) will help a lot in understanding how this concept is used in practice.

2. You are forbidden from using labels we put on things in mathematics to make our lives easier. You will replace them with “Suzy”. Instead of “Suzy” you are allowed to use formal definitions. If the argument does not hold in either of these cases, it does not hold. If you don’t understand the formal definitions, you certainly don’t understand what the theorem is about. You will not abuse labels, such as “computable” or “theory”, even though they might ring a bell. That’s not what they are for.

3. You are forbidden to infer anything from Church-Turing thesis. It’s a definition, that is what we use it for. We labelled it “thesis” instead of “definition” just to confuse you. The “thesis” part should have been a hint, as we rarely draw conclusions from hypotheses, and if we do, we are sure to mention it, in big red letters. It basically says “Those functions that satisfy blah blah we’ll call computable”. Nothing else. You will use “Frank” instead of “computable” (as per point 2), and whenever you need to figure out what “Frank” is, you’ll refer to the thesis. Hopefully you’ll not try to guess what we meant with “Frank” as you do with “computable”.

4. Until the amount of intuition you apply to mathematics is limited to “a set” and “a point”, you will not ask those who have the knowledge to debate with you. It’s like debating with a 3 year old and a waste of time. No amount of debating will help you figure out these things and without knowing formal definitions there is nothing to debate.

5. There are no shortcuts in mathematics. There are only shortcuts to mathematical ignorance. You will not use those.

Dear “Turing Machine”:

(Note to everyone else: This is an anonymous person who is so unwilling to give even the slightest indication of who he is, he didn’t even use a real email address.)

Your post consists of nothing but name-calling and accusations and is devoid of content, save for one lone statement:

“You are forbidden to infer anything from Church-Turing thesis.”

The burden of proof is on you to defend your view that a major proposition in mathematics has no implications, thus should be ignored.

I’m not afraid to sign my name to my work. What are you afraid of? If you have something to say, tell us your name and where you’re from and what your credentials are and I will be happy to hear you out.

I await your reply.

How does one defend from ignorance and intellectual dishonesty? Everyone knows how to drive a car, it’s easy, but not everyone knows how a car actually works. It does not matter how many car parts you can name, it’s painfully obvious you don’t have a clue what they do. If this was physics, I would recommend you build a car that you suppose would work, you would drive off a cliff, and that would be the end of it.

This is mathematics, however, and no car can be built. You either get it or you don’t, and you don’t.

Church-Turing thesis (thesis, look up the meaning) says:

if you have a function, and are able to construct a Turing machine that does what that function does, we’ll call it “computable”.

There is nothing magical about the word “computable”. It’s not different than:

if you have a function, and you’ll able to express it as f(x)=kx+n, we’ll call that function “linear”.

It’s a labeling system, that’s what definitions do. Nothing about linear functions (ie that they are bijections) comes from the name, it comes from the definition. Your label is by definition Perry. What could you possibly infer from that? If your label was Frank, you would infer that you are honest? How utterly ridiculous. Leave those labels alone, and stop abusing them.

Now, there are important (VERY important) things *behind* Church-Turing thesis (but since you didn’t do your homework, you miss the point). Why do you think it’s called that? When theory of computability was still in it’s infancy, we didn’t have a mathematical model of a computing machine, and therefore we could prove any result about it (and by prove I mean in the strictest mathematical sense, to rigorously prove something means it has to be this way and it cannot be any other way). So Church went on to develop lambda calculus, Turing his machine (it is a mathematical concept just as a function is), Godel recursive functions. THE important result was showing (proving) that all those models are equivalent (THAT is the important thing).

So it doesn’t matter if I use recursive functions, or lamdba calculus instead of Turing machine when I state the thesis, it’s still the exact same definition (it labels the exact same class of functions). We developed more models of machines (RAM machine etc) that are equal to Turing machine, we’ve shown that there are some that are weaker (finite automata), we’ve shown that the computer that you currently use is equal to Turing machine, and we have a lot that are stronger (actively researched). Not being able to construct something does not prevent us from doing mathematics on it (ie infinitely-dimensional spaces etc).

The interesting question, of course is, what are the limits of Turing machine-like models (are there any functions which are not “computable”)? There’s a whole bunch of problems (functions) which cannot be solved by a Turing machine, and we decided to call them “undecidable” (it’s also a label, keep your hands off). In other words, there is not an effective algorithm that would solve those problems on Turing machine-like models (are there other models we could actually build?).

One of these problems, halting problem, has weaker version of Godel’s incompleteness theorem as a direct consequence – you cannot build a Turing machine that would prove every true statement (how many are there?) about natural numbers (and do it in finite time) – Godel’s theorem does not care about whether the statement is true.

Be aware of limitations of Godel’s theorem – the requirement is that the set of axioms be at most countable (as many as natural numbers), we have tons of sets with more elements (real numbers), and you have no problems working with them. So Godel’s theorem states that, satisfying all conditions, you’ll find a statement that cannot be proven (rigorously), and when you add it as axiom, another one will pop up because of that addition.

In order to apply this pretty advanced result of mathematics to ie physics, you’d have to show that it follows the same rules. But it doesn’t (and it’s an absolutely silly proposition) – proof in mathematics is not the same as proof in physics, Godel’s theorem is all about mathematical proof, and please don’t abuse those labels. If you don’t get the difference, let me spell it out for you: mathematical proof means it’s this way and it is impossible to be any other way. Proof in physics does not mean that, it means we measured it to the best of our ability and result fits our predictions. That’s why you can always say “God did it”, and nobody can disprove that. You’re basically cutting the branch you’re sitting on.

Please leave mathematics (especially the parts you don’t understand) out of this.

You explained the Church Turing Thesis this way:

if you have a function, and are able to construct a Turing machine that does what that function does, we’ll call it “computable”.

No, that is not what the Church-Turing Thesis says.

What it says is that if some method or algorithm exists to carry out a calculation, then the same calculation can also be carried out by a Turing machine.

If you can write a proof of Godel’s incompleteness theorem on a piece of paper, a machine calculation will confirm the same result.

It also logically follows that if incompleteness is a property of all logical systems, and the universe is a logical system (=Turing machine) then the universe is also incomplete.

You said:

“In order to apply this pretty advanced result of mathematics to ie physics, you’d have to show that it follows the same rules.”

I cannot prove that the universe follows the same rules as math. You are welcome to believe that it doesn’t. I’m here to remind you that if you reject this proposition that the universe is mathematical, you reject all of science and physics. Do so at your own peril.

The universe is incomplete, therefore God exists. And that is the only way that it is possible to postulate that the universe is rational. You can believe in an irrational universe if you prefer. You decide what you want to believe.

Strict mathematical definition of algorithm IS “something that the Turing machine can solve in finitely many steps”. That is why we developed all those models – you can’t prove anything about algorithms unless you have a mathematical model for it. “A procedure” is not a mathematical model. Do you understand why it’s a definition (and what it’s a definition of)? So “solvable by Turing machine” == “algorithm” == “computable” == “can express via recursive function” == “can express via lambda calculus”. Stop writing nonsense.

Writing proof on a piece of paper has nothing to do with machine computation. How utterly silly. There are no results for machine to check. Can you explain, in detail, how that algorithm would look like and what it would do? What exactly is your input and what exactly are you expecting as your output?

..

It also logically follows that if incompleteness is a property of all logical systems, and the universe is a logical system (=Turing machine) then the universe is also incomplete.

There is no such thing as a logical system, so stop pulling labels out of your hat. If you want to use Godel’s theorem, you’ll have to stay within limits of what it exactly states. And it does not operate on Turing machine as elements of that set do not satisfy conditions required for Godel’s theorems (it does not have necessary elements). And theories affected by Godel’s theorems are only those that have up to countable many axioms (not all by a long shot).

Your whole argument is half ignorance and half label abuse. Let me indulge a bit in it.

It also logically follows that if incompleteness is a property of all theories, and the universe is a theory then the universe is also incomplete.

This is actually mathematically correct. Still, universe is not a theory (neither mathematically nor philosophically). Whether universe follows the same rules as math remains to be seen, while your reasoning has broken tons of them. Is it irrational then?

you can’t prove anything about algorithms unless you have a mathematical model for it

The point of the C-T Thesis is that if you have a mathematical model for the algorithm, then your model and the algorithm are equivalent.

Writing proof on a piece of paper has nothing to do with machine computation. How utterly silly.Quoting Wiki:

“Several independent attempts were made in the first half of the 20th century to formalize the notion of computability:

American mathematician Alonzo Church created a method for defining functions called the lambda-calculus,

British mathematician Alan Turing created a theoretical model for machines, now called Turing machines, that could carry out calculations from inputs,

Kurt Gödel, with Jacques Herbrand, created a formal definition of a class of functions whose values could be calculated by recursion.

All three computational processes (recursion, the lambda-calculus, and the Turing machine) were shown to be equivalent—all three approaches define the same class of functions.”

So yes, the proof you write on a piece of paper can be carried out by a machine, and both will give the same answer.

Turing Machine,

Your posts are a classic example of something I have witnessed over and over and over:

Most online atheists are anonymous cowards. You are one of them.

I do not say this lightly. I mean this in the most serious possible way. And I hope it gets under your skin and bugs you until you work it out.

You do not possess the courage to sign your real name to your beliefs or statements. You hide behind a screen name, call me names, casts insults, and disappear as soon as it gets too hot in the kitchen.

I have observed this pattern for 8 years. Ever since I got pulled into a debate on the Infidels forum (details at http://www.cosmicfingerprints.com/infidels) where I had to endure insults and derision from a bunch of anonymous people. I was one of the only people there with the guts to use my real name.

When someone uses a screen name and a throwaway email address, why should you take ANYTHING they say seriously?After all, all they have to do is click the “X” on the upper right corner of some screen and they’re out. I have seen this happen on my own blogs at least 100 times. Maybe 500.

I cannot respect that.

Go to any atheist website or blog or go to any article on any major newspaper about religion where a debate has sprung up, and most of the insults and derision and hurtful remarks from EITHER side of the aisle are from anonymous cowards hiding behind screen names.

I can respect a guy like Ken Koskinen who actually signs his name to what he says. I can have a real conversation with him. Despite the fact that we disagree, I appreciate him. At least he is a real man.

In the 1990′s I had a friend named Mark Vuletic who is now a college professor and his website is http://vuletic.com/. He is an ardent atheist, grew up Roman Catholic. We would have conversations that ran hours and hours in Oak Park Illinois coffee shops. Mark deeply challenged my beliefs, asked great questions and was always respectful and cordial. In the end he did not win me over but I always valued his friendship.

Our paths diverged. Then during the last 10 years the “New Atheism” sprang up. A cabal of raging, egotistical, narcissistic loud-mouths like PZ Myers and Richard Dawkins replaced the thoughtful dialogues of people like Bertrand Russell, Antony Flew, David Hume and Fred Hoyle. It became impossible to have a thoughtful exploration and dialogue. Even the “Friendly Atheist” website brims with hostility and name-calling.

I can promise you, a movement based on prejudice and insults, and a built-in presumption that “We’re smart and they’re all idiots” can only cave in upon itself. We have seen that in the last couple of years as the atheist community has exhibited egregious misogyny and clear inability to work things out in a civil way. Jen McCreight tells her story in excruciating detail on her blog.

All this stems from the same roots of cowardice and presumed superiority over others. God or no God, you cannot build any kind of community on those values.

I can’t have a serious conversation with you. You are not a real man. I don’t even know whether you’re a man or a woman. (Although generally I find women to be less snarky than men in these online conversations.)

I will be HAPPY to continue this conversation with you – on one condition. You tell us who you really are. Why? Because these questions MATTER.

I also suggest that you wipe the smirk off your face and familiarize yourself with some of the most important discoveries of 20th century mathematics. Do your homework and then continue the conversation. I’ve done mine.

From now on, posts from anonymous drive-by detractors will be deleted without comment or apology. Real conversations with real people only.

Perry, you still do not get it. In previous chats I showed how there are logical systems that are complete, in the Godel and any other sense. Yet you still write:

“It also logically follows that if incompleteness is a property of all logical systems, and the universe is a logical system (=Turing machine) then the universe is also incomplete.”

At the same time you seem to be saying that god mysteriously completes all the in-completions and thereby one can claim the universe is complete. This follows as you imply the universe can still be understood by rational inquiry.

Much can be understood even in rational systems that are incomplete in the Godel sense. Many of these have added much to our present understandings. It seems to me you have to show or prove how these systems have specifically failed and your god hypothesis has improved the understanding of anything. Note this: Godel’s Incompleteness theorems do not necessarily determine UN-truth.

It seems to me that you are using your belief in the biblical gods as the standard against which all divergent things are deemed to fail. This is self insistent thinking and I fail to see what anyone can say that can/would change your thinking.

The existence of your anthropomorphic god, has not been detected. You are filling in the blanks with an unknown and using it prove your conclusion. You can say that other theories also include unknowns but you are suggesting this as the grand theory of everything. The standards for determining such a final physical theory are far higher and cannot include anyone’s theological biases.

However such ideas are and can be used in religious literature where they aim to market to true believers and/or to convert people. THIS is what you are doing and since you are a true believing marketing person, I think nothing that is posted to the contrary will cause you to change your primary motive.

I addressed all your concerns point by point. I still await your reply from my previous post.