His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.

Natalie Wolchover writes in an article in Quanta Magazine:

In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.

Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.

But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.

His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.

Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality.

**The article next outlines a “simplified, informal rundown of how Gödel proved his theorems.”** **What the results imply is discussed next.**

## No Proof of Consistency

We’ve learned that if a set of axioms is consistent, then it is incomplete. That’s Gödel’s first incompleteness theorem. The second — that no set of axioms can prove its own consistency — easily follows.

What would it mean if a set of axioms could prove it will never yield a contradiction? It would mean that there exists a sequence of formulas built from these axioms that proves the formula that means, metamathematically, “This set of axioms is consistent.” By the first theorem, this set of axioms would then necessarily be incomplete. But “The set of axioms is incomplete” is the same as saying, “There is a true formula that cannot be proved.”

Gödel’s proof killed the search for a consistent, complete mathematical system. The meaning of incompleteness “has not been fully fathomed,” Nagel and Newman wrote in 1958. It remains true today.

Full article available at Quanta Magazine.

“**Truth is bigger than proof.”**

Good article. My only minor qualm is that Gödel’s proofs apply to axiomatic systems that are rich enough to capture arithmetic. Less rich axiomatic systems don’t have that problem. (I

thinkEuclidean geometry is complete but I’m not sure.)As to: “Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality.”

Natalie Wolchover links to a paper which discusses the undecidability of the Spectral Gap,

And although Natalie Wolchover characterized their finding as afflicting physical reality in “some ill-understood way”, in the following article they are quite clear as to how their finding ‘afflicts’ physical reality.

Specifically they state, “even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.,,,” and that “the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description.”

Moreover, per wikipedia, in 2020 they made their proof more robust,

And since attempting to mathematically unify the macroscopic descriptions of General Relativity with the microscopic descriptions of Quantum Mechanics lies at the heart of all (failed) attempts to find a single overarching mathematical ‘theory of everything’,,,

,,, then their proof clearly shows that the microscopic descriptions of quantum mechanics can never be successfully extended to the account for the macroscopic descriptions of General Relativity.

Moreover, their finding should not really be all that surprising to find out. It is already known that an unbridgeable ‘infinite mathematical divide’ exists between General Relativity and Quantum Mechanics.

But before we get into that it is necessary to clarify a few things.

First off, as Gregory Chaitin has shown, there are an infinite number of mathematical theorems that could have described the universe, but don’t.

This presents an irremediably difficult situation for those who hope to find a purely mathematical theory of everything that makes no reference to God. As the late Steven Weinberg, an atheist, confessed to Richard Dawkins, “I don’t think one should underestimate the fix we are in. That in the end we will not be able to explain the world. That we will have some set of laws of nature (that) we will not be able to derive them on the grounds simply of mathematical consistency. Because we can already think of mathematically consistent laws that don’t describe the world as we know it. And we will always be left with a question ‘why are the laws nature what they are rather than some other laws?’. And I don’t see any way out of that.”

And although atheists are, self-admittedly, in a pretty bad ‘fix’, the Christian Theist has a ready explanation. As Bruce Gordon explains, “the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them.”

And it is not as if ID proponents do not already have sufficient reason to believe that free will must be involved in choosing among an “infinite variety of mathematical descriptions and bring(ing) into existence a reality that corresponds to a consistent subset of them.”

As Douglas S. Robertson explains, “Human mathematicians are able to create axioms, but a computer program cannot do this without violating information conservation. Creating new axioms and free will are shown to be different aspects of the same phenomena: the creation of new information.”

In fact, modern science was born out of the belief that any mathematics that might describe this universe are ‘God’s thoughts’.

As Johannes Kepler stated shortly after discovering the third law of planetary motion,

And as Edward Feser explains, “Mathematical truths exhibit infinity, necessity, eternity, immutability, perfection, and immateriality because they are God’s thoughts, and they have such explanatory power in scientific theorizing because they are part of the blueprint implemented by God in creating the world.”

And you don’t have to take Kepler, Poythress, and Feser’s word for it, Eugene Wigner, (who’s insights into quantum mechanics continue to drive breakthroughs into quantum mechanics; per A. Zeilinger), and Albert Einstein, (who needs no introduction), are both on record as to regarding it as a ‘miracle’ that math should even be applicable to the universe in the first place.

Moreover, Wigner questioned Darwinism in his process of calling it a miracle. Whereas Einstein went so far as to chastise ‘professional atheists’ in his process of calling it a miracle.

And the last time I checked, a miracle is considered to be the sole province of God,

Thus, (before we even get into the ‘infinite mathematical divide’ that exists between General Relativity and Quantum Mechanics), the fact that mathematics should even be applicable to the universe in the first place is, (in spite of any a-priori philosophical biases that atheists may have against it), by all rights, already to be considered a miracle of God.

Now to the ‘infinite mathematical divide’ that exists between General Relativity and Quantum Mechanics.

Professor Jeremy Bernstein states the ‘infinite mathematical divide’ between the two theories as such, “there remains an irremediable difficulty. Every order reveals new types of infinities, and no finite number of renormalizations renders all the terms in the series finite.The theory is not renormalizable.”

And as theoretical physicist Sera Cremonini noted, “You would need to add infinitely many counterterms in a never-ending process. Renormalization would fail.,,,”

And as Michio Kaku stated in the following video, when you try to combine General Relativity with Quantum Mechanics, “you get an infinite sequence of infinities, (which is) infinitely worse than the divergences of Einstein’s original theory (i.e. General Relativity).”

Dr. William Dembski in this following comment, although he was not directly addressing the ‘infinite mathematical divide’ that exists between General Relativity and Quantum Mechanics, offers this insight into what the ‘unification’ of infinite God with finite man might look like mathematically:, Specifically he states, “The Cross is a path of humility in which the infinite God becomes finite and then contracts to zero, only to resurrect and thereby unite a finite humanity within a newfound infinity.”

And indeed, when rightly allow the Agent causality of God ‘back’ into physics as the Christian founders of modern science, such as Sir Isaac Newton, originally envisioned,

And when we rightly allow the agent causality of God ‘back’ into physics as is now empirically demanded by the closing of the ‘freedom of choice’ loophole by Anton Zeilinger and company,

,,, then that very reasonable concession on our part to rightly allow the Agent causality of God ‘back’ into physics provides us with a very plausible resolution for the much sought after ‘theory of everything’ in that Christ’s resurrection from the dead bridges that infinite mathematical divide that exists between General Relativity and Quantum Mechanics and provides us with an empirically backed reconciliation, (via the Shroud of Turin), between Quantum Mechanics and General Relativity into the much sought after ‘Theory of Everything”

Thus in conclusion, and to repeat, when we rightly allow the Agent causality of God ‘back’ into physics, (as the Christian founders of modern science originally envisioned, Isaac Newton, Michael Faraday, James Clerk Maxwell, and Max Planck, to name a few of the Christian founders,,,, and as quantum mechanics itself now empirically demands with the closing of the ‘freedom of choice’ loophole by Anton Zeilinger and company), then rightly allowing the Agent causality of God ‘back’ into physics provides us with a very plausible resolution for the much sought after ‘theory of everything’ in that Christ’s resurrection from the dead bridges the infinite mathematical divide that exists between General Relativity and Quantum Mechanics and provides us with a very plausible, empirically backed, reconciliation, (via the Shroud of Turin), between Quantum Mechanics and General Relativity into the much sought after ‘Theory of Everything”.

Wow, BA sure knows how to overwhelm a potential discussion!

Ba77,

I think gravity is very well understood. The so-called “strong force” and “weak force” in atoms is well understood. The various subatomic particles like quarks and leptons are well understood. Over the decades, the distances between various electron states, or levels, has become well understood. The quantum world is being mapped out and put to practical use at the same time.

I propose that a Theory of Everything exists at this moment. And that it is being put to use already in various experiments. Even though the math has not been all worked out, a series of trial and error experiments are revealing more and more pieces of the puzzle. The end result will be new techniques for doing certain things and for use in new propulsion systems.

God did create a universe that is rational and intelligible and given us the creativity to make discoveries about it. The same is true of the quantum world. However, if such knowledge exists, and I think it does, it will be kept highly secret by whoever has it.

The OP doesn’t seem to have a link to the article that is quoted from. I’d like to see it.

@7:

Here it is: How Gödel’s Proof Works.

Thanks. What you typed doesn’t work as a link, but I found it by googling her name and the title.

Whoops! My bad. Anyway, here it is — correctly this time — for anyone who wants it:

How Gödel’s Proof Works.

Relatd @6,

These are pretty ambitious assertions. Do you have any references?

Consider just gravity, for example.

-Q

Querius at 11,

What do you want to know about gravity? There is the macro version that we all experience, and the subatomic version.

Relatd,

Please understand, I’m not asking for your explanation. I’m simply asking about your references from where you’re getting your information.

Thanks,

-Q