A Fragment From the Archives

The ancient Greeks grappled in vain with three geometrical problems: the duplication of the cube, the trisection of the angle, and the squaring of the circle. What drove them to these endeavors? Divine inspiration? Well, yes—of a sort. The origin of the duplication of the cube is well-known. The story behind the trisection of the angle however has been lost to history—until now.

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Now the people of Atlantis worshiped nine gods, but they had only three temples. These they had spaced equally around the rim of an enormous circular bed of glassy black stone in the center of the island. Wondrous works of architecture were these temples: one tall and spindly, and of gleaming smooth marble; one golden, and of soft curves and sinuous contours that reflected the sun in bright serpentine patterns; and one of intricately cut crystal, here flat, here in a thousand facets, so that the eye was dazzled by swirling fragments of color, or lost in deep pools of light.

This was the custom of the land: for three tolod, the Atlantans worshiped the gods Avas, Kotar, and Síen, each god choosing one temple to himself. (One tolod = 9 days.) But when their term had passed, Avas, Kotar, and Síen departed, giving place to Vabas, Selon, and Ģitan. And these gods had their three tolod of adoration. And they gave way to Morope, Ygam, and Kheddar, who drank the spilled wine and ate the burning meat for three tolod, till Avas, Kotar, and Síen should return.

Long the gods endured this, for the Atlantans were generous with their sacrifices. Still, the gods smarted under the tauntings of foreign deities. “Six tolod of fasting! Is it not the mortals who should fast at the will of the gods?”

So at length the gods of Atlantis grew weary of this raillery, and wearier still of six dry tolod, and found themselves dreading the end of their brief space of adulation even as it began. Thus one day spoke Avas to his priests: “Harken to my will, and to the will of all your gods! Nine tolod hence let there be nine temples, spaced with perfect evenness round the rim of the great stone—for no temple may have greater dominion than another. And let them be equal among themselves even as they surpass all other temples of the world. Such is our will: see that you obey!” So, too, spoke Kotar to his priests, and Síen to hers. And all the priests of Atlantis trembled and replied, “It shall be done, O gods!”

Now the people of Atlantis numbered among themselves the greatest artists, engineers, architects, and stonemasons that the world had ever seen. Also among them were many mathematicians. And all set busily to work when they heard the priests pronouncement: the stonemasons to quarrying, the engineers to building great engines of lifting and hauling, the artists to painting and scupting, and the architects to planning the form and construction of the new temples. All the populace found needful work for the fulfillment of the gods’ command.

But the mathematicians said, “Now this is a very interesting problem. It is necessary (and sufficient) to trisect each of the angles that separate the old temples from one another, viewing them from the center of the Great Stone, then to place one new temple on each of the trisecting lines.” And they busied themselves with their instruments, and drew many pictures in the sand, and consumed much wine and ablakhar in thinking on the problem.

The work proceeded apace. Soon the Great Stone grew cluttered with machines the engineers had gathered, and the air became thick with dust from the stones the masons dressed.

So it was when Vabas, Selon, and Ģitan came down. These gods inquired of their priests, who replied that the engineers and stonemasons were ready to build, and waited only for the architects to finish their plans and the mathematicians to mark the new sites. And the gods were satisfied, for a time.

Three tolod passed. The Great Stone was littered with hundreds of sculptures and frescoes. The architects exulted in their finished plans, knowing that whosoever should stand on the Great Stone gazing on the nine temples would perish from excess of splendor, and not regret it.

Yet when Morope, Ygam, and Kheddar descended, they saw no sign of foundations. And they inquired of the priests, who inquired of the architects, who replied: “All stands in readiness; we wait only for the determination of the mathematicians.” Then said the priests to the mathematicians, “Look, maybe it will be all right, even if it’s just a little bit off—” “No, no!” answered the mathematicians. “It’s a very interesting problem! We’ll get it any day now!”

So the priests increased the offerings threefold, and then ninefold, and reminded the gods how well they had been served for uncounted millenia, and chanted of the glory of the new temples, and altogether somehow managed to keep the nine gods mollified for three, nine, and at last twenty-seven tolod more. And they questioned (nine thousand times) the mathematicians, who kept their fingers so busy in the sand that the shores of Atlantis resembled fields plowed by drunken oxen. But they received no other answer than, “Any day now!”

So at length the nine gods came down together—something never before or since seen, except perhaps at the creation. And the priests quailed before them, but said only: “O gods, we burn to do your bidding, but we know not how to trisect the angle!”

“Then we will show you,” replied the nine gods as one.

Nine crevices appeared in the smooth floor of the plaza: nine cracks that met in the center of the Great Stone and streamed out to the rim, past the rim to the shores, descending deep into the rock.

And Atlantis split in nine parts and sank beneath the waves.

 

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The fragment does not mention what method the gods used to trisect the angle.

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Non-standard Models of Arithmetic 14

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MW: Recap: we showed that PAT implies ΦT, where ΦT is the set of all formulas

\{\varphi\rightarrow\text{Con}(T_n+\varphi^\mathbb{N}):\varphi\in\text{L(PA)},n\in\omega\}

Now we have to show the converse, that PA+ΦT  implies PAT. But first let’s wave our hands, hopefully shaking off some intuition, like a dog shaking off water.

Below is a diagrammatic description of PAT. The outer rectangle contains all models of PA, the inner oval contains all those that are (isomorphic to) ω’s of models of T. In other words, all models of PAT. Now let’s say φ is a sentence that cannot be proved using only PA, but can be proved from T. So for T=ZF, φ could be the Paris-Harrington principle, or Goodstein’s theorem, or Con(PA). The blue region represents all the models where φ holds. So it includes the oval, but not all of the rectangle. The oval is the intersection of all such blue regions, as φ ranges over all of PAT. If φ were provable from PA, then it would paint the entire rectangle blue.

Next I want to take a second look at ΦT. I’m going to turn its formulas inside-out and upside-down. Start with

\varphi\rightarrow\text{Con}(T_n+\varphi^\mathbb{N})

The contrapositive is

\neg\text{Con}(T_n+\varphi^\mathbb{N})\rightarrow\neg\varphi

Basic results from proposition logic tells us that for any sentence φ and any theory T, ¬Con(T+φ) is equivalent to T⊢¬φ. (“Proof by contradiction”.) So we have

\text{``}T_n\vdash\neg\varphi^\mathbb{N}\text{''}\rightarrow\neg\varphi

Since φ ranges over all sentences of the language of PA, we might as well replace ¬φ with φ. Upshot: ΦT is equivalent to

\{\text{``}T_n\vdash\varphi^\mathbb{N}\text{''}\rightarrow\varphi : \varphi\in\text{L(PA)},n\in\omega\}

Let me explain the quote marks. Provability is a syntactic property, an assertion about strings of symbols. Or maybe I should say strings of strings of symbols. We can code syntactic claims into PA; the quoted stuff stands for the formalization of that claim into the language of PA. Kind of like the corner brackets for Gödel numbers, but more general.

The sentences of ΦT say “Trust T.” In other words, “If T proves it, then it’s true.”

Let’s take a second look at the implication \text{PA}^T\Rightarrow  \text{PA}+\Phi_T. We want to show that the formula \text{``}T_n\vdash\varphi^\mathbb{N}\text{''}\rightarrow\varphi “paints the oval blue”, i.e., holds in all models of T. Well, if T_n\vdash\varphi^\mathbb{N}, then “it stands to reason” that φ should hold in the \mathbb{N} of a model of T. That’s the intuition, at least.

As before, some subtleties lurk, but not enough to derail the conclusion. We must interpret the assumption “T_n\vdash\varphi^\mathbb{N}inside the model \mathbb{N}. The proof of \varphi^\mathbb{N} could have non-standard length! The trick is to bring the whole intuitive argument inside the model of T, which we can do thanks to the items mentioned last time: the Reflection Principle of ZF, and the existence of a formal truth predicate for formulas with up to d quantifiers (for any given d).

The reverse implication, \text{PA}+\Phi_T\Rightarrow\text{PA}^T, takes less work, though there is one fine point. Suppose T yields \varphi^\mathbb{N}. Then some finite fragment does, say

T_n\vdash\varphi^\mathbb{N}.

In fact (this is the fine point),

\text{PA}\vdash\text{``}T_n\vdash\varphi^\mathbb{N}\text{''}.

Why? Well, if Tn proves something, then PA can prove that it proves it. People often argue (correctly) that if the Goldbach conjecture is false, then it’s provably false in PA: if we have a counterexample, we can check that it is a counterexample with a finite calculation, which we can code into PA. The same holds for T_n\vdash\varphi^\mathbb{N}. Technically this is known as the Σ1-completeness of PA. Both “Tn proves φ” and “the Goldbach conjecture is false” are Σ1 statements, unlike “there are infinitely many prime pairs”, which is Π2. It’s quite conceivable that the prime pair conjecture could go either way without PA having a proof.

As we saw earlier, ΦT is equivalent to the collection of all statements of this form:

\text{``}T_n\vdash\varphi^\mathbb{N}\text{''}\rightarrow\varphi.

So in PA+ΦT, we can prove both \text{``}T_n\vdash\varphi^\mathbb{N}\text{''} and \text{``}T_n\vdash\varphi^\mathbb{N}\text{''}\rightarrow\varphi, for any φ in PAT. Modus ponens takes it from there!

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Year Zero

Awhile back, the BBC website History Extra had a post that included this tidbit:

AD 0… the date that never was

The AD years of the Christian calendar are counted from the year of Jesus Christ’s birth, and, as the number zero was then unknown to the west, Dionysius began his new Christian era as AD 1, not AD 0. …

This evoked the ire of the noted historian Thony Christie. In a post Something is Wrong on the Internet, he explained:

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Non-standard Models of Arithmetic 13

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MW: OK, back to the main plotline. Enayat asks for a “natural” axiomatization of PAT. Personally, I don’t find PAT all that “unnatural”, but he needs this for Theorem 7. (It’s been a while, so remember that Enayat’s T is a recursively axiomatizable extension of ZF.)

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First-Order Categorical Logic 6

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MW: An addendum to the last post. I do have an employment opportunity for one of those pathological scaffolds: the one where B(0) is the 2-element boolean algebra, and all the B(n)’s with n>0 are trivial. It’s perfect for the semantics of a structure with an empty domain.

The empty structure has a vexed history in model theory. Traditionally, authors excluded it from the get-go, but more recently some have rescued it from the outer darkness. (Two data points: Hodges’ A Shorter Model Theory allows it, but Marker’s Model Theory: An Introduction forbids it.)

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First-Order Categorical Logic 5

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JB: Okay, let me try to sketch out a more categorical approach to Gödel’s completeness theorem for first-order theories. First, I’ll take it for granted that we can express this result as the model existence theorem: a theory in first-order logic has a model if it is consistent. From this we can easily get the usual formulation: if a sentence holds in all models of a theory, it is provable in that theory.

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Epstein Relativity Diagrams

[This post is available in pdf format, sized for small and medium screens.]

Lewis Carroll Epstein wrote a book Relativity Visualized. It’s been called “the gold nugget of relativity books”. I wouldn’t go quite that far, but Epstein has devised a completely new way to explain relativity. The key concept: the Epstein diagram. (I should mention that Relativity Visualized is a pop-sci treatment.)

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