by | August 9, 2026 · 7:26 pm

Set Theory Jottings 30. Definability Tools

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Definability Tools

We will need some familiarity with the the “definability powers” of Peano arithmetic. Without giving an exhaustive treatment, here are some of the highlights.

Recursive Tupling Functions

We have a 1–1 correspondence ℕ×ℕ→ℕ. Lots of them! We’ll pick one in a moment; we’ll denote it by

(x,y)↦〈x,y〉

Here (x,y)∈ℕ×ℕ is an ordered pair, and 〈x,y〉∈ℕ is the number coding it.

We want 〈·,·〉 to be arithmetically definable. The following formula does the trick:

〈x,y〉= ½(x+y)(x+y+1)+y

It’s easy to see where it comes from. If you list the lattice points in ℕ×ℕ by “cross diagonals” (i.e., (0,0), (1,0), (0,1), (1,1), …), then the predecessors of (x,y) consist of a triangle, plus a “tail” lying along the diagonal. Count them up! You get the above formula. So 〈·,·〉 is not just arithmetic, but recursive.

You can iterate this to get recursive k-tupling correspondences ℕk→ℕ for every k. For example,

〈x,y,z〉= 〈〈x,y〉, z〉

The tupling functions allow us to translate formulas like this:

∀x1∀x2∃y1∃y2…

into this:

∀x∃y…

Using the tupling functions, we can extend the notions of arithmetic and hyperarithmetic to n-ary relations and functions. For relations this is obvious. For a function, we look at its graph: f(x̄) is arithmetic (respectively hyperarithmetic) if the relation f(x̄)=y is.

Our definition of ℒ2(PA) allowed quantification over set variables. Sometimes there are technical advantages to using function variables instead. These can be “translated away” in a mechanical fashion. Admittedly the details can get a little hairy; for example, try expressing ∀x f(g(x))=g(f(x)) using the relations φf(x,y) ≡ f(x)=y and φg(x,y) ≡ g(x)=y. I will use both freely in ℒ2(PA).

Recursion

Recursion is a basic tool. Here’s the simplest form. Suppose we have a function f:ℕ→ℕ, and let a be any number. Define F by

F(0) = a
F(n+1) = f(F(n))

If f is arithmetic (or hyperarithmetic), does that make F arithmetic (hyperarithmetic)? Here are two ways we can write a formula for F(x)=y, provided we can quantify over arbitrary finite sequences a0,…,ax:

F(x)=y ≡ (∃a0,…,ax)[
a0=a
∧ (∀z<x)(az+1=f(az))
∧ y=ax]

and

F(x)=y ≡ (∀a0,…,ax)[
a0=a
∧ (∀z<x)(az+1=f(az))
→y=ax]

So we have a choice of ∃ or ∀, good news for the hyperarithmetic case. We’ll see soon how to handle arbitrary finite sequences in PA.

More complicated forms of recursion? No problem. For example, say f(x,ȳ,z) is a function of n+2 variables and g(ȳ) of n variables. Define

F(0,ȳ) = g(ȳ)
F(n+1,ȳ) = f(F(n,ȳ),ȳ,n)

So we’re allowing the new value to depend on a bunch of parameters ȳ, on the argument n, and of course on the previous value F(n,ȳ). This scheme is called primitive recursion. It’s handled the same way as the simpler case.

With this in our toolbox, we can define any so-called primitive recursive function. That’s any function you get by starting with a few very basic ones (projection, constant functions, successor) and applying composition and primitive recursion as many times as you like. So-called “course of values” recursions, where the new value depends on all values previously computed, also are available, since we can code the list of previous values using the finite sequence technique (see below). Sets and relations can be coded via functions taking the values 0 and 1; then we can use such relations in boolean conditionals (or more generally “definition by cases”) when defining new functions.

Loosely speaking, primitive recursive functions coincide with the computable functions where you can “clearly see” that the computations always terminate. To get the full panoply of recursive functions, we add the so-called μ-operator. This defines a partial function x̄↦y, like so:

μy(φ(y,x̄)) = the least y such that φ(y,x̄)
undefined if there is no such y

It’s easy to express this in the language of PA:

μy(φ(y,x̄))=u ≡ φ(u,x̄)∧(∀z<u)¬φ(z,x̄)

Any function defined by applying the μ-operator to a primitive recursive relation is a partial recursive function. If a partial recursive function just happens to be total, it’s a recursive function. But we can’t (computably) tell in general if a partial recursive function is total—that’s the Halting Problem on steroids.

The “sequence trick” (see below) and the μ-operator together tell us that any partial recursive function is arithmetic. (That is, its graph is an arithmetic relation.) But these techniques go beyond this: if f and g are arithmetic or hyperarithmetic, then so is the F defined by the primitive recursion scheme or the μ-operator, even if f and g are horribly, hideously uncomputable.

Finite Sequences

As we’ve just seen, we need a way to code finite sequences of arbitrary length as single numbers. Gödel gave one technique: code (r1,…,rk) as p1r1···pkrk, where pi is the i-th prime. But this works only once we have a definition of exponentiation, and of the function i↦pi. Both these yield easily to recursion, but that requires a coding of finite sequences!

Gödel broke the vicious circle with the Chinese remainder theorem. This says that given any sequence of positive integers d1,…,dk, all pairwise coprime, and any sequence r1,…,rk with 0≤ri<di for i=1,…,k, there is an a such that a%di=ri for all i. Here % is the remainder function. We call the di’s divisors and the ri’s remainders.

The Chinese remainder theorem is actually stronger: it says that there is a unique such a satisfying 0≤a<d1···dk. Although we won’t need the stronger version, it falls right out of the easiest proof of the theorem. Let d=d1···dk. Write [d] for the set {0,…,d−1}, likewise for [di]. We have a mapping [d]→[d1]×…×[dk] defined by a↦(a%d1,…,a%dk). The mapping is injective because the di’s are pairwise coprime: if a%di=a′ %di for all i, then di|(a−a′) for all i and so d|(a−a′). Since the domain and codomain both have d elements, the mapping is surjective. qed.

So if we are given a finite sequence (r1,…,rk), we just have to find divisors di such that they are pairwise coprime and ri<di for all i. The sequence of divisors has to be “orderly”, in the sense that if we had to specify all the di’s individually, we wouldn’t have gained anything. Gödel chose the arithmetic progression di=bi+1, for a suitable b. It turns out that b works if it is a sufficiently large multiple of k!.

How do we use this to handle a quantifier like ∃(r1,…,rk)? Answer: first we define

β(a,b,i) = a%(bi+1)

Formally representing the remainder function is a piece of cake. Then

∃(r1,…,rk)…ri…

is equivalent to

∃b∃a…β(a,b,i)…

I’ve been a bit sloppy with notation. The ∃b∃a causes no hiccups. I did not write ∃(r1,…,rk)φ(r1,…,rk). That would suggest a concrete list of k variables. But we want the subscript i in ri to be variable as well. Gödel’s β function makes that possible. We’ll see an example next.

Bitstrings

Forcing in arithmetic uses bitstrings. First, we will regard a subset A⊆ℕ as an infinite bitstring, or equivalently, a function A:ℕ→{0,1}. Second, finite bitstrings will play the role of conditions: if p is a condition (i.e., finite bitstring) and an initial segment of A matches p, then we say A satisfies p. If the condition q extends p (or equals it), we write p≤q.

Suppose we have a sequence p0≤p1≤…, defined inductively, starting with an arbitrary condition p0=c. In other words, we have a function f:ℕ×C→C, where C is the set of conditions, and for all n∈ℕ, pn+1=f(n,pn). Let A=⋃pn; if the pn’s stop growing at some point (i.e., for some n0, we have pn=pn0 for all n≥n0), then we make the rest of A all 0’s after that (i.e., n∉A for all n past the end of pn0). Question: if f is arithmetic, or implicitly defined, or hyperarithmetic, can we say the same for A?

First, how do we code conditions? Using binary notation doesn’t quite do the trick: consider 0001 vs. 01, for example. Pairing a binary number with its length works: 〈k,b〉 where k is the length of the bitstring, and b is the bitstring in binary, padded out with leading 0’s if necessary.

A sketch of the formula ψS that represents x∈S:

ψS(x)≡ (∃ p0,…,pl)[
p0=c
∧ (∀n<l)[pn+1=f(n,pn)]
∧ x∈pl]

Let’s look at the individual pieces:

  • (∃ p0,…,pl): we make use of Gödel’s β function. Also, all the pn’s are conditions; we have to make this explicit. I omit details.
  • p0=c: c will be a constant, incorporated into our formula.
  • pn+1=f(n,pn): see below.
  • x∈pl: i.e., bit x in pl is 1. This translates into a statement about the size of a remainder: b has a 1 in position x iff b=2x+1·q+r with 2x≤r<2x+1.

Now, how about pn+1=f(n,pn), or in general, w=f(u,v)? If f is an arithmetic function, defined by a formula φ(w,u,v), then we just transcribe φ into our definition ψS(x) for A. So if f is arithmetic, then so is A. (Because of c, we have a definition of A for each initial condition c, with A satisfying c.)

Next, say f is implicitly defined, say by φF; F is a new function symbol added to ℒ(PA). (So φF is a closed formula of ℒ(PA+F).) Transcribing φF into the sketch above gives us a “paired” implicit definition. That is, {(f,A)} is definable by a closed formula in ℒ(PA+F+S), where we’ve added two new symbols. As you can imagine, it’s easy to combine f and A into a single function (or set), but that’s not the same as having an implicit definition for A by itself.

Finally, if f is hyperarithmetic, then so is A. We transcribe the Σ11 or the Π11 formula for f into the sketch, getting a formula ψ(x) in ℒ2(PA). The function (or set) quantifier can be migrated to the front, using basic facts of logic. So we have a Σ11 and a Π11 formula defining A.

Summary: if f is arithmetic, so is A; if f is hyperarithmetic, so is A; but if f is implicitly defined, then the best we can do is a paired implicit definition of A with f.

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by | August 5, 2026 · 5:37 pm

Set Theory Jottings 29. Definability

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Three Kinds of Definability

A⊆ℕ is arithmetic if there is a formula φ(x) in ℒ(PA) such that

For all n [n∈A if and only if ℕ⊧φ(n)]

Tarski’s undefinability theorem says that truth in ℕ is not arithmetic: the set of Gödel numbers of sentences of ℒ(PA) that hold in ℕ is not an arithmetic set. Using the Quine bracket notation where ⌜ψ⌝ is the Gödel number of ψ, we can say: there is no φ(x)∈ℒ(PA) such that for all closed ψ∈ℒ(PA),

ℕ⊧ψ if and only if ℕ⊧φ(⌜ψ⌝)

On the other hand, Tarski gave a recursive definition of satisfaction for first-order logic: if ℒ is a language and M is a structure for ℒ, then M⊧φ∧ψ if and only if M⊧φ and M⊧ψ, etc. We can formalize this for Peano arithmetic by adding a new predicate letter, say T, to ℒ(PA). I’ll write ℒ(PA+T) for this language. We can now express Tarski’s recursive definition this way:

T(⌜ψ⌝) ↔
⌜ψ⌝ = ⌜s=t⌝ ∧ s and t are closed terms with equal values
∨ ⌜ψ⌝ = ⌜¬α⌝ ∧ ¬T(⌜α⌝)
∨ ⌜ψ⌝ = ⌜α∧β⌝ ∧ T(⌜α⌝) ∧ T(⌜β⌝)
∨ ⌜ψ⌝ = ⌜∃x φ(x)⌝ ∧ ∃n T(⌜φ(n)⌝)

This is the outline of a sentence in ℒ(PA+T). With a large but bounded amount of work, it can be translated completely into the formal language. I will only hint at why this is true—a convincing explanation would take many pages. You can find that in many textbooks, and my  Logic Notes also have a bit to say. Anyway: one can show that Peano arithmetic can represent any recursive function. It follows that PA can handle all purely syntactical matters. For example, “n is a Gödel number”; or “n is the Gödel number of a formula of the form α∧β, and k is the Gödel number of α and l is the Gödel number of β’’; or “s is the Gödel number of a closed term whose value is v.”

Say A⊆ℕ. (ℕ,+,·,<,A) (or (ℕ,A) for short) is a structure for ℒ(PA+T), if we interpret T(n) as meaning n∈A. Then our sentence holds in (ℕ,A) precisely when A is the set of Gödel numbers of true sentences ψ of ℒ(PA).

I generalize. Write ℒ(PA+S) for ℒ(PA) augmented with a new predicate letter S. (ℕ,A) is a structure for ℒ(PA+S), using A to interpret S. Say we have a class 𝒜 of subsets of ℕ, and a closed formula ψS in ℒ(PA+S). Suppose that

For all A⊆ℕ [A∈𝒜 if and only if (ℕ,A)⊧ψS]

Then we say that 𝒜 is an arithmetic class.

For example, consider the formula

ψS ≡ ∀x (S(x)→S(x·x))

In other words, (ℕ,A)⊧ψS means that A is closed under squaring. The class 𝒜 determined by ψS contains many sets: the set of all squares, the set of all powers of a prime p, the set of numbers whose prime factors all belong to a fixed set of primes, etc.

If {A} is an arithmetic class, then we say that A is implicitly definable by the formula ψS defining its singleton class. So truth in ℕ is implicitly definable.

Observe that any arithmetic set is implicitly definable: just let ψS be ∀x[S(x)↔φ(x)], where φ(x) is the formula in ℒ(PA) defining the arithmetic set.

The class of arithmetic sets enjoys some nice properties: it is closed under intersection, union, and complement. This falls out just from using the logical connectives. The existential quantifier gives us another operation, projection. Let 〈y,z〉 be a recursive pairing function, that is, a recursive bijection from ℕ×ℕ to ℕ. If A is defined by φ(x), then the formula ∃y φ(〈y,z〉) defines the projection of the relation 〈y,z〉∈A onto its second coordinate. (If this relation is a function, then the projection is its range. Likewise, if the relation is a partial function, we can project on the first coordinate to find its domain.)

The class of arithmetic sets is precisely the smallest class of subsets of ℕ closed under intersection, union, complement, projection, and containing the singletons {0} and {1} and the graphs of the addition and multiplication operations. (Take a moment to see why this is true.)

As we will see later, the class of implicitly definable sets is less tidy. For one thing, it’s not closed under intersection.

We recover some of the nice features by passing to a yet more inclusive class, the hyperarithmetic sets. First we expand our language to so-called second-order arithmetic: instead of adding a new predicate letter, we allow variables ranging over subsets of ℕ. Quantification is allowed. I will use capital letters for subsets and lowercase for numbers1. I’ll write S(x) and x∈S to mean the same thing; you can decide which is vernacular. I’ll write ℒ2(PA) for second-order arithmetic.

Suppose A is implicitly definable via a formula ψS in ℒ(PA+S). Then A is definable in ℒ2(PA) in two ways, existentially:

∃S (ψS ∧ x∈S)

and universally:

∀S (ψS → x∈S)

since in both forms, the clause “ψS’’ guarantees that S is the set A. Note that each formula has a sole free number variable, x.

A formula in ℒ2(PA) is called a Σ11 formula if it has an existential set quantifier out in front, no universal set quantifiers, and as many number quantifiers as your heart desires. Likewise, a Π11 has a universal set quantifier out front, no existential set quantifiers, and as many number quantifiers as needed. Thus:

Σ11: ∃S ψ(S,x)
Π11: ∀S ψ(S,x)

where ψ(S,x) can contain number quantifiers in abundance, but no other set quantifiers.

A set is hyperarithmetic or Δ11 if it possesses both a Σ11 and a Π11 definition. (That is, Σ11 and a Π11 formulas with one free number variable, but no other free variables.)

An example: the set of even numbers, arithmetically defined by a formula φ(x)∈ℒ(PA), implicitly defined by a formula Φ(S)∈ℒ(PA+S), and hyperarithmetically defined by formulas α(u)∈Π11 and ε(u)∈Σ11.

Arithmetic definition: φ(x) ≡ ∃y (y+y=x)

Implicit definition: Φ(S) ≡ ∀x (S(x)↔∃y(y+y=x))

Π11 definition: α(u) ≡ ∀S [∀x(S(x)↔∃y(y+y=x))→S(u)]

Σ11 definition: ε(u) ≡ ∃S [∀x(S(x)↔∃y(y+y=x))∧S(u)]

The pattern is clear:

Arithmetic definition: φ(x) ≡ ∃y (y+y=x)

Implicit definition: Φ(S) ≡ ∀x (S(x)↔φ(x))

Π11 definition: α(u) ≡ ∀S [Φ(S)→S(u)]

Σ11 definition: ε(u) ≡ ∃S [Φ(S)∧S(u)]

We see from this that every arithmetic set is implicitly definable, and every implicitly definable set is hyperarithmetic.

The set of Gödel numbers of true sentences in ℒ(PA) provides an example of an implicitly definable set that is not arithmetic, as we’ve just seen. Feferman’s theorem provides an example of a hyperarithmetic set that is not implicitly definable.

[1] So this is a two-sorted first-order language. Alternately, add predicates set and number, and treat ∃X… as vernacular for ∃x(set(x) ∧ …), etc.

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by | August 2, 2026 · 4:45 pm

From Kepler to Ptolemy 26

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The Vicarious Hypothesis

Part II of the Astronomia nova bears the title “Imitation of the Ancients”. Kepler obtains a model for Mars with an eccentric circle and an equant. But the eccentricity is not bisected. Instead, the center is closer to the equant than to the sun. See the figure below. (Notation: S=Sun, C=Center, E=Equant, r=radius, eS=SC/r, and eE=EC/r.)

The Vicarious Model

As this post is a bit long, let me highlight the main points.

  • The general model has parameters eS and eE. We always have eS+eE=2e, where e is the eccentricity.
  • We get the best fit for heliocentric longitudes using the ratio eS:eE=5:3. This is quite close to Kepler’s vicarious hypothesis.
  • We get the best fit for latitudes and “off to the side” observations with eS=eE, the famous “bisection of eccentricity”. It’s longitudes or latitudes, you can’t have both! At least with a circular orbit.
  • Kepler determined the parameters for his vicarious hypothesis by an intricate double iteration, using four acronychal observations.
  • The whirlpool force hypothesis requires bisection of eccentricity.
  • Although Kepler ultimately rejected the vicarious hypothesis as the correct model, he continued to use it to compute heliocentric longitudes. This is an implicit appeal to his “zeroth law”.

A Modern Perspective

I will refer to the scheme of the figure above as the vicarious model; Kepler used this term for a particular choice of parameters. By the “elliptical model”, I mean the usual elliptical orbit with the planet obeying the area speed law. I assume throughout that the period is the same for both models.

Before considering what Kepler wrote and did, we’ll examine the two models with modern techniques.

Suppose all we care about are heliocentric longitudes. We want to choose the parameters in the vicarious model to match these as accurately as possible. We don’t care about latitudes or distances. Observationally, this means we look only at acronychal measurements of the geocentric longitude. For as noted in post 24, when the planet is in opposition, the heliocentric and geocentric longitudes are equal.

What parameters will make the vicarious model best approximate the elliptical model for heliocentric longitudes? Evans and Linton (p.179) derive the answer: eS+eE=2e where e is the eccentricity of the ellipse, and eS:eE=5:3.

Evans obtains this result several ways. He imposes three conditions on the motion. Let P be perihelion, and Q quadrature (i.e., the point such that PSQ is a right angle). First condition: the time from perihelion P to Q must be the same for both models. To an excellent approximation when e is small, this implies that eS+eE=2e.

Second condition: the angular speed at aphelion must be the same for both models. This gives the equation

\frac{1-e_E}{1+e_S}=\frac{\sqrt{1-e^2}}{(1+e^2)^2};

retaining only terms up to order e2, this becomes

eS+eE = 2e+2eeS−5/2e2

Third condition: the angular speed at perihelion must be the same for both models. By similar reasoning, we have

\frac{1+e_E}{1-e_S}=\frac{\sqrt{1-e^2}}{(1-e^2)^2};

approximated to

eS+eE = 2e−2eeS+5/2e2

So we have three equations: eS+eE=2e, and the aphelion and perihelion angular speed results. Any two of these equations yields eS:eE=5:3.

Evens gives another derivation, using an expression for the heliocentric longitude measured from perihelion as a function of time. (Linton gives essentially the same derivation, but using aphelion instead of perihelion.) Let this be θ, and let ω be the average angular speed, i.e., 2π/T. Up to second order in e, the expression is

θ(t) = ωt+2e sin ωt+5/4e2 sin 2ωt

for the elliptical model, and

θ(t) = ωt+(eS+eE) sin ωt+½eS(eS+eE) sin 2ωt

for the vicarious model. So first: when eS+eE=2e, the expressions agree to first order in e. Second: when also eS:eE=5:3, we have eS=5/4e, and so they agree to second order.

In short, the vicarious model gives very good heliocentric longitudes when eS:eE=5:3. To quote Evans: “In fact, with the precision achievable with naked-eye observations even in Kepler’s day, there is no perceptible error in the angular position anywhere around the orbit—as Kepler himself verified.”

Okay, what if we do care about distances and latitudes? Geocentric latitudes provide a means to determine distances. I’ll explain this in more detail below, but intuitively: the closer you are to Mars, the bigger the latitude looks. An optical effect, just like approaching a vertical pole and seeing it look taller. Viewing Mars out of opposition, “from the side”, furnishes another way to check actual distances.

Perihelion and aphelion are minimal and maximal Sun-Mars distances, so a vicarious model that tries to do a good job with distances will share the apsidal line with the elliptical model. That means they will also have the same center and same r, where r is the vicarious radius and the elliptical semi-major axis. In the elliptical model, the Sun-Mars distance at aphelion is r+e and at perihelion is r−e. In the vicarious model, these distances are respectively r+eS and r−eS. Conclusion: the vicarious model gives the correct Sun-Mars distance at aphelion and/or perihelion only when eS=e. In other words, the eccentricity must be bisected.

The upshot: using the scheme of the above figure, you are faced with Scylla and Charybdis. You can have an excellent model for finding heliocentric longitudes at any time, all around the orbit. Just choose eS:eE=5:3. Or you can have a model for distances that is accurate at aphelion and perihelion. Choose eS=eE=e. You can’t have both.

One more observation: if we do bisect the eccentricity, then the two expressions for θ(t) will first disagree at order e2. That is, in the term with the factor sin 2ωt. The maximum (absolute) value occurs when 2ωt=±π/2. Since 2ω=4π/T, that means at t=±T/8. For small eccentricity, this is close to the octants.

What Kepler Did

After some preliminaries, Kepler gets down to business in Chapter 16 of the Astronomia nova. It takes four parameters to specify a vicarious model: the two eccentricities eS and eE, the direction of aphelion, and a time when the planet is at aphelion (known as the epoch). Kepler used four acronychal observations to derive the parameters.

Determining the Vicarious Parameters

The left of the figure above is Kepler’s diagram; the right side removes some clutter. A is the sun, B is the center of the orbit, and C is equant. D, E, F, and G are the four acronychal observations. They give us heliocentric longitudes: the directions of the lines AD, AE, AF, and AG. The directions from the equant (CD, CE, CF, and CG) are called mean longitudes. Since we know the times of the observations, we would know the mean longitudes if we knew the epoch and the direction of aphelion. That’s because the planet moves uniformly when viewed from the equant, and we know the period. Referring to the right side of the figure, you can see that we’d have two angles of the triangle AGC; of course, this also holds for the other three triangles. Temporarily chose the distance unit so that AC has length 1. Then we can solve all these triangles and find the positions D, E, F, and G.

Kepler imposed two constraints. First, the four acronychals must lie on a circle. This is easily checked, since a quadrilateral is cyclic if and only if opposite angles add to 180°. Also, the center B of the circle must lie on the line segment AC. If these are met, then we can determine both the radius and the center of the circle. That gives us the two parameters of the vicarious model, AB/r=eS and CB/r=eE.

Unfortunately, there is no direct way to find the epoch and direction of aphelion from the acronychals. Kepler proceeded iteratively. Starting with an estimate for these two quantities, he computed the positions DEFG. If these failed the circle requirement, he adjusted the direction of aphelion. “Repeat until done”, i.e., until DEFG lie on a circle. Gingerich (Ch.22) calls this the “inner iteration”. Then Kepler checked that B lay on the line segment from A to C. If not, he adjusted the epoch. This is the “outer iteration”, since the aphelion direction must now be recomputed. Eventually the process converged on values with both requirements satisfied. Kepler’s final result: eS:eE=18564:11332≈4.91:3.

Gingerich (Ch.22) and Voelkel (pp.106–107,111,114–121) present the whole messy story, based on Kepler’s copious manuscripts. In contrast to the account in the Astronomia nova, Kepler began in 1600 by tackling the earth’s orbit (i.e., revising Tycho’s solar theory). In the same year he made his first assault on the orbit of Mars. The two investigations continued in a tangled fashion, with interruptions, until sometime in 1602.

Downfall: Down but not Out

In Chapter 18, Kepler proclaims victory:

You see then, O studious reader, that the hypothesis found by the method developed above, is able in its calculations to account, in turn, for the four observations upon which it was founded, but also to comprehend all the other observations within two minutes…

And Chapter 19 begins:

Who would have thought it possible? This hypothesis so closely in agreement with the acronychal observations, is nonetheless false…

Chapters 19 and 20 then show, “with great thoroughness and almost masochistic delight” (Koestler (p.322)) that the vicarious hypothesis disagrees with geocentric latitudes and observations out of opposition.

Geocentric Latitude

We’ve already seen all the ingredients. First, latitudes. Consider the Sun-Earth-Mars triangle (see figure above). At opposition this lies in a plane perpendicular to the earth’s orbit (the ecliptic). The inclination of Mars’s orbital plane is the angle Earth-Sun-Mars. The geocentric latitude is the supplement of the angle Sun-Earth-Mars. If you know the Earth-Sun distance, then you can solve the triangle and get the Sun-Mars distance.

Kepler does this in Chapter 19, and concludes that the Martian eccentricity is between 0.08000 and 0.09943; his vicarious hypothesis sets it at 0.11332. He also notes that “combined eccentricity” (i.e., from the sun to the equant) is about 0.18564; half that is 0.9282, or just about the mean between 0.08000 and 0.09943.

Chapter 20 looks at some longitudes when Mars at perihelion and aphelion, but Earth is “off to the side”—that is, away from opposition. Again the results refute the vicarious model, and fit a bisected eccentricity rather well.

Puzzling. Kepler anticipates the resolution: “the orbit of the star is not a perfect circle, but an oval…”

Although the vicarious hypothesis failed, it still provides Martian longitudes to an accuracy of 2′. Kepler uses it for this purpose in subsequent chapters. In other words, Kepler assumes that if the earth were in opposition, instead of off to the side, then the vicarious longitudes would be correct.

Note the implicit appeal to the zeroth law: No special treatment for the earth. The solar longitude of Mars can’t depend on Earth’s position. Okay, but why prefer the acronychal (opposition) data to the off-to-the-side data?

Answer: the longitudes in question are the Sun-Mars directions. An acronychal observation directly measures this, since the Earth-Mars and Sun-Mars directions are identical. This doesn’t hold for an off-to-the-side measurement. So we should trust the acronychal values, and the vicarious hypothesis gives those values with good accuracy around the whole of Mars’s orbit.

Feetnote

(1) The Mars model by Longomontanus, mentioned in post 24, used a Copernican epicyclet instead an equant (see post 20). But we can convert it into a nearly equivalent equant model; Evans shows how to translate the parameters. Under this transformation, the Longomontanus model had eS:eE=5:3.

(2) Kepler complains in Chapter 16:

If this wearisome method has filled you with loathing, it should more properly fill you with compassion for me, as I have gone through it at least seventy times at the expense of a great deal of time, and you will cease to wonder that the fifth year has now gone by since I took up Mars, although the year 1603 was nearly all given over to optical investigations.

Gingerich (Ch.21) wondered why it took Kepler 70 iterations. He programmed it, and it took the computer only 9 trials! He guessed initially (and wrongly) that “Kepler was horribly plagued by numerical errors”. But when he gained access to Kepler’s manuscripts (Ch.22), he found two causes: bad data, and Kepler’s treatment of redundant observations. Gingerich writes:

Thus we see that in the vicarious orbit solution Kepler worked always with the same four oppositions, but the results were repeatedly tested against additional oppositions. In the course of five years’ work, the reduction of the basic data was continually improved. His 70 iterations were spent (probably) in five separate solutions differing only in the values chosen for the initial times and angles.

Gingerich also notes a fine point I glossed over in my account:

Tycho Brahe had no operational way of knowing precisely when Mars was at opposition. Tycho’s raw observations only approximated the opposition places, and Kepler was obliged to correct and interpolate them to obtain the acronychal positions he required.

(3) The whirlpool force explanation for the equant requires bisecting the eccentricity (see post 4 and post 20). So in considering the vicarious model at all, Kepler tempered his commitment to this hypothesis.

Voelkel (pp.107–111) found confirmation in a document Kepler wrote around 1602. We’ve noted how the inverse distance speed law works tolerably well within an orbit, but fails when comparing different orbits (see post 20). In the document, Kepler notices his earlier errors in this matter.Voelkel suggests that this realization freed Kepler to study other divisions of eccentricity. It must have come as a relief when the latitude and off-side computations indicated that eccentricity should be bisected.

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by | July 29, 2026 · 11:02 am

Set Theory Jottings 28. Forcing Roadmap

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Cohen introduced forcing in 1963 to prove the independence of the continuum hypothesis from ZFC. The next few years witnessed simplifications, other applications, and other perspectives. Despite the simplifications, forcing in set theory comes encrusted with technicalities.

In set theory, the basic forcing concepts occur entangled with the cumulative hierarchy, the level-by-level build-up of the universe of sets from the empty set. This makes for a messy situation. The most “barnacle-free” setting for forcing is Peano arithmetic. Only three levels demand attention:

  1. Numbers.
  2. Sets of numbers.
  3. Classes of sets of numbers. (I use the word ‘class’ just for smoothness; these are bona fide sets in ZF, not proper classes.)

Both in ZF and in the cases treated here, forcing deals with definability. Gödel had shown that if all sets are constructible (V=L), then AC and GCH hold. So Cohen started off by trying to show the relative consistency of V≠L. As we’ve seen, constructibility is all about definability.

We’ll look at four kinds of definability for Peano arithmetic (or really three):

  • Arithmetic sets of numbers. A set is arithmetic if it can be defined by a formula of ℒ(PA).
  • Implicitly definable sets of numbers. A set is implicitly definable if it can be defined by a formula in the language of PA augmented by a predicate symbol (say S) standing for the set. I’ll write ℒ(PA+S) for the augmented language.
  • Arithmetic classes of sets of numbers. A class is arithmetic if it can be defined by a formula of ℒ(PA+S). Implicit definability is the special case where a singleton {A} is an arithmetic class.
  • Hyperarithmetic sets of numbers (aka Δ11 subsets). A set is hyperarithmetic if it can be defined by a certain kind of formula in the second-order language of PA, denoted by ℒ2(PA).

Using forcing, Feferman showed the existence of a hyperarithmetic set that is not implicitly definable, and Addison showed that the class of arithmetic sets is not an arithmetic class.

Our roadmap:

  1. We define the four kinds of definability.
  2. In the context of PA, we offer intuition for the notions generic set and forcing.
  3. We proceed with the formal development, and prove the Feferman and Addison theorems.
  4. We generalize the notions of forcing and generic sets.
  5. We turn to ZF set theory.

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by | July 26, 2026 · 3:38 pm

From Kepler to Ptolemy 25

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The heliometer (meridian line) at the Basilica of San Petronio

Earth’s Orbit

Ptolemy’s solar model gives the sun an eccentric orbit, with uniform speed—no equant. Both Copernicus and Tycho adopted the uniform speed for the earth’s/sun’s orbit. As a consequence, they end up with twice the correct eccentricity. (See post 8.)

The Ptolemaic system further disguises the earth’s orbit as the epicycles of the outer planets and the deferent of Venus. (Mercury is sui generis.) The epicycles have neither equants nor eccentricities, which amounts to giving the earth uniform speed in a Sun-centered circular orbit. Venus’s deferent has both an eccentric circle and an equant.

Kepler of course would have none of this. No special treatment for the earth! He gave it an equant, with the center of the orbit situated midway between the equant and the true sun. So Kepler assigned the earth half its former eccentricity. This is known as the “bisection of eccentricity”.

How does this matter for Mars? Earth is our observing platform. If you are mistaken about your own celestial position, you will inevitably screw up how you interpret the data.

Kepler devised an ingenious triangulation scheme to determine the earth’s orbit. Mars has a period of 687 days. Observations separated by that interval thus involve Earth-Mars vectors from the variable position of the earth to a fixed position for Mars. Mentally reversing the direction of the vector gives us essential information about the earth’s orbit. I will look at how Kepler used this insight below. First, though, some more historical context.

Voelkel writes:

Kepler undertook to write the Astronomia nova in October or November 1602, long before the discovery of the ellipse and elucidation of the area law… At the time he resolved to write the book, he had written to Herwart von Hohenburg that his work on Mars had allowed him to solve the problem of the orbit of the sun (or earth). And in many ways, the Astronomia nova was as much about the earth’s orbit as it was about Mars’s orbit. [Voelkel (p.212)]

Kepler develops his theory of the earth in the first few chapters of Part III. Part II contains the vicarious hypothesis. In fact, this almost inverts the timeline. Not only the zeroth law, but the whirlpool force hypothesis pushed Kepler to investigate the earth’s orbit.

Presenting the vicarious hypothesis first served a rhetorical purpose. Kepler could say, “See? I tried to make the traditional scheme work. The failure of my Imitation of the Ancients, and the way in which it failed, indicate that there is something amiss with the existing model of Earth’s orbit.”

This innovation with the earth’s orbit met with considerable resistance. Tycho disliked it, as did Kepler’s pen pal Fabricius. Tycho felt, “If it ain’t broke, don’t fix it!” Likewise Fabricius, who wrote, “We ought not heedlessly to depart from Tycho’s well-developed solar hypothesis.” As Voelkel puts it:

…this unprecedented innovation, clashing as it did with Tycho’s solar theory and meeting Fabricius’s deep skepticism, needed an unusually sure foundation, especially because it was the ultimate justification for his physical astronomy. [Voelkel (p.236)]

The disapproval lasted—more so than for the elliptical orbit. The astronomer Giovanni Cassini, using a so-called heliometer (aka meridian line) at the Basilica of San Petronio, finally confirmed Kepler’s bisection in 1655. Heilbron (p.102–112) recounts the story. He writes:

Many astronomers objected to a solar equant. Thus began what Flamsteed, writing a generation after its resolution, called a “controversy … of no small moment”. It was this great and obscure controversy between adherents of Ptolemy’s traditional solar theory and proponents of Kepler’s “bisection of the eccentricity”, with its Copernican associations, that Cassini proposed to settle at San Petronio.

…to put the point in a few words (they are those of Astronomer Royal Flamsteed), “the Suns Excentricity is bisected as the Copernicans affirme.” This connection of ideas—the bisection of the eccentricity implies that the earth is a planet—became commonplace among Copernicans.

The Great Martian Catastrophe

In 1593 Tycho observed Mars. It was in the wrong position, by about 10 Moon diameters! The Ptolemaic and Copernican tables gave longitudes differing by about 5°, in opposite directions. The values remained off for several weeks. Gingerich called this “The Great Martian Catastrophe”.

Fixing the earth’s orbit reduced the size of the “great Martian catastrophe” tenfold. Specifically, it reduced the maximum longitudinal discrepancy from about 5° to about 0.5°, or 30′. Kepler’s area speed law plus the elliptical orbit reduced it further to about 2′.

Other simplification flowed from the modified earth’s orbit. Kepler called it “the Key to a Deeper Astronomy”.

The Two Methods

OK, now let’s look at how Kepler determined the position of Earth’s equant. The old theory had the earth move uniformly around the center of the orbit, which is displaced from the sun. Effectively, this makes the equant and the center identical. Kepler replaced this with the center midway between the sun and the equant. The figure below illustrates the first method, from Chapters 22 and 23. In Chapter 22 he shows that the equant is not the same as the center.

Earth’s Orbit: First Method

In the figure, S, C, and E are the sun, the center, and the equant; and O1 and O2 are two positions of the earth. Recall his triangulation trick: if the time interval between O1 and O2 is a multiple of 687 days, then Mars will occupy the same position M both times. The times of O1 and O2 are chosen so that the two angles θ are equal. Since motion is uniform about the equant, angles about the equant are measured by times. If O1 and O2 are separated by equal times from the aphelion and perihelion respectively, then the two angles will be equal.

Kepler also needed the line EM to be perpendicular to the line of apsides (the diameter in the figure). Again this is achieved by picking the right epochs. Altogether, this method requires observations at very particular times. We see the importance of Tycho’s treasure trove.

Now suppose that E and C were the same. Then the diagram would be symmetrical about the line EM, and angles α and β would be equal. (Mentally slide the line EM left until E coincides with C.) But Kepler found that α exceeded β by more than one degree. “Witness the great difference” he exclaimed.

In Chapter 23 he computes the relative positions of S, C, and E. The directions O1M and O2M are known from observation. That gives us the angles α and β. The times of the observations give him the angle θ (see above). So all relevant angles are known. I won’t go through the intricate trigonometry that computes the position of E in the line of apsides from this data. But intuitively, it is clear it uniquely determines the position: sliding E from perihelion to aphelion continuously increases α up from zero and decreases β down to zero, so there is exactly one spot where they have the right values.

Kepler found that the eccentricity (CE over the orbital radius) was 0.01834. Tycho’s value for the eccentricity was 0.03584. Half that is 0.01792, close enough to 0.01834. The bisection of eccentricity confirmed!

Chapter 24 sets out with the bold declaration, “Now that we have once made a hazard of this, we are buoyed by audacity and will begin to move more free on this battlefield.” Kepler proposes to take triplets of observations, separated by multiples of 687 days, and from them find the distances of the equant to three points of the orbit. Three points determine a circle. “If a fourth observation will be at hand, it will serve as a test.”

Earth’s Orbit: Second Method

The figure above reproduces the diagram from Chapter 24. Here κ is the fixed position of Mars. The variable Earth positions are θ,η,ε,ζ. The dashed circle is the earth’s new orbit, with the new center β bisecting the eccentricity.

The observations give the Earth-Mars directions θκ, ηκ, εκ, and ζκ, but not the distances. Earth-Equant directions are given by the times (because uniform motion). Mars-Equant directions are given by Tycho’s theory of Mars (or by the vicarious hypothesis). Consider an Earth-Equant-Mars triangle, such as △ηακ. We have all the directions, and thus all the angles. The law of sines tells us all ratios of sides. In particular, it gives us the distance αη in terms of ακ. We have this for all four Earth positions, which means we have the distances αθ, αη, αε, αζ in terms of some common unit. (For another exposition of this material, see Stephenson (pp.53–55), Voelkel (pp.105–106), or Donahue (pp.45–50).)

The distances differed, again disproving the identity of the equant with the center. The results once more agreed with the bisection of eccentricity. Kepler now had empirical proof of one of his deductions from his whirlpool force hypothesis.

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by | July 22, 2026 · 12:36 am

Set Theory Jottings 27. V=L Implies GCH

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Now we turn to the proof that GCH holds in L. Reflection principles again come to the rescue. First note that the cardinality of Lα equals the cardinality of α for infinite α. (Just observe that the number of formulas with parameters from Lα is #Lα. So #Lα+1=#Lα. The rest is routine transfinite induction.)

Let’s just look at CH; this gives the flavor. Now, ω∈Lω+1, so the question is, how high up do we have to go in the constructible hierarchy to get all the constructible subsets of ω? Answer: no higher than Lω1, where ω1 is the first uncountable cardinal. Given this, CH holds in L: 𝒫L(ω)⊆Lω1 and #Lω1=#ω1=ℵ1, so #𝒫L(ω)≤ℵ1. The reverse inequality holds because #𝒫L(ω)>ℵ0 and ℵ1 is the next larger cardinal after ℵ0. (Remember that we have AC in L, so all cardinals are comparable. Also, these cardinality computations are done in L. For example, 𝒫L(ω) is uncountable in L; we don’t care about its cardinality according to V.)

Cohen is worth quoting (with a couple of tiny changes):

Let us attempt to give some intuitive justification for why all sets of integers are constructible by countable ordinals. If x⊆ω, x∈Lα, then one can ask what are the essential properties of α which imply that x∈Lα. Now x is determined by the truth values of the countably many statements “n∈x’’. For each n we can think of this as imposing one condition on α. Thus it is not unreasonable that these countably many conditions, if they can be satisfied by any α, can also be satisfied by a countable α. The mechanism for making this precise will be furnished by the Löwenheim-Skolem theorem which allows us to construct smaller sets having the same properties as larger sets.

Here’s the argument. Let x be a constructible subset of ω, belonging to Lβ. Recall from post 23 that the relation y=Lβ is definable (in fact Δ1ZF). In other words, there is a formula Λ(u,y) that says that u is an ordinal and y is the set Lu.

We now apply the reflection principle from post 25:

If K is a countable set containing a transitive subset K0, and Φ is a finite set of closed formulas in ℒK(ZF) (so allowing parameters from K), then there is a countable transitive set N⊇K0 reflecting all the formulas in Φ. In reflecting the formulas of Φ, the parameters from K are replaced with images under an ∈-isomorphism F; this ∈-isomorphism is the identity on K0.

Our set K is ω ∪ {ω,x,β,Lβ}. K0=ω. Our formulas Φ are:

Λ(β,Lβ)
x⊆ω
x∈Lβ

Let’s examine N along with the map F sending K into N. F is the identity on ω. So it’s also the identiy on x: F(x)⊆ω because “x⊆ω’’ is reflected, and n∈x ⇔ F(n)∈F(x) ⇔ n∈F(x). We also have

N⊧x∈F(Lβ) (reflection)
L⊧x∈F(Lβ) (absoluteness)
N⊧Λ(F(β),F(Lβ)) (reflection)
L⊧Λ(F(β),F(Lβ)) (absoluteness)

In other words, N reflects x∈Lβ, so x is in F(Lβ) according to N. But by absoluteness of ∈, x really is in F(Lβ). Likewise, β constructs Lβ, so by reflection F(β) constructs F(Lβ) in N. But by absoluteness of Λ(u,y), F(β) really does construct F(Lβ). (When I say “really” I mean according to L, although it’s also true according to V.)

So x∈F(Lβ)=LF(β). Since F(β) is an ordinal contained in the countable transitive set N, it’s countable. We’ve now shown that x is constructed by the countable ordinal F(β), as claimed.

Again I quote Cohen (editing to match our notation):

One should compare the above proof with the intuitive motivation. The “collapsing” of the set K by the isomorphism F means that we have extracted from the ordinal β all the lower ordinals which played a role in the formation of x in Lβ, and then discarded the other lower ordinals. The result is a much smaller ordinal F(β) which also constructs x.

For the full GCH, we need an extension of the reflection principle. Instead of a countable K, we have to allow K to have arbitrary cardinality, with the reflecting M⊇K having the same cardinality. The proof of this is not difficult, and the proof that V=L implies GCH is pretty much the same.

Next up: Forcing!

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by | July 19, 2026 · 8:53 am

From Kepler to Ptolemy 24

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Here Comes the (True) Sun

I mentioned earlier “Kepler’s zeroth law”, a corollary to the motto:

Refer everything to the true sun, and not the mean sun. Nor should the earth receive any special treatment.

Kepler plucked several fruits from this tree. Here are two fruits of the zeroth law.

Apsides

Wrong Apsides

The solid circle is the true orbit for Mars. The dashed circle has the wrong apsidal line, passing through the mean sun instead of the true sun. X and Y are the positions of Mars in the two models at the same time. Note that the longitudes of X and Y differ little, whether viewed from the true sun or the mean sun. But the distances differ significantly.

Gingerich summarizes the situation nicely:

[T]he eccentricities assigned by Ptolemy to each of the planets are essentially vector sums of the earth’s and the planet’s eccentricities, with the result that in each case the line of apsides is slightly wrong…Copernicus did not bother to sort out the individual eccentricities, so each planetary eccentricity implicitly combines the earth’s. In addition, his apsidal lines are drawn through the mean sun… [Gingerich, Chap.19 “Kepler’s Place in Astronomy”]

So far as heliocentric longitudes are concerned, the wrong apsidal line won’t pose too serious a problem. See the figure above, which shows that solar distances are not so forgiving.

To understand this better, we need to look at acronychal risings. These are observations taken when the planet (here Mars) is in opposition to the sun. (See the figure below. ‘Acronychal’ means ‘night rising’.)

Key point: for an acronychal observation, the heliocentric and geocentric longitudes are the same. The astronomer can directly measure only geocentric longitudes. Except at opposition and conjunction, deducing the heliocentric longitude requires an assumption about the earth’s position. At conjunction, the brightness of the sun makes the planet invisible.

Kepler analysed the situation in Chapters 5 and 6. (These make slow reading; Stephenson (pp.31–39) gives a detailed discussion.)

This was a “must pass” test for the switch to the true sun. Tycho’s chief assistant Longomontanus (aka Christian Severinus) had worked out a model of Mars’s orbit that accounted for the acronychal data with an accuracy of 2 minutes, for a period of over 40 years. As Kepler explains in Chapter 7:

A hypothesis was invented which, it was proclaimed, represented all these oppositions within a distance of two minutes in longitude… It was only in the latitude at achronychal positions and also the parallax of the annual orb [i.e., observations out of opposition] that Christian got stuck… At the beginning there was great controversy between us as to whether it were possible to set up another sort of hypothesis which would express to a hair’s breadth so many positions of the planet, and whether it were possible for the former hypothesis to be false despite its having accomplished this so far over the entire circuit of the zodiac.

Chapter 6 showed the near agreement of the true sun and mean sun models for acronychal longitudes. But observations “from the side” (i.e., out of opposition) could amount to as much as 1° 20′, fourteen times the error at opposition and easily observable.

Latitudes

In post 13, I noted (following Swerdlow) that poor data bore most of the blame for Ptolemy’s complicated latitude theory. But having the orbital planes pass through the mean sun is just asking for trouble. The figure below illustrates the problem.

Latitudes and the Mean Sun

The diagram shows a cross-section of the three-dimensional geometry. Oaphelion and Operihelion are Earth’s aphelion and perihelion, so the line through them is the ecliptic, and the mean sun is midway between Oaphelion and Operihelion. The solid line at an angle is the orbital plane of Mars, with the inclination greatly exaggerated.

Shifting Mars’s orbital plane to pass through the mean sun, while keeping the inclination the same, gives the dashed line. Mars shifts from M to M; exactly where M lands depends on the longitude theory. Viewed from Earth’s aphelion its predicted latitude would be too large, from Earth’s perihelion too small, and from other points in the earth’s orbit also incorrect (except for two points). To fix this problem, one might naturally change the inclination, resulting in the dotted line. This dotted line represents a plane passing through the true position of Mars (M) and the mean sun. However, this fixes things only at one position M; at other points in Mars’s orbit, like M′, you’d need a different inclination. We see the cause of the rocking orbital plane.

This scheme still has a lot of leeway, since it takes three (non-collinear) points to determine a plane; an infinite number of planes pass through M and the mean sun. But it is impossible to avoid a varying inclination, simply because the mean sun does not lie in the actual orbital plane of Mars. Any plane passing through M and the mean sun will intersect Mars’s orbital plane in a line, which means it can correct matters for at most one other point of Mars’s orbit.

Copernicus’s latitude theory did not follow this scheme, but was essentially a transcription of the Ptolemy theory into heliocentric terms. It suffered from an additional drawback: the inclination depended on Earth’s position. This violates Kepler’s zeroth law.

Kepler showed if the orbital plane of Mars passes through the true sun, then you can use a fixed inclination. (The line of nodes does rotate very slowly. Kepler knew this.) He demonstrated this in three ways. I’ll outline only the first two.

Both involve finding the geocentric latitude for observations when this will equal the heliocentric latitude. When Mars is “at a limit”—that is, as far above or below the ecliptic as it gets—this will give the inclination.

Kepler noted that the highest accuracy is not required for these methods. Say M is Mars, and M′ is M projected onto the ecliptic. The distance MM′ doesn’t change that much for M near a limit.

Determining the Inclination, First Method

For the first method, suppose that the earth and the sun are equidistant from Mars, hence likewise from M′. Then MM′ appears the same whether viewed from the sun or the earth. So the geocentric and heliocentric latitudes are equal. (See Kepler’s diagram, reproduced above. Here A=Earth, B=Sun, E=Mars (my M), and C=projection of E to the ecliptic (my M′).)

How do we know when Sun-Mars-Earth form an isosceles triangle? Given the ratio of the Sun-Earth and Sun-Mars sides, (i.e., the orbital radii), basic trigonometry tells us the Sun-Earth-Mars angle. It turns out that you can use any observation where that angle is between (roughly) 60° and 72°.

Determining the Inclination, Second Method

For the second method, Kepler noted that when two planes cut one another, any two lines drawn in the respective planes to a point on the line of intersection, and perpendicular to that line, always include the same angle. (See the diagram above, also Kepler’s.) So if B is the earth, F is Mars, and D is its projection to the ecliptic, then ∠ DBF is the inclination. Therefore we need an observation when the earth lies in the line of nodes, Mars is at a limit, and the sun and Mars appear at right angles from the earth. For this method, we make no assumptions about the orbital radii. However, it requires very special observations.

Kepler consistently found an inclination close to 1.8° with all three methods and many observations. He concluded that not only is the inclination fixed, but it’s rather small. Tycho’s assistants had made a mess of things, finding not only that the orbit of Mars wobbled, but was also “fractured”, with a maximum latitude of 4.6° at one limit and 6.4° at the other.

I noted at the beginning of post 13 that Ptolemy “decoupled” longitudinal computations from latitudinal ones. Now, the speed of planet differs from the speed of its projection to the ecliptic. Moreover, the ratio of these two speeds is not constant. Ptolemy could safely ignore this issue because the effect is small; that’s true because because the inclinations are small. But with Tycho’s much greater accuracy, the matter could no longer be swept under the rug. (It’s known as the “reduction to the ecliptic”.) So in cleaning up the latitude theory, Kepler laid a crucial foundation stone for everything else.

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by | July 15, 2026 · 8:42 pm

Set Theory Jottings 26. Relative Consistency of V=L

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Let’s put it all together. Recall that Gödel proved three main results about L: Continue reading →

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by | July 8, 2026 · 2:33 pm

Set Theory Jottings 25. Mostowski Collapsing Lemma

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The Collapse of the Tacoma Bridge

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by | July 1, 2026 · 9:33 am

Set Theory Jottings 24. Reflection Principles

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Mount Hood Reflected in Mirror Lake (Public Domain)

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