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Infinity symbol II October 20, 2009

Posted by David Pierce in Uncategorized.
1 comment so far


infinity sign and hexagram

infinity sign and hexagram

Completions and the Archimedean property August 28, 2009

Posted by David Pierce in Uncategorized.



The main point of this article is to observe that an ordered


    subset with an upper bound has a least upper bound.

    sequence (with respect to the absolute value function) converges.

complete field. The complex numbers compose another such field,
Every valued field has a completion.

I have encountered at least one mathematician who seemed not to
confused completeness of valued fields with completeness of
ordered fields.

Possibly the distinction between ordered fields and valued fields
is like that between induction and completion: a distinction that
may be overlooked in one’s early education and then never
returned to.

At the end of his book Calculus, Michael Spivak constructs
the field of real numbers and proves its uniqueness (up to
first learned of the uniqueness of the real field. Spivak
Dedekind’s notion of a cut. Yet Spivak is disparaging of

I revisited the construction recently, in a course on
non-standard analysis at the Nesin Mathematics Village. If the
construction of the real numbers was going to be drudgery, I
wanted to see what more general results could be found in the

The Dedekind cut construction gives a completion to every

  1. The set c(A) is ordered by inclusion
    and is complete with
    respect to this ordering.

Definition 4 of Book V of Euclid’s Elements. Indeed, the
positive part of an Archimedean ordered Abelian group would seem
magnitude: the set is closed under addition, and under
subtraction of a lesser magnitude from a greater.

the completion of the underlying ordered set, and then extend the


Among Archimedean ordered Abelian groups, the group of integers
If A and B are complete dense ordered Abelian groups, with
positive elements a and b respectively, then there is a unique
ordered field of rational numbers embeds in A under the map
of the rational field, which is the real field.

Consequently, for every real number a greater than 1, the
additive group of real numbers is isomorphic to the
multiplicative group of positive real numbers under a map taking
shows that multiplication distributes over addition on the
complete ordered field, which must be unique, because the
underlying group is unique as a dense complete ordered Abelian

Infinity symbol April 8, 2009

Posted by David Pierce in Uncategorized.

Why is infinity denoted by a lemniscate?

Lemniscates (figure-eights)



[picture: an empty circle]

This heaven is so vast no message can stain it.
How may a snowflake exist in a raging fire?
Here are the footprints of the patriarchs.


(English text by Reps and Senzaki.)






Induction and recursion II May 28, 2008

Posted by David Pierce in Uncategorized.

Formal logic provides examples of structures that allow proof by induction, but not necessarily definition by recursion. Let us consider, for example, the propositional logic that is apparently due to Łukasiewicz. We first define the (well-formed) formulas of the logic. We start with some set of so-called propositional variables. Then the set of formulas is the smallest set of strings that contains these variables and is closed under certain formation rules, namely:

  1. the singulary operation converting a string S to its negation, ~S;
  2. the binary operation converting a pair (S, T) of strings to the implication, (S → T).

The set of theorems of the logic is the smallest set of formulas that contains all of the axioms and is closed under a certain rule of inference. The axioms are the formulas of any of three forms:

  1. (F → (G → F))
  2. ((F → (G → H)) → ((F → G) → (F → H)))
  3. ((~F → ~G) → (G → F))

Here F and G stand for formulas. The rule of inference is detachment (or modus ponens), the binary operation that converts a pair (F, (F → G)) of formulas into the formula G. As stated, this is only a partial operation; we can make it total by having it convert (F, H) to F whenever H does not have the form (F → G) for some formula G.
The “inductive” definition of theorems is thus similar to the inductive definition of formulas. In each case, we start with a set and close under one or more operations. Immediately, proof by induction is possible on the set so defined. For example, we can prove inductively that each formula features as many left as right brackets. A feature of theorems that is proved by induction will be given in a moment.
However, unlike the formation rules for formulas, our rule of inference for theorems is not obviously well-defined. To see that it is well-defined, we must establish “unique readability” of formulas, namely:

  1. each formula is exactly one of the following: a variable, a negation, or an implication;
  2. the formation rules are injective.

Here injectivity of the formation rules corresponds to injectivity of the successor-operation on the set of natural numbers. The partition of the set of formulas into variables, negations, and implications corresponds to the partition of the set of natural numbers into the successors and the initial number (zero or one, as one prefers).
Unique readability of formulas allows recursive definitions. Indeed, the definition of the rule of detachment can be understood as recursive: If we denote this operation by D, then we have

  1. D(F, P) = F for all propositional variables P;
  2. D(F, ~G) = F;
  3. D(F, (F → G)) = G;
  4. D(F, (H → G)) = F if H is not F.

Another example of a recursively defined function on formulas is a truth-assignment. Consider the set {true, false} as the two-element field {1, 0}. Given a function φ from the set of propositional variables into the this field, we have a unique function Φ on the set of all formulas that agrees with φ on the variables, that takes ~F to 1 + Φ(F), and that takes (F → G) to 1 + Φ(F) + Φ(F)Φ(G).
By induction, every theorem takes the value 1 under every truth-assignment. But the truth-assignment is not recursively defined on the set of theorems: it is recursively defined on the set of formulas. Since the rule of detachment is not injective, we do not automatically have recursively defined functions on the set of theorems.
As far as I know, we do not have an efficient algorithm to determine whether a given formula is indeed a theorem: this lack of an algorithm would appear to be connected to the non-injectivity of the rule of detachment. A formula carries within itself the method of its construction; but a theorem does not carry within itself its proof, and indeed it will have infinitely many proofs.

Dante’s universe as 3-sphere April 22, 2008

Posted by David Pierce in Uncategorized.

The quotations about the universe as an infinite sphere remind me of another kind of sphere that is beyond our usual powers of imagination.
In the one-hundred cantos of the Divine Comedy, Dante travels from the central point of the earth out into the heavens. He passes through the spheres of

  1. the moon,
  2. Mercury,
  3. Venus,
  4. the sun,
  5. Mars,
  6. Jupiter,
  7. Saturn,
  8. the fixed stars,
  9. the Primum Mobile.

In this ninth sphere, he sees another point, surrounded by circles or spheres. These spheres are angels, named from inside to outside:

  1. Seraphim,
  2. Cherubim,
  3. Thrones,
  4. Dominions,
  5. Virtues,
  6. Powers,
  7. Principalities,
  8. Archangels,
  9. Angels.

These are named in Canto XXVIII of the Paradiso, where Dante reports (text from the Princeton Dante Project):

16 I saw a point that flashed a beam of light
17 so sharp the eye on which it burns
18 must close against its piercing brightness.
19 The star that, seen from here below, seems smallest
20 would seem a moon if put beside it,
21 as when one star is set beside another.
22 As near, perhaps, as a halo seems to be
23 when it encircles the light that colors it,
24 where the vapor that forms it is most dense,
25 there whirled about that point a ring of fire
26 so quick it would have easily outsped
27 the swiftest sphere circling the universe.
28 This point was encircled by another ring,
29 and that by the third, the third by the fourth,
30 the fourth by the fifth, and the fifth by the sixth.
31 Higher there followed the seventh, now spread so wide
32 that the messenger of Juno, in full circle,
33 would be unable to contain its size.
34 And so, too, the eighth and ninth,
35 each one revolving with diminished speed
36 the farther it was wheeling from the first.
37 And that one least removed from the blazing point of light
38 possessed the clearest flame, because, I think,
39 it was the one that is the most intruthed by it.
40 My lady, who saw me in grave doubt
41 yet eager to know and comprehend, said:
42 ‘From that point depend the heavens and all nature.
43 ‘Observe that circle nearest it,
44 and understand its motion is so swift
45 because it is spurred on by flaming love.’
46 And I to her: ‘If the universe were arranged
47 in the order I see here among these wheels
48 I would be content with what you’ve set before me.
49 ‘However, in the world of sense we see
50 the farther from the center they revolve
51 the more divinity is in their orbits.

Beatrice then explains that the innermost rings of angels correspond to the outermost heavenly spheres: so n corresponds to 10-n in the lists above.

In a lecture given in Ankara a while back, Piergiorgio Odifreddi suggested (as I recall) that the two spheres or rather balls—of the heavens, and of the angels—should be considered as identified along the 2-spheres that are their boundaries, so that a 3-sphere is obtained, with Lucifer and God as antipodal points.
I note Dante’s description of the latter Point in lines 11–12 of Canto XXX:

1 About six thousand miles away from here
2 the sixth hour burns and even now this world
3 inclines its shadow almost to a level bed,
4 when, deep in intervening air, above us,
5 begins such change that here and there,
6 at our depth, a star is lost to sight.
7 And, as that brightest handmaid of the sun advances,
8 the sky extinguishes its lights,
9 even the most beautiful, one by one.
10 Not otherwise the victory that revels
11 in eternal joy around the point that overcame me
12 and seems enclosed by that which it encloses
13 little by little faded from my sight,
14 so that, compelled by seeing nothing and by love,
15 I turned my eyes to gaze on Beatrice.

As a symbol for a world that cannot be fully comprehended, the three-dimensional surface of a four-dimensional body may serve as well as something infinite in extent.

Induction and recursion March 6, 2008

Posted by David Pierce in Uncategorized.













