Naive Set Theory_1 / Naive Set Theory_2

Completed in 1960, Paul R. Halmos’ NAÏVE SET THEORY is an introductory textbook for aspiring mathematicians to the complexities of Set Theory. This work is a re-interpretation of the Theory, filtered by Dexter Sinister´s view.


Caption One

Naïve Set Theory
DEXTER SINISTER

WHAT PRAGMATISM MEANS.

Some years ago, being with a camping party in the mountains, I returned from a solitary ramble to find every one engaged in a ferocious metaphysical dispute. The corpus of the dispute was a squirrel — a live squirrel supposed to be clinging to one side of a treetrunk; while over against the tree’s opposite side a human being was imagined to stand. This human witness tries to get sight of the squirrel by moving rapidly round the tree, but no matter how fast he goes: the squirrel moves as fast in the opposite direction, and always keeps the tree between himself and the man, so that never a glimpse of him is caught. The resultant metaphysical problem now is this:

Does the man go round the squirrel or not?

He goes round the tree, sure enough, and the squirrel is on the tree; but does he go round the squirrel? In the unlimited leisure of the wilderness, discussion had been worn threadbare. Everyone had taken sides, and was obstinate; and the numbers on both sides were even. Each side, when I appeared therefore appealed to me to make it a majority. Mindful of the scholastic adage that whenever you meet a contradiction you must make a distinction, I immediately sought and found one, as follows: “Which party is right,” I said, “depends on what you practically mean by ‘going round’ the squirrel. If you mean passing from the north of him to the east, then to the south, then to the west, and then to the north of him again, obviously the man does go round him, for he occupies these successive positions. But if on the contrary you mean being first in front of him, then on the right of him, then behind him, then on his left, and finally in front again, it is quite as obvious that the man fails to go round him, for by the compensating movements the squirrel makes, he keeps his belly turned towards the man all the time, and his back turned away. Make the distinction, and there is no occasion for any farther dispute. You are both right and both wrong according as you conceive the verb ‘to go round’ in one practical fashion or the other.”

Although one or two of the hotter disputants called my speech a shuffling evasion, saying they wanted no quibbling or scholastic hairsplitting, but meant just plain honest English ‘round,’ the majority seemed to think that the distinction had assuaged the dispute. I tell this trivial anecdote because it is a peculiarly simple example of what I wish now to speak of as the pragmatic method.

Now the particular difference of temperament that I have in mind in making these remarks is one that has counted in literature, art, government, and manners as well as in philosophy. In manners we find formalists and free-and-easy persons. In government, authoritarians and anarchists. In literature, purists or academicals, and realists. In art, classics and romantics. You recognize these contrasts as familiar; well, in philosophy we have a very similar contrast expressed in the pair of terms Rationalist and Empiricist — Empiricist, your lover of facts in all their crude variety, Rationalist, your devotee to abstract, eternal principles.

No one can live an hour without both facts and principles, so it is a difference rather of emphasis; yet it breeds antipathies of the most pungent character between those who lay the emphasis differently; and we shall find it extraordinarily convenient to express a certain contrast in men’s ways of taking their universe, by talking of the Empiricist and of the Rationalist temper. These terms make the contrast simple and massive.

Against rationalism as a pretension and a method pragmatism is fully armed and militant by empiricism. But, at the outset, at least, pragmatism stands for no particular results. It has no dogmas, and no doctrines save its method; but only an attitude of orientation. The attitude of looking away from first things, principles, ‘categories,’ supposed necessities; and of looking towards last things, fruits, consequences, facts.

The difference between pragmatism and rationalism is now in sight throughout its whole extent. The essential contrast is that for rationalism reality is ready-made and complete from all eternity, while for pragmatism it is still in the making, and awaits part of its complexion from the future


Caption Two

Naïve Set Theory
DEXTER SINISTER

THIS STATEMENT IS FALSE.

Kurt Gödel translated this paradoxical sentence into the language of mathematics and then designed a way out of it. To understand Gödel’s work requires a bit of background on the state of mathematics in 1930 — following from the Set Theory of Georg Cantor and Giuseppe Peano, Bertrand Russell and Alfred North Whitehead produced a totalizing Axiomatic system to (once and for all) resolve all of mathematics in Principia Mathematica (1910 – 1913). In the book, they presented a deductive system starting from a limited number of axioms (rules) and proceeding through each theorem, statement and mathematical sentence by a set manipulation of symbols or calculus.

But, this correspondence between a deductive, Axiomatic system and its calculus can be viewed in reverse. The same results can be gathered by mathematical in-duction rather than by mathematical de-duction. So, where an Axiomatic definition of the set of whole numbers might be:

The set of whole numbers = { 0, 1, 2, 3 . . . }

Then, a non-Axiomatic, or more precisely named Naïve definition of the same would be:

Starting with 0, 1 is defined as the immediate successor of 0, and so on and so on and so on . . .

Every number here is recursively defined in order, one by one, and must be constructed as you proceed through the sequence. Then a remark about the infinite sequence is never a remark about the infinite sequence but always only a remark about its construction. In the Naïve view, (mathematical) truth does not exist waiting to be uncovered but rather, must be produced by practice.

Following this Naïve reasoning, Kurt Gödel intuited that the Axiomatic system presented by in Principia Mathematica was incomplete. To demonstrate, he designed a second set of numbers on top of the set of whole numbers which he called the Gödel Numbers. Gödel Numbers are single numbers which stand for sentences of numbers. So, the mathematical sentence

1 + 1 = 2

is assigned a Gödel Number, in this case “3”. So that

“3” = (1 + 1 = 2)

Numbers in Gödel’s logic become two things at one time — they are both numbers and Gödel Numbers. And he uses the alphabet of whole numbers {0, 1, 2, 3 . . . } to form mathematical sentences that are also mathematical sentences about mathematical sentences. So

“3” + “3” = (1 + 1 = 2) + (1 + 1 = 2)

All this formality then to (mathematically) prove that any system of mathematics always contains at least one sentence which cannot be proved either true or false — mathematics is incomplete. So the self-referential logical-roundabout sentence that started this digression
This statement is false.

(which is true (false)) would be restated by Gödel as

This statement is neither true nor false.

(which is neither true nor false.) Paul R. Halmos’ Naïve Set Theory (1960) completes this stream of thought with the simple, profound assertion that there is no set that contains every other set. And I quote quote quote:

We have proved, in other words, that nothing contains everything, or more spectacularly, there is no universe.

100 × 70,7 cm
lithographic proof print + 2 captions
Exhibited in Pole Shift, Berlin