Thursday, April 26, 2012

Persons without infinities

There is a distant planet, Zeta, whose people are taught math without the concept of infinity. You see, unlike ours, the quantifiers (the existential ∃ and universal ∀) they have in their math are bounded, which means the variable governed by a quantifier ranges over a finite set (according to certain rules, explained below).

On Zeta — as opposed to here on Terra — they write

(∃Zix) instead of (∃x),   and

(∀Zjy) instead of (∀y).

We can think of the Z subindexes ranging over some ordered set ι of indefinite ordinals such that given two of them, i and j with j > i, we can find a k between them: i < k < j. (We can for our purposes assume them to be rational numbers, for example.)

Each Zi is thought of as some indefinitely large set: a set with an i-zillion elements. Zix means that Zi is large enough to contain x.

Furthermore, in formulas where a Zj occurs within the scope of a Zi, j > i. Such a Zj (also called a domain) is larger than Zi, but — and this is a key point — Zj doesn't have to include every element of Zi.

Removing the Zis from all their formulas gives our Terran formulas. There is a number of Zetan mathematicians though (the Platonists) who study this variation of math that looks like ours. But this number is small.

(The Zetans think in terms of the indefinitely large, not infinity, and the indefinitely small, not continuity.)

Their Axiom of Zillions (corresponding to our Axiom of Infinity) looks like this:

(∃Z0x)(∅∈x ∧ (∀Z1y)(y∈x → (y∪{y})∈x))

This axiom says that there is some indefinitely large set Z0 which contains a set x that contains the empty set, and from Z0 a larger indefinitely large set Z1 can be reached (that is, is in the scope of Z0) such that for its elements y that are also in the set x, the successor of y — defined as succ(y) = y∪{y} — is in x as well.

Now for Z0 and Z1 to be finite, we have to have a case where there is some Z1n such that succ(n)∈x but where this successor is not in Z1 itself: n∪{n}∉ Z1 — this is so x and Z1 can have some greatest element that breaks the progression to Terran infinity. (This is where it's the case that Z1 need not — and in this case must not — include all the elements Z0.)

If the Zs mentioned in the Axiom of Zillions can be superseded by larger Zs then there seems to be a clear sense in which all the numbers in the set x cannot be "all" the numbers (that is, in the Terran way of looking at math). We can get bigger and bigger Zetan numbers, some without an immediate successor or an immediate predecessor. Think of gaps in their numbers, and of very huge numbers out by themselves, all alone. (The Zetans are particularly fond of incredibly huge numbers. And incredibly small ones too, it turns out.)

Going further, in Zetan mathematics, they have their own Cantor's theorem: They also can prove their "integers" — they say there are a bazillion of them — cannot be put into a one-to-one matchup with their "reals" —there's a gazillion of them — which is really bigger). And their own Continuum Hypothesis, and so on ...

. . . . . .       .   ....  . ...        .

It turns out every theorem of Zetan math is a theorem of Terran math, and vice versa — the two maths are isomorphic (or congruent) — due to a theorem by Jan Mycielski*:

Zetan math     Terran math

* This post is based on §5 of and the references therein (e.g., see