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

**, which means the variable governed by a quantifier ranges over a finite set (according to certain rules, explained below).**

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

_{i}x) instead of (∃x), and

(∀Z

_{j}y) 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 Z_{i} is thought of as some indefinitely large set: a set with an i-zillion elements. Z_{i}x means that Z_{i} is large enough to contain x.

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

*Removing the Z _{i}s 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:

_{0}x)(∅∈x ∧ (∀Z

_{1}y)(y∈x → (y∪{y})∈x))

This axiom says that there is some indefinitely large set Z_{0} which contains a set x that contains the empty set, and from Z_{0} a larger indefinitely large set Z_{1} can be reached (that is, is in the scope of Z_{0}) 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 Z_{0} and Z_{1} to be finite, we have to have a case where there is some Z_{1}n such that *succ*(n)∈x but where this successor is not in Z_{1} itself: n∪{n}∉ Z_{1} — this is so x and Z_{1} can have some greatest element that breaks the progression to Terran infinity. (This is where it's the case that Z_{1} need not — and in this case must not — include all the elements Z_{0}.)

If the Zs mentioned in the Axiom of Zillions can be superseded by larger Zs then there seems to be a clear sense in which all the numbers in the set x cannot be "all" the numbers (that is, in the Terran way of looking at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 uni-duesseldorf.academia.edu/ManuelBremer/Papers/896122/Varieties_of_Finitism and the references therein (e.g., see jstor.org/discover/10.2307/2273942).