Death of infinity

 Definition (Finite Rule):

A rule R is finite if its output for any input is determined by a bounded set of distinctions.

The death of infinity.
By John Gavel 

Lemma (Closure under Self-Application):

If R is finite and contractive (output is a proper subset/simplification of input's distinctions), then there exists a maximal set of distinctions D^* such that:

R(D^*) \text{ introduces no new distinctions beyond } D^*

This D^* is not "reached" — it is encoded in R from the start.

What "Convergence" Actually Is:

The map n \mapsto R^{(n)}(D_0) (where R^{(n)} means n applications) has the property that for some finite N , all n \geq N yield outputs indistinguishable under the rule's distinction set.

That N is not a temporal endpoint — it's the recognition threshold for the rule's closure.

The Euler/Basel Example, Recast

Take:

S = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \dots

Sloppy story: "Summing infinitely many terms gives \pi^2/6 ."

The corrected story:

We have a finite rule for generating terms: a_n = 1/n^2 .

We have a finite rule for summing: partial sums S_N = \sum_{n=1}^N a_n .

The closed form \pi^2/6 is not produced by infinity. It is already latent in the structural relationship between:

· The generating rule a_n = 1/n^2 

· The known identity \sum 1/n^2 = \pi^2/6 (via Fourier analysis, etc.)

The "infinite sum" notation is just a refusal to stop early — but the value is determined by the finite rules + known identities.

Thus:

1. A finite, contractive rule defines its own closure set D^* .

2. The sequence of applications is just enumeration, not generation.

3. The "limit" is just D^* — already present, not progressively built.

Category Error: Treating \lim_{n\to\infty} as a process rather than a description of closure.

Infinity in these contexts is not a engine of creation — it's a rhetorical device for "not stopping."

All the creative work is in the finite rules and their closure properties.

This is why you can know \sum 1/n^2 = \pi^2/6 without summing infinitely — because the equality is a structural fact about the rules, not a result of accumulation.

A finite, contractive rule determines a unique maximal set of distinctions closed under its application. What is conventionally called the "limit of iteration" is simply this closure set. No temporal process or infinite completion is required — only recognition of the rule's inherent boundedness.

This eliminates the "double move" entirely:

Finite rule → finite distinctions → closure encoded ab initio.

No infinity needed. No stabilization narrative. Just structural inevitability.