Foundation Pinch Algebra Finite Not Onto Infinite

The finite-capacity veto

A finite set cannot be mapped onto an infinite one. This is a basic fact of set theory: if you have a function from a finite set to an infinite set, there will always be elements in the target set that the function never reaches. The theorem finite_not_onto_infinite in the framework's machine-checked library of formal theorems establishes exactly this, in full generality, for any finite domain and any infinite codomain. It is a statement about the impossibility of a certain kind of function, not about the size of any particular physical system.

The proof is a short argument by contradiction. Suppose a function from a finite set to an infinite set were surjective, meaning it hits every element of the target. Then the image of the finite domain would be the whole infinite target. But the image of a finite set under any function is finite. A finite set cannot equal an infinite one, so the assumption fails. The theorem is a direct consequence of the definition of finiteness and surjectivity, with no additional assumptions about the nature of the sets involved.

In the framework, this result is called the finite-capacity veto. It is used as a basic structural check: if a model claims to describe an infinite range of phenomena with only finitely many states or resources, this theorem shows the claim is inconsistent. The veto does not say that infinite phenomena do not exist, nor that finite models are always wrong. It only says that a finite description cannot cover an infinite range in the sense of a surjective function. The theorem is a tool for identifying a specific kind of overreach, not a general statement about the limits of knowledge.

The theorem also connects to a companion result about budgets. If each operation costs a positive amount and the total budget is finite, then only finitely many operations can be performed. This is a simple inequality: the number of operations times the cost per operation must be at most the budget. Together, these two results form a pair of constraints: one about the size of a set, the other about the size of a resource. Both are elementary, but they serve as the foundation for more complex arguments about what can and cannot be done within a finite framework.

/-- **F5.3.2**: A function that is not surjective cannot map finite sets
    onto infinite sets. (Basic set-theoretic obstruction.)
    This is the finite-capacity veto in its simplest form. -/
theorem finite_not_onto_infinite {α β : Type*} (f : α → β)
    [Finite α] (hβ : Infinite β) : ¬Function.Surjective f := by
  intro hsurj
  have : Finite β := Finite.of_surjective f hsurj
  exact not_finite β

/-- **F5.3.2**: A function that is not surjective cannot map finite sets
    onto infinite sets. (Basic set-theoretic obstruction.)
    This is the finite-capacity veto in its simplest form. -/
theorem finite_not_onto_infinite {α β : Type*} (f : α → β)
    [Finite α] (hβ : Infinite β) : ¬Function.Surjective f := by
  intro hsurj
  have : Finite β := Finite.of_surjective f hsurj
  exact not_finite β

/-- **F5.3.1/3.2**: If the cost per operation is positive and the budget is finite,
    only finitely many operations can be performed. -/
theorem finite_operations_from_budget {n : ℕ} {cost budget : ℝ}
    (hcost : 0 < cost) (hbudget : 0 ≤ budget)
    (hfit : n * cost ≤ budget) :
    (n : ℝ) ≤ budget / cost := by
  rwa [le_div_iff₀ hcost]

What this page does not claim

The theorem does not claim that infinite sets do not exist or that finite models are always inadequate. The theorem does not say anything about the physical size of any system, only about the cardinality of sets. The finite-capacity veto does not by itself rule out any particular physical theory; it only identifies a specific kind of logical inconsistency.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PinchAlgebra.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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