bibkey: kreh2015minimalsets
authors: Martin Kreh
year: 2015
title: "Minimal Sets, Journal of Integer Sequences 18 (2015), Article 15.5.3"
doi: null
url: https://cs.uwaterloo.ca/journals/JIS/VOL18/Kreh/kreh2.pdf
claim: "Definition 16. For a given set M ⊂ N define a sequence δⁿ(M) of sets recursively by δ⁰(M) := M, δ(M) := δ¹(M) := M \\ S(M), δⁿ⁺¹(M) := δ(δⁿ(M)) and let ηⁿ(M) := |S(δⁿ(M))|, η(M) := η¹(M). Conjecture 18. There are only countably many infinite sets M ⊂ N with η(M) ≤ η⁰(M). For all other sets we have ηⁿ(M) → ∞."
strata_touched:
  - D5/S1/Digit/KrehMinimalSetLayerGrowthRefutation
  - D5/S1/Digit/KrehMinimalSetCountabilityRefutation
license: citation-only
triage: anchor

Kreh’s minimal sets

Kreh orders positive integers by decimal-string subsequence: x ⊳ y means that the decimal string of x is obtained from that of y by deleting zero or more digits. The set S(M) consists of the elements of M that are minimal for this order.

Verified locator

• URL: https://cs.uwaterloo.ca/journals/JIS/VOL18/Kreh/kreh2.pdf
• Publisher PDF: page 14.
• Publisher TeX: https://cs.uwaterloo.ca/journals/JIS/VOL18/Kreh/kreh2.tex
• Definition 16 (Unicode transcription, preserving the printed superscripts): Definition 16. For a given set M ⊂ N define a sequence δⁿ(M) of sets recursively by δ⁰(M) := M, δ(M) := δ¹(M) := M \ S(M), δⁿ⁺¹(M) := δ(δⁿ(M)) and let ηⁿ(M) := |S(δⁿ(M))|, η(M) := η¹(M).
• Conjecture 18 (Unicode transcription, preserving the printed superscripts): Conjecture 18. There are only countably many infinite sets M ⊂ N with η(M) ≤ η⁰(M). For all other sets we have ηⁿ(M) → ∞.

Here N denotes the positive integers, x ⊳ y is the decimal-string subsequence order, and S(M) is the set of its minimal elements in M. The divergence result in KrehMinimalSetLayerGrowthRefutation refutes the second sentence: the set M* = {1, 10, 11} ∪ {110·10ʲ : j ≥ 0} has successive minimal-layer sizes 1, 2, 1, 1, ....

Countability sentence

The companion theorem D5/S1/Digit/KrehMinimalSetCountabilityRefutation.result refutes the first sentence for the same decimal-subsequence definitions. It states

¬ Set.Countable {M : Set ℕ |
  (∀ n ∈ M, 0 < n) ∧ M.Infinite ∧ eta M 1 ≤ eta M 0}

Kreh’s Theorem 14 and Example 15 already give the two-seed chain mechanism; Example 17 explicitly exhibits minimal-layer sizes 2, 1, 1, .... The additional argument is the powerset encoding and its uncountability deduction, not the original two-seed mechanism.

For arbitrary A : Set ℕ, set u(j)=16·10ʲ, T_A={u(2n) : n ∈ ℕ} ∪ {u(2n+1) : n ∈ A}, and F_A={1,6} ∪ T_A. The exact equalities minimal F_A={1,6} and minimal (peel F_A 1)={16} give finite layer sizes 2 and 1. Every member is positive, and the mandatory even indices make every F_A infinite. The equivalence u(2j+1) ∈ F_A ↔ j ∈ A makes A ↦ F_A injective. An assumed natural-number encoding of the displayed collection would therefore inject Set ℕ into ℕ, contrary to Cantor’s theorem.

This is an additional clause of Problems/kreh-2015-minimal-sets-conjecture-18.md, whose existing formal resolution binding remains anchored to the divergence theorem. It does not constitute a second distinct published question.

Literature scope

The 2015 source statement and its two-seed construction are literature-attested; the explicit powerset deduction above is repo-derived. The complete supplied text of Baoulina, Kreh and Steuding’s Deleting Digits, arXiv:1607.01548v1 (2016), contains no explicit resolution of Conjecture 18 or its countability sentence. Its structural section concerns unions and intersections of minimal sets. The 2017 version of record is identified by DOI 10.1017/mag.2017.6, but its publisher PDF returned HTTP 403; its full contents remain ASSUMED-UNVERIFIED. The negative finding is limited to the texts read, and neither historical novelty nor first-publication priority is certified.