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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | bj-findisg 16601* | Version of bj-findis 16600 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 16600 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝜃 & ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) & ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜏 & ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) ⇒ ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) | ||
| Theorem | bj-findes 16602 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 16600 for explanations. From this version, it is easy to prove findes 4701. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
| ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) | ||
In this section, we state the axiom scheme of strong collection, which is part of CZF set theory. | ||
| Axiom | ax-strcoll 16603* | Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function on 𝑎, or equivalently a collection of nonempty classes indexed by 𝑎, and the axiom asserts the existence of a set 𝑏 which "collects" at least one element in the image of each 𝑥 ∈ 𝑎 and which is made only of such elements. That second conjunct is what makes it "strong", compared to the axiom scheme of collection ax-coll 4204. (Contributed by BJ, 5-Oct-2019.) |
| ⊢ ∀𝑎(∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
| Theorem | strcoll2 16604* | Version of ax-strcoll 16603 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
| ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
| Theorem | strcollnft 16605* | Closed form of strcollnf 16606. (Contributed by BJ, 21-Oct-2019.) |
| ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))) | ||
| Theorem | strcollnf 16606* |
Version of ax-strcoll 16603 with one disjoint variable condition
removed,
the other disjoint variable condition replaced with a nonfreeness
hypothesis, and without initial universal quantifier. Version of
strcoll2 16604 with the disjoint variable condition on
𝑏, 𝜑 replaced
with a nonfreeness hypothesis.
This proof aims to demonstrate a standard technique, but strcoll2 16604 will generally suffice: since the theorem asserts the existence of a set 𝑏, supposing that that setvar does not occur in the already defined 𝜑 is not a big constraint. (Contributed by BJ, 21-Oct-2019.) |
| ⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
| Theorem | strcollnfALT 16607* | Alternate proof of strcollnf 16606, not using strcollnft 16605. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑏𝜑 ⇒ ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)) | ||
In this section, we state the axiom scheme of subset collection, which is part of CZF set theory. | ||
| Axiom | ax-sscoll 16608* | Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. The antecedent means that 𝜑 represents a multivalued function from 𝑎 to 𝑏, or equivalently a collection of nonempty subsets of 𝑏 indexed by 𝑎, and the consequent asserts the existence of a subset of 𝑐 which "collects" at least one element in the image of each 𝑥 ∈ 𝑎 and which is made only of such elements. The axiom asserts the existence, for any sets 𝑎, 𝑏, of a set 𝑐 such that that implication holds for any value of the parameter 𝑧 of 𝜑. (Contributed by BJ, 5-Oct-2019.) |
| ⊢ ∀𝑎∀𝑏∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑)) | ||
| Theorem | sscoll2 16609* | Version of ax-sscoll 16608 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.) |
| ⊢ ∃𝑐∀𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 → ∃𝑑 ∈ 𝑐 (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑑 𝜑 ∧ ∀𝑦 ∈ 𝑑 ∃𝑥 ∈ 𝑎 𝜑)) | ||
| Axiom | ax-ddkcomp 16610 | Axiom of Dedekind completeness for Dedekind real numbers: every inhabited upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then Axiom ax-ddkcomp 16610 should be used in place of construction specific results. In particular, axcaucvg 8120 should be proved from it. (Contributed by BJ, 24-Oct-2021.) |
| ⊢ (((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ∧ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 < 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝐵) → 𝑥 ≤ 𝐵))) | ||
| Theorem | nnnotnotr 16611 | Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 857, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.) |
| ⊢ ¬ ¬ (¬ ¬ 𝜑 → 𝜑) | ||
| Theorem | ss1oel2o 16612 | Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4288 which more directly illustrates the contrast with el2oss1o 6611. (Contributed by Jim Kingdon, 8-Aug-2022.) |
| ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ 1o → 𝑥 ∈ 2o)) | ||
| Theorem | 3dom 16613* | A set that dominates ordinal 3 has at least 3 different members. (Contributed by Jim Kingdon, 12-Feb-2026.) |
| ⊢ (3o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧)) | ||
| Theorem | pw1ndom3lem 16614 | Lemma for pw1ndom3 16615. (Contributed by Jim Kingdon, 14-Feb-2026.) |
| ⊢ (𝜑 → 𝑋 ∈ 𝒫 1o) & ⊢ (𝜑 → 𝑌 ∈ 𝒫 1o) & ⊢ (𝜑 → 𝑍 ∈ 𝒫 1o) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑋 ≠ 𝑍) & ⊢ (𝜑 → 𝑌 ≠ 𝑍) ⇒ ⊢ (𝜑 → 𝑋 = ∅) | ||
| Theorem | pw1ndom3 16615 | The powerset of 1o does not dominate 3o. This is another way of saying that 𝒫 1o does not have three elements (like pwntru 4289). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.) |
| ⊢ ¬ 3o ≼ 𝒫 1o | ||
| Theorem | pw1ninf 16616 | The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7102), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7098. (Contributed by Jim Kingdon, 14-Feb-2026.) |
| ⊢ ¬ ω ≼ 𝒫 1o | ||
| Theorem | nnti 16617 | Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ω) ⇒ ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢 E 𝑣 ∧ ¬ 𝑣 E 𝑢))) | ||
| Theorem | 012of 16618 | Mapping zero and one between ℕ0 and ω style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
| ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (◡𝐺 ↾ {0, 1}):{0, 1}⟶2o | ||
| Theorem | 2o01f 16619 | Mapping zero and one between ω and ℕ0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
| ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝐺 ↾ 2o):2o⟶{0, 1} | ||
| Theorem | 2omap 16620* | Mapping between (2o ↑𝑚 𝐴) and decidable subsets of 𝐴. (Contributed by Jim Kingdon, 12-Nov-2025.) |
| ⊢ 𝐹 = (𝑠 ∈ (2o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2o ↑𝑚 𝐴)–1-1-onto→{𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥}) | ||
| Theorem | 2omapen 16621* | Equinumerosity of (2o ↑𝑚 𝐴) and the set of decidable subsets of 𝐴. (Contributed by Jim Kingdon, 14-Nov-2025.) |
| ⊢ (𝐴 ∈ 𝑉 → (2o ↑𝑚 𝐴) ≈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥}) | ||
| Theorem | pw1map 16622* | Mapping between (𝒫 1o ↑𝑚 𝐴) and subsets of 𝐴. (Contributed by Jim Kingdon, 9-Jan-2026.) |
| ⊢ 𝐹 = (𝑠 ∈ (𝒫 1o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹:(𝒫 1o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴) | ||
| Theorem | pw1mapen 16623 | Equinumerosity of (𝒫 1o ↑𝑚 𝐴) and the set of subsets of 𝐴. (Contributed by Jim Kingdon, 10-Jan-2026.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝒫 1o ↑𝑚 𝐴) ≈ 𝒫 𝐴) | ||
| Theorem | pwtrufal 16624 | A subset of the singleton {∅} cannot be anything other than ∅ or {∅}. Removing the double negation would change the meaning, as seen at exmid01 4288. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4286), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.) |
| ⊢ (𝐴 ⊆ {∅} → ¬ ¬ (𝐴 = ∅ ∨ 𝐴 = {∅})) | ||
| Theorem | pwle2 16625* | An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.) |
| ⊢ 𝑇 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) ⇒ ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → 𝑁 ⊆ 2o) | ||
| Theorem | pwf1oexmid 16626* | An exercise related to 𝑁 copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.) |
| ⊢ 𝑇 = ∪ 𝑥 ∈ 𝑁 ({𝑥} × 1o) ⇒ ⊢ ((𝑁 ∈ ω ∧ 𝐺:𝑇–1-1→𝒫 1o) → (ran 𝐺 = 𝒫 1o ↔ (𝑁 = 2o ∧ EXMID))) | ||
| Theorem | subctctexmid 16627* | If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.) |
| ⊢ (𝜑 → ∀𝑥(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝑥) → ∃𝑔 𝑔:ω–onto→(𝑥 ⊔ 1o))) & ⊢ (𝜑 → ω ∈ Markov) ⇒ ⊢ (𝜑 → EXMID) | ||
| Theorem | domomsubct 16628* | A set dominated by ω is subcountable. (Contributed by Jim Kingdon, 11-Nov-2025.) |
| ⊢ (𝐴 ≼ ω → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴)) | ||
| Theorem | sssneq 16629* | Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
| ⊢ (𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧) | ||
| Theorem | pw1nct 16630* | A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.) |
| ⊢ (∀𝑟(𝑟 ⊆ (𝒫 1o × ω) → (∀𝑝 ∈ 𝒫 1o∃𝑛 ∈ ω 𝑝𝑟𝑛 → ∃𝑚 ∈ ω ∀𝑞 ∈ 𝒫 1o𝑞𝑟𝑚)) → ¬ ∃𝑓 𝑓:ω–onto→(𝒫 1o ⊔ 1o)) | ||
| Theorem | pw1dceq 16631* | The powerset of 1o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.) |
| ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦) | ||
| Theorem | 0nninf 16632 | The zero element of ℕ∞ (the constant sequence equal to ∅). (Contributed by Jim Kingdon, 14-Jul-2022.) |
| ⊢ (ω × {∅}) ∈ ℕ∞ | ||
| Theorem | nnsf 16633* | Domain and range of 𝑆. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.) |
| ⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))) ⇒ ⊢ 𝑆:ℕ∞⟶ℕ∞ | ||
| Theorem | peano4nninf 16634* | The successor function on ℕ∞ is one to one. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 31-Jul-2022.) |
| ⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))) ⇒ ⊢ 𝑆:ℕ∞–1-1→ℕ∞ | ||
| Theorem | peano3nninf 16635* | The successor function on ℕ∞ is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.) |
| ⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))) ⇒ ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅)) | ||
| Theorem | nninfalllem1 16636* | Lemma for nninfall 16637. (Contributed by Jim Kingdon, 1-Aug-2022.) |
| ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o) & ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o) & ⊢ (𝜑 → 𝑃 ∈ ℕ∞) & ⊢ (𝜑 → (𝑄‘𝑃) = ∅) ⇒ ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑃‘𝑛) = 1o) | ||
| Theorem | nninfall 16637* | Given a decidable predicate on ℕ∞, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which 𝑄 is a decidable predicate is that it assigns a value of either ∅ or 1o (which can be thought of as false and true) to every element of ℕ∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.) |
| ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝑥 ∈ ω ↦ 1o)) = 1o) & ⊢ (𝜑 → ∀𝑛 ∈ ω (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o) ⇒ ⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o) | ||
| Theorem | nninfsellemdc 16638* | Lemma for nninfself 16641. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.) |
| ⊢ ((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o) | ||
| Theorem | nninfsellemcl 16639* | Lemma for nninfself 16641. (Contributed by Jim Kingdon, 8-Aug-2022.) |
| ⊢ ((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → if(∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ∈ 2o) | ||
| Theorem | nninfsellemsuc 16640* | Lemma for nninfself 16641. (Contributed by Jim Kingdon, 6-Aug-2022.) |
| ⊢ ((𝑄 ∈ (2o ↑𝑚 ℕ∞) ∧ 𝐽 ∈ ω) → if(∀𝑘 ∈ suc suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅) ⊆ if(∀𝑘 ∈ suc 𝐽(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅)) | ||
| Theorem | nninfself 16641* | Domain and range of the selection function for ℕ∞. (Contributed by Jim Kingdon, 6-Aug-2022.) |
| ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) ⇒ ⊢ 𝐸:(2o ↑𝑚 ℕ∞)⟶ℕ∞ | ||
| Theorem | nninfsellemeq 16642* | Lemma for nninfsel 16645. (Contributed by Jim Kingdon, 9-Aug-2022.) |
| ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → ∀𝑘 ∈ 𝑁 (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o) & ⊢ (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = ∅) ⇒ ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) | ||
| Theorem | nninfsellemqall 16643* | Lemma for nninfsel 16645. (Contributed by Jim Kingdon, 9-Aug-2022.) |
| ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) & ⊢ (𝜑 → 𝑁 ∈ ω) ⇒ ⊢ (𝜑 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))) = 1o) | ||
| Theorem | nninfsellemeqinf 16644* | Lemma for nninfsel 16645. (Contributed by Jim Kingdon, 9-Aug-2022.) |
| ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) ⇒ ⊢ (𝜑 → (𝐸‘𝑄) = (𝑖 ∈ ω ↦ 1o)) | ||
| Theorem | nninfsel 16645* | 𝐸 is a selection function for ℕ∞. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.) |
| ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) & ⊢ (𝜑 → 𝑄 ∈ (2o ↑𝑚 ℕ∞)) & ⊢ (𝜑 → (𝑄‘(𝐸‘𝑄)) = 1o) ⇒ ⊢ (𝜑 → ∀𝑝 ∈ ℕ∞ (𝑄‘𝑝) = 1o) | ||
| Theorem | nninfomnilem 16646* | Lemma for nninfomni 16647. (Contributed by Jim Kingdon, 10-Aug-2022.) |
| ⊢ 𝐸 = (𝑞 ∈ (2o ↑𝑚 ℕ∞) ↦ (𝑛 ∈ ω ↦ if(∀𝑘 ∈ suc 𝑛(𝑞‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1o, ∅))) = 1o, 1o, ∅))) ⇒ ⊢ ℕ∞ ∈ Omni | ||
| Theorem | nninfomni 16647 | ℕ∞ is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.) |
| ⊢ ℕ∞ ∈ Omni | ||
| Theorem | nninffeq 16648* | Equality of two functions on ℕ∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, ⊢ (𝜑 → ∀𝑛 ∈ suc ω...). (Contributed by Jim Kingdon, 4-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ∞⟶ℕ0) & ⊢ (𝜑 → 𝐺:ℕ∞⟶ℕ0) & ⊢ (𝜑 → (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑥 ∈ ω ↦ 1o))) & ⊢ (𝜑 → ∀𝑛 ∈ ω (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
| Theorem | nnnninfen 16649 | Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.) |
| ⊢ (ω ≈ ℕ∞ ↔ ω ∈ Omni) | ||
| Theorem | nnnninfex 16650* | If an element of ℕ∞ has a value of zero somewhere, then it is the mapping of a natural number. (Contributed by Jim Kingdon, 4-Aug-2022.) |
| ⊢ (𝜑 → 𝑃 ∈ ℕ∞) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → (𝑃‘𝑁) = ∅) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) | ||
| Theorem | nninfnfiinf 16651* | An element of ℕ∞ which is not finite is infinite. (Contributed by Jim Kingdon, 30-Nov-2025.) |
| ⊢ ((𝐴 ∈ ℕ∞ ∧ ¬ ∃𝑛 ∈ ω 𝐴 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → 𝐴 = (𝑖 ∈ ω ↦ 1o)) | ||
| Theorem | exmidsbthrlem 16652* | Lemma for exmidsbthr 16653. (Contributed by Jim Kingdon, 11-Aug-2022.) |
| ⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)))) ⇒ ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) | ||
| Theorem | exmidsbthr 16653* | The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.) |
| ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) | ||
| Theorem | exmidsbth 16654* |
The Schroeder-Bernstein Theorem is equivalent to excluded middle. This
is Metamath 100 proof #25. The forward direction (isbth 7166) is the
proof of the Schroeder-Bernstein Theorem from the Metamath Proof
Explorer database (in which excluded middle holds), but adapted to use
EXMID as an antecedent rather
than being unconditionally true, as in
the non-intuitionistic proof at
https://us.metamath.org/mpeuni/sbth.html 7166.
The reverse direction (exmidsbthr 16653) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.) |
| ⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) | ||
| Theorem | sbthomlem 16655 | Lemma for sbthom 16656. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.) |
| ⊢ (𝜑 → ω ∈ Omni) & ⊢ (𝜑 → 𝑌 ⊆ {∅}) & ⊢ (𝜑 → 𝐹:ω–1-1-onto→(𝑌 ⊔ ω)) ⇒ ⊢ (𝜑 → (𝑌 = ∅ ∨ 𝑌 = {∅})) | ||
| Theorem | sbthom 16656 | Schroeder-Bernstein is not possible even for ω. We know by exmidsbth 16654 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ω? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.) |
| ⊢ ((∀𝑥((𝑥 ≼ ω ∧ ω ≼ 𝑥) → 𝑥 ≈ ω) ∧ ω ∈ Omni) → EXMID) | ||
| Theorem | qdencn 16657* | The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 11767 (and also would hold for ℝ × ℝ with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.) |
| ⊢ 𝑄 = {𝑧 ∈ ℂ ∣ ((ℜ‘𝑧) ∈ ℚ ∧ (ℑ‘𝑧) ∈ ℚ)} ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ 𝑄 (abs‘(𝑥 − 𝐴)) < 𝐵) | ||
| Theorem | refeq 16658* | Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐺:ℝ⟶ℝ) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑥 < 0 → (𝐹‘𝑥) = (𝐺‘𝑥))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (0 < 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥))) & ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
| Theorem | triap 16659 | Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ DECID 𝐴 # 𝐵)) | ||
| Theorem | isomninnlem 16660* | Lemma for isomninn 16661. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.) |
| ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))) | ||
| Theorem | isomninn 16661* | Omniscience stated in terms of natural numbers. Similar to isomnimap 7336 but it will sometimes be more convenient to use 0 and 1 rather than ∅ and 1o. (Contributed by Jim Kingdon, 30-Aug-2023.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0 ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1))) | ||
| Theorem | cvgcmp2nlemabs 16662* | Lemma for cvgcmp2n 16663. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting (seq1( + , 𝐺)‘𝑁) as the sum of (seq1( + , 𝐺)‘𝑀) and a term which gets smaller as 𝑀 gets large. (Contributed by Jim Kingdon, 25-Aug-2023.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀)) | ||
| Theorem | cvgcmp2n 16663* | A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘)) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘))) ⇒ ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ ) | ||
| Theorem | iooref1o 16664 | A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (1 / (1 + (exp‘𝑥)))) ⇒ ⊢ 𝐹:ℝ–1-1-onto→(0(,)1) | ||
| Theorem | iooreen 16665 | An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.) |
| ⊢ (0(,)1) ≈ ℝ | ||
Omniscience principles refer to several propositions, most of them weaker than full excluded middle, which do not follow from the axioms of IZF set theory. They are: (0) the Principle of Omniscience (PO), which is another name for excluded middle (see exmidomni 7341), (1) the Limited Principle of Omniscience (LPO) is ω ∈ Omni (see df-omni 7334), (2) the Weak Limited Principle of Omniscience (WLPO) is ω ∈ WOmni (see df-womni 7363), (3) Markov's Principle (MP) is ω ∈ Markov (see df-markov 7351), (4) the Lesser Limited Principle of Omniscience (LLPO) is not yet defined in iset.mm. They also have analytic counterparts each of which follows from the corresponding omniscience principle: (1) Analytic LPO is real number trichotomy, ∀𝑥 ∈ ℝ∀𝑦 ∈ ℝ(𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) (see trilpo 16673), (2) Analytic WLPO is decidability of real number equality, ∀𝑥 ∈ ℝ∀𝑦 ∈ ℝDECID 𝑥 = 𝑦 (see redcwlpo 16686), (3) Analytic MP is ∀𝑥 ∈ ℝ∀𝑦 ∈ ℝ(𝑥 ≠ 𝑦 → 𝑥 # 𝑦) (see neapmkv 16699), (4) Analytic LLPO is real number dichotomy, ∀𝑥 ∈ ℝ∀𝑦 ∈ ℝ(𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) (most relevant current theorem is maxclpr 11787). | ||
| Theorem | trilpolemclim 16666* | Lemma for trilpo 16673. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) ⇒ ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ ) | ||
| Theorem | trilpolemcl 16667* | Lemma for trilpo 16673. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
| Theorem | trilpolemisumle 16668* | Lemma for trilpo 16673. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℕ) ⇒ ⊢ (𝜑 → Σ𝑖 ∈ 𝑍 ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ 𝑍 (1 / (2↑𝑖))) | ||
| Theorem | trilpolemgt1 16669* | Lemma for trilpo 16673. The 1 < 𝐴 case. (Contributed by Jim Kingdon, 23-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ⇒ ⊢ (𝜑 → ¬ 1 < 𝐴) | ||
| Theorem | trilpolemeq1 16670* | Lemma for trilpo 16673. The 𝐴 = 1 case. This is proved by noting that if any (𝐹‘𝑥) is zero, then the infinite sum 𝐴 is less than one based on the term which is zero. We are using the fact that the 𝐹 sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) & ⊢ (𝜑 → 𝐴 = 1) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) | ||
| Theorem | trilpolemlt1 16671* | Lemma for trilpo 16673. The 𝐴 < 1 case. We can use the distance between 𝐴 and one (that is, 1 − 𝐴) to find a position in the sequence 𝑛 where terms after that point will not add up to as much as 1 − 𝐴. By finomni 7339 we know the terms up to 𝑛 either contain a zero or are all one. But if they are all one that contradicts the way we constructed 𝑛, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) & ⊢ (𝜑 → 𝐴 < 1) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0) | ||
| Theorem | trilpolemres 16672* | Lemma for trilpo 16673. The result. (Contributed by Jim Kingdon, 23-Aug-2023.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) & ⊢ (𝜑 → (𝐴 < 1 ∨ 𝐴 = 1 ∨ 1 < 𝐴)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0 ∨ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) | ||
| Theorem | trilpo 16673* |
Real number trichotomy implies the Limited Principle of Omniscience
(LPO). We expect that we'd need some form of countable choice to prove
the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 16671 (which means the sequence contains a zero), trilpolemeq1 16670 (which means the sequence is all ones), and trilpolemgt1 16669 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 16659) or that the real numbers are a discrete field (see trirec0 16674). LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10501 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ω ∈ Omni) | ||
| Theorem | trirec0 16674* |
Every real number having a reciprocal or equaling zero is equivalent to
real number trichotomy.
This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 16673). (Contributed by Jim Kingdon, 10-Jun-2024.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ∨ 𝑥 = 0)) | ||
| Theorem | trirec0xor 16675* |
Version of trirec0 16674 with exclusive-or.
The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑥 ∈ ℝ (∃𝑧 ∈ ℝ (𝑥 · 𝑧) = 1 ⊻ 𝑥 = 0)) | ||
| Theorem | apdifflemf 16676 | Lemma for apdiff 16678. Being apart from the point halfway between 𝑄 and 𝑅 suffices for 𝐴 to be a different distance from 𝑄 and from 𝑅. (Contributed by Jim Kingdon, 18-May-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑄 ∈ ℚ) & ⊢ (𝜑 → 𝑅 ∈ ℚ) & ⊢ (𝜑 → 𝑄 < 𝑅) & ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅))) | ||
| Theorem | apdifflemr 16677 | Lemma for apdiff 16678. (Contributed by Jim Kingdon, 19-May-2024.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑆 ∈ ℚ) & ⊢ (𝜑 → (abs‘(𝐴 − -1)) # (abs‘(𝐴 − 1))) & ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (abs‘(𝐴 − 0)) # (abs‘(𝐴 − (2 · 𝑆)))) ⇒ ⊢ (𝜑 → 𝐴 # 𝑆) | ||
| Theorem | apdiff 16678* | The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.) |
| ⊢ (𝐴 ∈ ℝ → (∀𝑞 ∈ ℚ 𝐴 # 𝑞 ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) # (abs‘(𝐴 − 𝑟))))) | ||
| Theorem | qdiff 16679* | The rationals are exactly those reals for which there exist two distinct rationals that are the same distance from the original number. Similar to apdiff 16678 but by stating the result positively we can completely sidestep the issue of not equal versus apart in the statement of the result. From an online post by Ingo Blechschmidt. (Contributed by Jim Kingdon, 24-Apr-2026.) |
| ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℚ ↔ ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))))) | ||
| Theorem | iswomninnlem 16680* | Lemma for iswomnimap 7365. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.) |
| ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) | ||
| Theorem | iswomninn 16681* | Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7365 but it will sometimes be more convenient to use 0 and 1 rather than ∅ and 1o. (Contributed by Jim Kingdon, 20-Jun-2024.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1)) | ||
| Theorem | iswomni0 16682* | Weak omniscience stated in terms of equality with 0. Like iswomninn 16681 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0)) | ||
| Theorem | ismkvnnlem 16683* | Lemma for ismkvnn 16684. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
| ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))) | ||
| Theorem | ismkvnn 16684* | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ ({0, 1} ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1 → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 0))) | ||
| Theorem | redcwlpolemeq1 16685* | Lemma for redcwlpo 16686. A biconditionalized version of trilpolemeq1 16670. (Contributed by Jim Kingdon, 21-Jun-2024.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ⇒ ⊢ (𝜑 → (𝐴 = 1 ↔ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1)) | ||
| Theorem | redcwlpo 16686* |
Decidability of real number equality implies the Weak Limited Principle
of Omniscience (WLPO). We expect that we'd need some form of countable
choice to prove the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 16685). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10505 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ω ∈ WOmni) | ||
| Theorem | tridceq 16687* | Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 16673 and redcwlpo 16686). Thus, this is an analytic analogue to lpowlpo 7367. (Contributed by Jim Kingdon, 24-Jul-2024.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦) | ||
| Theorem | redc0 16688* | Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0) | ||
| Theorem | reap0 16689* | Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℝ DECID 𝑧 # 0) | ||
| Theorem | cndcap 16690* | Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ∀𝑧 ∈ ℂ ∀𝑤 ∈ ℂ DECID 𝑧 # 𝑤) | ||
| Theorem | dceqnconst 16691* | Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 16686 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.) |
| ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0)) | ||
| Theorem | dcapnconst 16692* |
Decidability of real number apartness implies the existence of a certain
non-constant function from real numbers to integers. Variation of
Exercise 11.6(i) of [HoTT], p. (varies).
See trilpo 16673 for more
discussion of decidability of real number apartness.
This is a weaker form of dceqnconst 16691 and in fact this theorem can be proved using dceqnconst 16691 as shown at dcapnconstALT 16693. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) |
| ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0)) | ||
| Theorem | dcapnconstALT 16693* | Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 16692 by means of dceqnconst 16691. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0)) | ||
| Theorem | nconstwlpolem0 16694* | Lemma for nconstwlpo 16697. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.) |
| ⊢ (𝜑 → 𝐺:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖)) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ (𝐺‘𝑥) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
| Theorem | nconstwlpolemgt0 16695* | Lemma for nconstwlpo 16697. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.) |
| ⊢ (𝜑 → 𝐺:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖)) & ⊢ (𝜑 → ∃𝑥 ∈ ℕ (𝐺‘𝑥) = 1) ⇒ ⊢ (𝜑 → 0 < 𝐴) | ||
| Theorem | nconstwlpolem 16696* | Lemma for nconstwlpo 16697. (Contributed by Jim Kingdon, 23-Jul-2024.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℤ) & ⊢ (𝜑 → (𝐹‘0) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ≠ 0) & ⊢ (𝜑 → 𝐺:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐺‘𝑖)) ⇒ ⊢ (𝜑 → (∀𝑦 ∈ ℕ (𝐺‘𝑦) = 0 ∨ ¬ ∀𝑦 ∈ ℕ (𝐺‘𝑦) = 0)) | ||
| Theorem | nconstwlpo 16697* | Existence of a certain non-constant function from reals to integers implies ω ∈ WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.) |
| ⊢ (𝜑 → 𝐹:ℝ⟶ℤ) & ⊢ (𝜑 → (𝐹‘0) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝐹‘𝑥) ≠ 0) ⇒ ⊢ (𝜑 → ω ∈ WOmni) | ||
| Theorem | neapmkvlem 16698* | Lemma for neapmkv 16699. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) & ⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) & ⊢ ((𝜑 ∧ 𝐴 ≠ 1) → 𝐴 # 1) ⇒ ⊢ (𝜑 → (¬ ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1 → ∃𝑥 ∈ ℕ (𝐹‘𝑥) = 0)) | ||
| Theorem | neapmkv 16699* | If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦) → ω ∈ Markov) | ||
| Theorem | neap0mkv 16700* | The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.) |
| ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 ≠ 𝑦 → 𝑥 # 𝑦) ↔ ∀𝑥 ∈ ℝ (𝑥 ≠ 0 → 𝑥 # 0)) | ||
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