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Type | Label | Description |
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Statement | ||
Theorem | subctctexmid 13301* | 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.) |
⊔ Markov EXMID | ||
Theorem | sssneq 13302* | Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
Theorem | pw1nct 13303* | 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.) |
⊔ | ||
Theorem | 0nninf 13304 | The zero element of ℕ_{∞} (the constant sequence equal to ). (Contributed by Jim Kingdon, 14-Jul-2022.) |
ℕ_{∞} | ||
Theorem | nninff 13305 | An element of ℕ_{∞} is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nnsf 13306* | Domain and range of . Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.) |
ℕ_{∞} ℕ_{∞}ℕ_{∞} | ||
Theorem | peano4nninf 13307* | 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.) |
ℕ_{∞} ℕ_{∞}ℕ_{∞} | ||
Theorem | peano3nninf 13308* | The successor function on ℕ_{∞} is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfalllemn 13309* | Lemma for nninfall 13311. Mapping of a natural number to an element of ℕ_{∞}. (Contributed by Jim Kingdon, 4-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfalllem1 13310* | Lemma for nninfall 13311. (Contributed by Jim Kingdon, 1-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfall 13311* | 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 is a decidable predicate is that it assigns a value of either or (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.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfex 13312 | ℕ_{∞} is a set. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfsellemdc 13313* | Lemma for nninfself 13316. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.) |
ℕ_{∞} DECID | ||
Theorem | nninfsellemcl 13314* | Lemma for nninfself 13316. (Contributed by Jim Kingdon, 8-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfsellemsuc 13315* | Lemma for nninfself 13316. (Contributed by Jim Kingdon, 6-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfself 13316* | Domain and range of the selection function for ℕ_{∞}. (Contributed by Jim Kingdon, 6-Aug-2022.) |
ℕ_{∞} ℕ_{∞}ℕ_{∞} | ||
Theorem | nninfsellemeq 13317* | Lemma for nninfsel 13320. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsellemqall 13318* | Lemma for nninfsel 13320. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsellemeqinf 13319* | Lemma for nninfsel 13320. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsel 13320* | is a selection function for ℕ_{∞}. Theorem 3.6 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfomnilem 13321* | Lemma for nninfomni 13322. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} ℕ_{∞} Omni | ||
Theorem | nninfomni 13322 | ℕ_{∞} is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} Omni | ||
Theorem | nninffeq 13323* | Equality of two functions on ℕ_{∞} which agree at every integer and at the point at infinity. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 4-Aug-2023.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | exmidsbthrlem 13324* | Lemma for exmidsbthr 13325. (Contributed by Jim Kingdon, 11-Aug-2022.) |
ℕ_{∞} EXMID | ||
Theorem | exmidsbthr 13325* | The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.) |
EXMID | ||
Theorem | exmidsbth 13326* |
The Schroeder-Bernstein Theorem is equivalent to excluded middle. This
is Metamath 100 proof #25. The forward direction (isbth 6855) 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-intuitionist proof at
https://us.metamath.org/mpeuni/sbth.html 6855.
The reverse direction (exmidsbthr 13325) 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 | ||
Theorem | sbthomlem 13327 | Lemma for sbthom 13328. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.) |
Omni ⊔ | ||
Theorem | sbthom 13328 | Schroeder-Bernstein is not possible even for . We know by exmidsbth 13326 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is ? 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.) |
Omni EXMID | ||
Theorem | qdencn 13329* | 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 10986 (and also would hold for with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | refeq 13330* | 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.) |
Theorem | triap 13331 | Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.) |
DECID # | ||
Theorem | isomninnlem 13332* | Lemma for isomninn 13333. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.) |
frec Omni | ||
Theorem | isomninn 13333* | Omniscience stated in terms of natural numbers. Similar to isomnimap 7009 but it will sometimes be more convenient to use and rather than and . (Contributed by Jim Kingdon, 30-Aug-2023.) |
Omni | ||
Theorem | cvgcmp2nlemabs 13334* | Lemma for cvgcmp2n 13335. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting as the sum of and a term which gets smaller as gets large. (Contributed by Jim Kingdon, 25-Aug-2023.) |
Theorem | cvgcmp2n 13335* | A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.) |
Theorem | trilpolemclim 13336* | Lemma for trilpo 13343. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.) |
Theorem | trilpolemcl 13337* | Lemma for trilpo 13343. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemisumle 13338* | Lemma for trilpo 13343. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.) |
Theorem | trilpolemgt1 13339* | Lemma for trilpo 13343. The case. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemeq1 13340* | Lemma for trilpo 13343. The case. This is proved by noting that if any is zero, then the infinite sum is less than one based on the term which is zero. We are using the fact that the sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemlt1 13341* | Lemma for trilpo 13343. The case. We can use the distance between and one (that is, ) to find a position in the sequence where terms after that point will not add up to as much as . By finomni 7012 we know the terms up to either contain a zero or are all one. But if they are all one that contradicts the way we constructed , so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpolemres 13342* | Lemma for trilpo 13343. The result. (Contributed by Jim Kingdon, 23-Aug-2023.) |
Theorem | trilpo 13343* |
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 13341 (which means the sequence contains a zero), trilpolemeq1 13340 (which means the sequence is all ones), and trilpolemgt1 13339 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 13331) or that the real numbers are a discrete field (see trirec0 13344). (Contributed by Jim Kingdon, 23-Aug-2023.) |
Omni | ||
Theorem | trirec0 13344* |
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 13343). (Contributed by Jim Kingdon, 10-Jun-2024.) |
Theorem | trirec0xor 13345* |
Version of trirec0 13344 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.) |
Theorem | apdifflemf 13346 | Lemma for apdiff 13348. Being apart from the point halfway between and suffices for to be a different distance from and from . (Contributed by Jim Kingdon, 18-May-2024.) |
# # | ||
Theorem | apdifflemr 13347 | Lemma for apdiff 13348. (Contributed by Jim Kingdon, 19-May-2024.) |
# # # | ||
Theorem | apdiff 13348* | 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.) |
# # | ||
Theorem | supfz 13349 | The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
Theorem | inffz 13350 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Jim Kingdon, 15-Oct-2022.) |
inf | ||
Theorem | taupi 13351 | Relationship between and . This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.) |
Theorem | ax1hfs 13352 | Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.) |
Theorem | dftest 13353 |
A proposition is testable iff its negative or double-negative is true.
See Chapter 2 [Moschovakis] p. 2.
We do not formally define testability with a new token, but instead use DECID before the formula in question. For example, DECID corresponds to " is testable". (Contributed by David A. Wheeler, 13-Aug-2018.) For statements about testable propositions, search for the keyword "testable" in the comments of statements, for instance using the Metamath command "MM> SEARCH * "testable" / COMMENTS". (New usage is discouraged.) |
DECID | ||
These are definitions and proofs involving an experimental "allsome" quantifier (aka "all some"). In informal language, statements like "All Martians are green" imply that there is at least one Martian. But it's easy to mistranslate informal language into formal notations because similar statements like do not imply that is ever true, leading to vacuous truths. Some systems include a mechanism to counter this, e.g., PVS allows types to be appended with "+" to declare that they are nonempty. This section presents a different solution to the same problem. The "allsome" quantifier expressly includes the notion of both "all" and "there exists at least one" (aka some), and is defined to make it easier to more directly express both notions. The hope is that if a quantifier more directly expresses this concept, it will be used instead and reduce the risk of creating formal expressions that look okay but in fact are mistranslations. The term "allsome" was chosen because it's short, easy to say, and clearly hints at the two concepts it combines. I do not expect this to be used much in metamath, because in metamath there's a general policy of avoiding the use of new definitions unless there are very strong reasons to do so. Instead, my goal is to rigorously define this quantifier and demonstrate a few basic properties of it. The syntax allows two forms that look like they would be problematic, but they are fine. When applied to a top-level implication we allow ! , and when restricted (applied to a class) we allow ! . The first symbol after the setvar variable must always be if it is the form applied to a class, and since cannot begin a wff, it is unambiguous. The looks like it would be a problem because or might include implications, but any implication arrow within any wff must be surrounded by parentheses, so only the implication arrow of ! can follow the wff. The implication syntax would work fine without the parentheses, but I added the parentheses because it makes things clearer inside larger complex expressions, and it's also more consistent with the rest of the syntax. For more, see "The Allsome Quantifier" by David A. Wheeler at https://dwheeler.com/essays/allsome.html I hope that others will eventually agree that allsome is awesome. | ||
Syntax | walsi 13354 | Extend wff definition to include "all some" applied to a top-level implication, which means is true whenever is true, and there is at least least one where is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Syntax | walsc 13355 | Extend wff definition to include "all some" applied to a class, which means is true for all in , and there is at least one in . (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Definition | df-alsi 13356 | Define "all some" applied to a top-level implication, which means is true whenever is true and there is at least one where is true. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Definition | df-alsc 13357 | Define "all some" applied to a class, which means is true for all in and there is at least one in . (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsconv 13358 | There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.) |
! ! | ||
Theorem | alsi1d 13359 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsi2d 13360 | Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsc1d 13361 | Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! | ||
Theorem | alsc2d 13362 | Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.) |
! |
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