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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | wind 13501 | Syntax for inductive classes. |
Ind | ||
Definition | df-bj-ind 13502* | Define the property of being an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-indsuc 13503 | A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-indeq 13504 | Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) |
Ind Ind | ||
Theorem | bj-bdind 13505 | Boundedness of the formula "the setvar is an inductive class". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED Ind | ||
Theorem | bj-indint 13506* | The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.) |
Ind Ind | ||
Theorem | bj-indind 13507* | If is inductive and is "inductive in ", then is inductive. (Contributed by BJ, 25-Oct-2020.) |
Ind Ind | ||
Theorem | bj-dfom 13508 | Alternate definition of , as the intersection of all the inductive sets. Proposal: make this the definition. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-omind 13509 | is an inductive class. (Contributed by BJ, 30-Nov-2019.) |
Ind | ||
Theorem | bj-omssind 13510 | is included in all the inductive sets (but for the moment, we cannot prove that it is included in all the inductive classes). (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-ssom 13511* | A characterization of subclasses of . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-om 13512* | A set is equal to if and only if it is the smallest inductive set. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind Ind | ||
Theorem | bj-2inf 13513* | Two formulations of the axiom of infinity (see ax-infvn 13516 and bj-omex 13517) . (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ind Ind | ||
The first three Peano postulates follow from constructive set theory (actually, from its core axioms). The proofs peano1 4553 and peano3 4555 already show this. In this section, we prove bj-peano2 13514 to complete this program. We also prove a preliminary version of the fifth Peano postulate from the core axioms. | ||
Theorem | bj-peano2 13514 | Constructive proof of peano2 4554. Temporary note: another possibility is to simply replace sucexg 4457 with bj-sucexg 13497 in the proof of peano2 4554. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | peano5set 13515* | Version of peano5 4557 when is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
In the absence of full separation, the axiom of infinity has to be stated more precisely, as the existence of the smallest class containing the empty set and the successor of each of its elements. | ||
In this section, we introduce the axiom of infinity in a constructive setting (ax-infvn 13516) and deduce that the class of natural number ordinals is a set (bj-omex 13517). | ||
Axiom | ax-infvn 13516* | Axiom of infinity in a constructive setting. This asserts the existence of the special set we want (the set of natural numbers), instead of the existence of a set with some properties (ax-iinf 4547) from which one then proves, using full separation, that the wanted set exists (omex 4552). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.) |
Ind Ind | ||
Theorem | bj-omex 13517 | Proof of omex 4552 from ax-infvn 13516. (Contributed by BJ, 14-Nov-2019.) (Proof modification is discouraged.) |
In this section, we give constructive proofs of two versions of Peano's fifth postulate. | ||
Theorem | bdpeano5 13518* | Bounded version of peano5 4557. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | speano5 13519* | Version of peano5 4557 when is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
In this section, we prove various versions of bounded induction from the basic axioms of CZF (in particular, without the axiom of set induction). We also prove Peano's fourth postulate. Together with the results from the previous sections, this proves from the core axioms of CZF (with infinity) that the set of natural number ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers. | ||
Theorem | findset 13520* | Bounded induction (principle of induction when is assumed to be a set) allowing a proof from basic constructive axioms. See find 4558 for a nonconstructive proof of the general case. See bdfind 13521 for a proof when is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdfind 13521* | Bounded induction (principle of induction when is assumed to be bounded), proved from basic constructive axioms. See find 4558 for a nonconstructive proof of the general case. See findset 13520 for a proof when is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindis 13522* | Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4559 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4559, finds2 4560, finds1 4561. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindisg 13523* | Version of bj-bdfindis 13522 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 13522 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindes 13524 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13522 for explanations. From this version, it is easy to prove the bounded version of findes 4562. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-nn0suc0 13525* | Constructive proof of a variant of nn0suc 4563. For a constructive proof of nn0suc 4563, see bj-nn0suc 13539. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans 13526 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans2 13527 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnelirr 13528 | A natural number does not belong to itself. Version of elirr 4500 for natural numbers, which does not require ax-setind 4496. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnen2lp 13529 |
A version of en2lp 4513 for natural numbers, which does not require
ax-setind 4496.
Note: using this theorem and bj-nnelirr 13528, one can remove dependency on ax-setind 4496 from nntri2 6441 and nndcel 6447; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-peano4 13530 | Remove from peano4 4556 dependency on ax-setind 4496. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-omtrans 13531 |
The set is
transitive. A natural number is included in
.
Constructive proof of elnn 4565.
The idea is to use bounded induction with the formula . This formula, in a logic with terms, is bounded. So in our logic without terms, we need to temporarily replace it with and then deduce the original claim. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omtrans2 13532 | The set is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnord 13533 | A natural number is an ordinal class. Constructive proof of nnord 4571. Can also be proved from bj-nnelon 13534 if the latter is proved from bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-nnelon 13534 | A natural number is an ordinal. Constructive proof of nnon 4569. Can also be proved from bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-omord 13535 | The set is an ordinal class. Constructive proof of ordom 4566. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omelon 13536 | The set is an ordinal. Constructive proof of omelon 4568. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omsson 13537 | Constructive proof of omsson 4572. See also bj-omssonALT 13538. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged. |
Theorem | bj-omssonALT 13538 | Alternate proof of bj-omsson 13537. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | bj-nn0suc 13539* | Proof of (biconditional form of) nn0suc 4563 from the core axioms of CZF. See also bj-nn0sucALT 13553. As a characterization of the elements of , this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
In this section, we add the axiom of set induction to the core axioms of CZF. | ||
In this section, we prove some variants of the axiom of set induction. | ||
Theorem | setindft 13540* | Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindf 13541* | Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindis 13542* | Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) |
Axiom | ax-bdsetind 13543* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
BOUNDED | ||
Theorem | bdsetindis 13544* | Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-inf2vnlem1 13545* | Lemma for bj-inf2vn 13549. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem2 13546* | Lemma for bj-inf2vnlem3 13547 and bj-inf2vnlem4 13548. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem3 13547* | Lemma for bj-inf2vn 13549. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED Ind | ||
Theorem | bj-inf2vnlem4 13548* | Lemma for bj-inf2vn2 13550. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vn 13549* | A sufficient condition for to be a set. See bj-inf2vn2 13550 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-inf2vn2 13550* | A sufficient condition for to be a set; unbounded version of bj-inf2vn 13549. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Axiom | ax-inf2 13551* | Another axiom of infinity in a constructive setting (see ax-infvn 13516). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) |
Theorem | bj-omex2 13552 | Using bounded set induction and the strong axiom of infinity, is a set, that is, we recover ax-infvn 13516 (see bj-2inf 13513 for the equivalence of the latter with bj-omex 13517). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | bj-nn0sucALT 13553* | Alternate proof of bj-nn0suc 13539, also constructive but from ax-inf2 13551, hence requiring ax-bdsetind 13543. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
In this section, using the axiom of set induction, we prove full induction on the set of natural numbers. | ||
Theorem | bj-findis 13554* | Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 13522 for a bounded version not requiring ax-setind 4496. See finds 4559 for a proof in IZF. From this version, it is easy to prove of finds 4559, finds2 4560, finds1 4561. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-findisg 13555* | Version of bj-findis 13554 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13554 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-findes 13556 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13554 for explanations. From this version, it is easy to prove findes 4562. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
In this section, we state the axiom scheme of strong collection, which is part of CZF set theory. | ||
Axiom | ax-strcoll 13557* | 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 represents a multivalued function on , or equivalently a collection of nonempty classes indexed by , and the axiom asserts the existence of a set which "collects" at least one element in the image of each 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 4079. (Contributed by BJ, 5-Oct-2019.) |
Theorem | strcoll2 13558* | Version of ax-strcoll 13557 with one disjoint variable condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
Theorem | strcollnft 13559* | Closed form of strcollnf 13560. (Contributed by BJ, 21-Oct-2019.) |
Theorem | strcollnf 13560* |
Version of ax-strcoll 13557 with one disjoint variable condition
removed,
the other disjoint variable condition replaced with a nonfreeness
hypothesis, and without initial universal quantifier. Version of
strcoll2 13558 with the disjoint variable condition on replaced
with a nonfreeness hypothesis.
This proof aims to demonstrate a standard technique, but strcoll2 13558 will generally suffice: since the theorem asserts the existence of a set , supposing that that setvar does not occur in the already defined is not a big constraint. (Contributed by BJ, 21-Oct-2019.) |
Theorem | strcollnfALT 13561* | Alternate proof of strcollnf 13560, not using strcollnft 13559. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
In this section, we state the axiom scheme of subset collection, which is part of CZF set theory. | ||
Axiom | ax-sscoll 13562* | 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 represents a multivalued function from to , or equivalently a collection of nonempty subsets of indexed by , and the consequent asserts the existence of a subset of which "collects" at least one element in the image of each and which is made only of such elements. The axiom asserts the existence, for any sets , of a set such that that implication holds for any value of the parameter of . (Contributed by BJ, 5-Oct-2019.) |
Theorem | sscoll2 13563* | Version of ax-sscoll 13562 with two disjoint variable conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.) |
Axiom | ax-ddkcomp 13564 | 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 13564 should be used in place of construction specific results. In particular, axcaucvg 7820 should be proved from it. (Contributed by BJ, 24-Oct-2021.) |
Theorem | ss1oel2o 13565 | Any subset of ordinal one being an element of ordinal two is equivalent to excluded middle. A variation of exmid01 4159 which more directly illustrates the contrast with el2oss1o 6390. (Contributed by Jim Kingdon, 8-Aug-2022.) |
EXMID | ||
Theorem | nnti 13566 | Ordering on a natural number generates a tight apartness. (Contributed by Jim Kingdon, 7-Aug-2022.) |
Theorem | 012of 13567 | Mapping zero and one between and style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
frec | ||
Theorem | 2o01f 13568 | Mapping zero and one between and style integers. (Contributed by Jim Kingdon, 28-Jun-2024.) |
frec | ||
Theorem | pwtrufal 13569 | A subset of the singleton cannot be anything other than or . Removing the double negation would change the meaning, as seen at exmid01 4159. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4157), 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.) |
Theorem | pwle2 13570* | An exercise related to 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.) |
Theorem | pwf1oexmid 13571* | An exercise related to 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.) |
EXMID | ||
Theorem | exmid1stab 13572* | If any proposition is stable, excluded middle follows. We are thinking of as a proposition and as "x is true". (Contributed by Jim Kingdon, 28-Nov-2023.) |
STAB EXMID | ||
Theorem | subctctexmid 13573* | 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 13574* | Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
Theorem | pw1nct 13575* | 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 13576 | The zero element of ℕ_{∞} (the constant sequence equal to ). (Contributed by Jim Kingdon, 14-Jul-2022.) |
ℕ_{∞} | ||
Theorem | nnsf 13577* | 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 13578* | 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 13579* | 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 | nninfalllem1 13580* | Lemma for nninfall 13581. (Contributed by Jim Kingdon, 1-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfall 13581* | 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 | nninfsellemdc 13582* | Lemma for nninfself 13585. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.) |
ℕ_{∞} DECID | ||
Theorem | nninfsellemcl 13583* | Lemma for nninfself 13585. (Contributed by Jim Kingdon, 8-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfsellemsuc 13584* | Lemma for nninfself 13585. (Contributed by Jim Kingdon, 6-Aug-2022.) |
ℕ_{∞} | ||
Theorem | nninfself 13585* | Domain and range of the selection function for ℕ_{∞}. (Contributed by Jim Kingdon, 6-Aug-2022.) |
ℕ_{∞} ℕ_{∞}ℕ_{∞} | ||
Theorem | nninfsellemeq 13586* | Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsellemqall 13587* | Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsellemeqinf 13588* | Lemma for nninfsel 13589. (Contributed by Jim Kingdon, 9-Aug-2022.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | nninfsel 13589* | 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 13590* | Lemma for nninfomni 13591. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} ℕ_{∞} Omni | ||
Theorem | nninfomni 13591 | ℕ_{∞} is omniscient. Corollary 3.7 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 10-Aug-2022.) |
ℕ_{∞} Omni | ||
Theorem | nninffeq 13592* | 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, . (Contributed by Jim Kingdon, 4-Aug-2023.) |
ℕ_{∞} ℕ_{∞} | ||
Theorem | exmidsbthrlem 13593* | Lemma for exmidsbthr 13594. (Contributed by Jim Kingdon, 11-Aug-2022.) |
ℕ_{∞} EXMID | ||
Theorem | exmidsbthr 13594* | The Schroeder-Bernstein Theorem implies excluded middle. Theorem 1 of [PradicBrown2022], p. 1. (Contributed by Jim Kingdon, 11-Aug-2022.) |
EXMID | ||
Theorem | exmidsbth 13595* |
The Schroeder-Bernstein Theorem is equivalent to excluded middle. This
is Metamath 100 proof #25. The forward direction (isbth 6911) 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 6911.
The reverse direction (exmidsbthr 13594) 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 13596 | Lemma for sbthom 13597. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.) |
Omni ⊔ | ||
Theorem | sbthom 13597 | Schroeder-Bernstein is not possible even for . We know by exmidsbth 13595 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 13598* | 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 11102 (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 13599* | 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 13600 | Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.) |
DECID # |
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