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Type | Label | Description |
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Statement | ||
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 9401) and deduce that the class of finite ordinals is a set (bj-omex 9402). | ||
Axiom | ax-infvn 9401* | 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 4254) from which one then proves, using full separation, that the wanted set exists (omex 4259). "vn" is for "Von Neumann". (Contributed by BJ, 14-Nov-2019.) |
Ind Ind | ||
Theorem | bj-omex 9402 | Proof of omex 4259 from ax-infvn 9401. (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 9403* | Bounded version of peano5 4264. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | speano5 9404* | Version of peano5 4264 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 finite ordinals satisfies the five Peano postulates and thus provides a model for the set of natural numbers. | ||
Theorem | findset 9405* | Bounded induction (principle of induction when is assumed to be a set) allowing a proof from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See bdfind 9406 for a proof when is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdfind 9406* | Bounded induction (principle of induction when is assumed to be bounded), proved from basic constructive axioms. See find 4265 for a nonconstructive proof of the general case. See findset 9405 for a proof when is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindis 9407* | 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 4266 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4266, finds2 4267, finds1 4268. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindisg 9408* | Version of bj-bdfindis 9407 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 9407 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-bdfindes 9409 | Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 9407 for explanations. From this version, it is easy to prove the bounded version of findes 4269. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-nn0suc0 9410* | Constructive proof of a variant of nn0suc 4270. For a constructive proof of nn0suc 4270, see bj-nn0suc 9424. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans 9411 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nntrans2 9412 | A natural number is a transitive set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnelirr 9413 | A natural number does not belong to itself. Version of elirr 4224 for natural numbers, which does not require ax-setind 4220. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnen2lp 9414 |
A version of en2lp 4232 for natural numbers, which does not require
ax-setind 4220.
Note: using this theorem and bj-nnelirr 9413, one can remove dependency on ax-setind 4220 from nntri2 6012 and nndcel 6016; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-peano4 9415 | Remove from peano4 4263 dependency on ax-setind 4220. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-omtrans 9416 |
The set is
transitive. A natural number is included in
.
Constructive proof of elnn 4271.
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 9417 | The set is transitive. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-nnord 9418 | A natural number is an ordinal. Constructive proof of nnord 4277. Can also be proved from bj-nnelon 9419 if the latter is proved from bj-omssonALT 9423. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-nnelon 9419 | A natural number is an ordinal. Constructive proof of nnon 4275. Can also be proved from bj-omssonALT 9423. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) |
Theorem | bj-omord 9420 | The set is an ordinal. Constructive proof of ordom 4272. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omelon 9421 | The set is an ordinal. Constructive proof of omelon 4274. (Contributed by BJ, 29-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-omsson 9422 | Constructive proof of omsson 4278. See also bj-omssonALT 9423. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged. |
Theorem | bj-omssonALT 9423 | Alternate proof of bj-omsson 9422. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | bj-nn0suc 9424* | Proof of (biconditional form of) nn0suc 4270 from the core axioms of CZF. See also bj-nn0sucALT 9438. 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 9425* | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindf 9426* | Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
Theorem | setindis 9427* | Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) |
Axiom | ax-bdsetind 9428* | Axiom of bounded set induction. (Contributed by BJ, 28-Nov-2019.) |
BOUNDED | ||
Theorem | bdsetindis 9429* | Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-inf2vnlem1 9430* | Lemma for bj-inf2vn 9434. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem2 9431* | Lemma for bj-inf2vnlem3 9432 and bj-inf2vnlem4 9433. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vnlem3 9432* | Lemma for bj-inf2vn 9434. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED Ind | ||
Theorem | bj-inf2vnlem4 9433* | Lemma for bj-inf2vn2 9435. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Ind | ||
Theorem | bj-inf2vn 9434* | A sufficient condition for to be a set. See bj-inf2vn2 9435 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-inf2vn2 9435* | A sufficient condition for to be a set; unbounded version of bj-inf2vn 9434. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) |
Axiom | ax-inf2 9436* | Another axiom of infinity in a constructive setting (see ax-infvn 9401). (Contributed by BJ, 14-Nov-2019.) (New usage is discouraged.) |
Theorem | bj-omex2 9437 | Using bounded set induction and the strong axiom of infinity, is a set, that is, we recover ax-infvn 9401 (see bj-2inf 9397 for the equivalence of the latter with bj-omex 9402). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | bj-nn0sucALT 9438* | Alternate proof of bj-nn0suc 9424, also constructive but from ax-inf2 9436, hence requiring ax-bdsetind 9428. (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 9439* | 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 9407 for a bounded version not requiring ax-setind 4220. See finds 4266 for a proof in IZF. From this version, it is easy to prove of finds 4266, finds2 4267, finds1 4268. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.) |
Theorem | bj-findisg 9440* | Version of bj-findis 9439 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 9439 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-findes 9441 | Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 9439 for explanations. From this version, it is easy to prove findes 4269. (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 9442* | Axiom scheme of strong collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.) |
Theorem | strcoll2 9443* | Version of ax-strcoll 9442 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
Theorem | strcollnft 9444* | Closed form of strcollnf 9445. Version of ax-strcoll 9442 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.) |
Theorem | strcollnf 9445* | Version of ax-strcoll 9442 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. (Contributed by BJ, 21-Oct-2019.) |
Theorem | strcollnfALT 9446* | Alternate proof of strcollnf 9445, not using strcollnft 9444. (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 9447* | Axiom scheme of subset collection. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. (Contributed by BJ, 5-Oct-2019.) |
Theorem | sscoll2 9448* | Version of ax-sscoll 9447 with two DV conditions removed and without initial universal quantifiers. (Contributed by BJ, 5-Oct-2019.) |
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 9449 | 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 9450 | 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 9451 | 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 9452 | 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 9453 | There is an equivalence between the two "all some" forms. (Contributed by David A. Wheeler, 22-Oct-2018.) |
! ! | ||
Theorem | alsi1d 9454 | 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 9455 | 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 9456 | 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 9457 | 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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