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Statement | ||
This is an ongoing project to define bounded formulas, following a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein). In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Δ_{0}) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ_{0}) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitonistic, for instance to state the axiom scheme of Δ_{0}-induction. To formalize this in Metamath, there are several choices to make. A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph_{0} ...) and an axiom "$a wff ph_{0} " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph_{0} -> ps_{0} )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED " is a formula meaning that is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.) A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded. A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 9202. Indeed, if we posited it in closed form, then we could prove for instance BOUNDED and BOUNDED which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.) Having ax-bd0 9202 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 9203 through ax-bdsb 9211) can be written either in closed or inference form. The fact that ax-bd0 9202 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness. Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that is a bounded formula. However, since can be defined as "the such that PHI" a proof using the fact that is bounded can be converted to a proof in iset.mm by replacing with everywhere and prepending the antecedent PHI, since is bounded by ax-bdel 9210. For a similar method, see bj-omtrans 9344. Note that one cannot add an axiom BOUNDED since by bdph 9239 it would imply that every formula is bounded. For CZF, a useful set of notes is Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf) and an interesting article is Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. (available at https://arxiv.org/abs/1808.05204) | ||
Syntax | wbd 9201 | Syntax for the predicate BOUNDED. |
BOUNDED | ||
Axiom | ax-bd0 9202 | If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdim 9203 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdan 9204 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdor 9205 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Axiom | ax-bdn 9206 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdal 9207* | A bounded universal quantification of a bounded formula is bounded. Note the DV condition on . (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdex 9208* | A bounded existential quantification of a bounded formula is bounded. Note the DV condition on . (Contributed by BJ, 25-Sep-2019.) |
BOUNDED BOUNDED | ||
Axiom | ax-bdeq 9209 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Axiom | ax-bdel 9210 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Axiom | ax-bdsb 9211 | A formula resulting from proper substitution in a bounded formula is bounded. This probably cannot be proved from the other axioms, since neither the definiens in df-sb 1643, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdeq 9212 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bd0 9213 | A formula equivalent to a bounded one is bounded. See also bd0r 9214. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bd0r 9214 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 9213) biconditional in the hypothesis, to work better with definitions ( is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdbi 9215 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdstab 9216 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED STAB | ||
Theorem | bddc 9217 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED DECID | ||
Theorem | bd3or 9218 | A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED BOUNDED | ||
Theorem | bd3an 9219 | A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED BOUNDED | ||
Theorem | bdth 9220 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED | ||
Theorem | bdtru 9221 | The truth value is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdfal 9222 | The truth value is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdnth 9223 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED | ||
Theorem | bdnthALT 9224 | Alternate proof of bdnth 9223 not using bdfal 9222. Then, bdfal 9222 can be proved from this theorem, using fal 1249. The total number of proof steps would be 17 (for bdnthALT 9224) + 3 = 20, which is more than 8 (for bdfal 9222) + 9 (for bdnth 9223) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdxor 9225 | The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bj-bdcel 9226* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdab 9227 | Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcdeq 9228 | Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED CondEq | ||
In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 9230. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas. As will be clear by the end of this subsection (see for instance bdop 9264), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, BOUNDED BOUNDED . The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like BOUNDED BOUNDED . | ||
Syntax | wbdc 9229 | Syntax for the predicate BOUNDED. |
BOUNDED | ||
Definition | df-bdc 9230* | Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceq 9231 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceqi 9232 | A class equal to a bounded one is bounded. Note the use of ax-ext 2019. See also bdceqir 9233. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdceqir 9233 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 9232) equality in the hypothesis, to work better with definitions ( is the definiendum that one wants to prove bounded; see comment of bd0r 9214). (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdel 9234* | The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdeli 9235* | Inference associated with bdel 9234. Its converse is bdelir 9236. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdelir 9236* | Inference associated with df-bdc 9230. Its converse is bdeli 9235. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcv 9237 | A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdcab 9238 | A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdph 9239 | A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bds 9240* | Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 9211; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 9211. (Contributed by BJ, 19-Nov-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcrab 9241* | A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdne 9242 | Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdnel 9243* | Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdreu 9244* |
Boundedness of existential uniqueness.
Remark regarding restricted quantifiers: the formula need not be bounded even if and are. Indeed, is bounded by bdcvv 9246, and (in minimal propositional calculus), so by bd0 9213, if were bounded when is bounded, then would be bounded as well when is bounded, which is not the case. The same remark holds with . (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdrmo 9245* | Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcvv 9246 | The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ_{0}". (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdsbc 9247 | A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 9248. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdsbcALT 9248 | Alternate proof of bdsbc 9247. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bdccsb 9249 | A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcdif 9250 | The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdcun 9251 | The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdcin 9252 | The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED BOUNDED | ||
Theorem | bdss 9253 | The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcnul 9254 | The empty class is bounded. See also bdcnulALT 9255. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED | ||
Theorem | bdcnulALT 9255 | Alternate proof of bdcnul 9254. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 9233, or use the corresponding characterizations of its elements followed by bdelir 9236. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdeq0 9256 | Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bj-bd0el 9257 | Boundedness of the formula "the empty set belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcpw 9258 | The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcsn 9259 | The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdcpr 9260 | The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdctp 9261 | The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdsnss 9262* | Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdvsn 9263* | Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdop 9264 | The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bdcuni 9265 | The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.) |
BOUNDED | ||
Theorem | bdcint 9266 | The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdciun 9267* | The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdciin 9268* | The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED BOUNDED | ||
Theorem | bdcsuc 9269 | The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
BOUNDED | ||
Theorem | bdeqsuc 9270* | Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.) |
BOUNDED | ||
Theorem | bj-bdsucel 9271 | Boundedness of the formula "the successor of the setvar belongs to the setvar ". (Contributed by BJ, 30-Nov-2019.) |
BOUNDED | ||
Theorem | bdcriota 9272* | A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.) |
BOUNDED BOUNDED | ||
In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory. | ||
Axiom | ax-bdsep 9273* | Axiom scheme of bounded (or restricted, or Δ_{0}) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3866. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsep2 9274* | Version of ax-bdsep 9273 with one DV condition removed and without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnft 9275* | Closed form of bdsepnf 9276. Version of ax-bdsep 9273 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. (Contributed by BJ, 19-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnf 9276* | Version of ax-bdsep 9273 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 9277. (Contributed by BJ, 5-Oct-2019.) |
BOUNDED | ||
Theorem | bdsepnfALT 9277* | Alternate proof of bdsepnf 9276, not using bdsepnft 9275. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
BOUNDED | ||
Theorem | bdzfauscl 9278* | Closed form of the version of zfauscl 3868 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.) |
BOUNDED | ||
Theorem | bdbm1.3ii 9279* | Bounded version of bm1.3ii 3869. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bj-nalset 9280* | nalset 3878 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vprc 9281 | vprc 3879 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-nvel 9282 | nvel 3880 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-vnex 9283 | vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bdinex1 9284 | Bounded version of inex1 3882. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex2 9285 | Bounded version of inex2 3883. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdinex1g 9286 | Bounded version of inex1g 3884. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssex 9287 | Bounded version of ssex 3885. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexi 9288 | Bounded version of ssexi 3886. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexg 9289 | Bounded version of ssexg 3887. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdssexd 9290 | Bounded version of ssexd 3888. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED | ||
Theorem | bdrabexg 9291* | Bounded version of rabexg 3891. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
BOUNDED BOUNDED | ||
Theorem | bj-inex 9292 | The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intexr 9293 | vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-intnexr 9294 | vnex 3881 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Theorem | bj-zfpair2 9295 | Proof of zfpair2 3936 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-prexg 9296 | Proof of prexg 3938 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-snexg 9297 | snexg 3927 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-snex 9298 | snex 3928 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) |
Theorem | bj-sels 9299* | If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.) |
Theorem | bj-axun2 9300* | axun2 4138 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.) |
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