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Intuitionistic Logic (Wikipedia [accessed 19-Jul-2015], Stanford Encyclopedia of Philosophy [accessed 19-Jul-2015]) can be thought of as a constructive logic in which we must build and exhibit concrete examples of objects before we can accept their existence. Unproved statements in intuitionistic logic are not given an intermediate truth value, instead, they remain of unknown truth value until they are either proved or disproved. Intuitionist logic can also be thought of as a weakening of classical logic such that the law of excluded middle (LEM), (φ ∨ ¬ φ), doesn't always hold. Specifically, it holds if we have a proof for φ or we have a proof for ¬ φ, but it doesn't necessarily hold if we don't have a proof of either one. There is also no rule for double negation elimination. Brouwer observed in 1908 that LEM was abstracted from finite situations, then extended without justification to statements about infinite collections.
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(Placeholder for future use)
(By Gérard Lang, 7-May-2018)
Mario Carneiro's work (Metamath database) "iset.mm" provides in Metamath a development of "set.mm" whose eventual aim is to show how many of the theorems of set theory and mathematics that can be derived from classical first order logic can also be derived from a weaker system called "intuitionistic logic." To achieve this task, iset.mm adds (or substitutes) intuitionistic axioms for a number of the classical logical axioms of set.mm.
Among these new axioms, the 6 first (ax-ia1, ax-ia2, ax-ia3, ax-io, ax-in1 and ax-in2), when added to ax-1, ax-2 and ax-mp, allow for the development of intuitionistic propositional logic. We omit the classical axiom ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) (which is ax-3 in set.mm). Each of our new axioms is a theorem of classical propositional logic, but ax-3 cannot be derived from them. Similarly, other basic classical theorems, like the third middle excluded or the equivalence of a proposition with its double negation, cannot be derived in intuitionistic propositional calculus. Glivenko showed that a proposition φ is a theorem of classical propositional calculus if and only if ¬¬φ is a theorem of intuitionistic propositional calculus.
The next 4 new axioms (ax-ial, ax-i5r, ax-ie1 and ax-ie2) together with the set.mm axioms ax-4, ax-5, ax-7 and ax-gen do not mention equality or distinct variables.
The ax-i9 axiom is just a slight variation of the classical ax-9. The classical axiom ax-12 is strengthened into first ax-i12 and then ax-bnd (two results which would be fairly readily equivalent to ax-12 classically but which do not follow from ax-12, at least not in an obvious way, in intuitionistic logic). The substitution of ax-i9, ax-i12, and ax-bnd for ax-9 and ax-12 and the inclusion of ax-8, ax-10, ax-11, ax-13, ax-14 and ax-17 allow for the development of the intuitionistic predicate calculus.
Each of the new axioms is a theorem of classical first order logic with equality. But some axioms of classical first order logic with equality, like ax-3, cannot be derived in the intuitionistic predicate calculus.
One of the major interests of the intuitionistic predicate calculus is that its use can be considered as a realization of the program of the constructivist philosophical view of mathematics.
As with the classical axioms we have propositional logic and predicate logic.
The axioms of intuitionistic propositional logic consist of some of the axioms from classical propositional logic, plus additional axioms for the operation of the 'and', 'or' and 'not' connectives.
|Axiom Simp||ax-1||⊢ (φ → (ψ → φ))|
|Axiom Frege||ax-2||⊢ ((φ → (ψ → χ)) → ((φ → ψ) → (φ → χ)))|
|Rule of Modus Ponens||ax-mp||⊢ φ & ⊢ φ → ψ => ⊢ψ|
|Left 'and' elimination||ax-ia1||⊢ ((φ ∧ ψ) → φ)|
|Right 'and' elimination||ax-ia2||⊢ ((φ ∧ ψ) → ψ)|
|'And' introduction||ax-ia3||⊢ (φ → (ψ → (φ ∧ ψ)))|
|Definition of 'or'||ax-io||⊢ (((φ ∨ χ) → ψ) ↔ ((φ → ψ) ∧ (χ → ψ)))|
|'Not' introduction||ax-in1||⊢ ((φ → ¬ φ) → ¬ φ)|
|'Not' elimination||ax-in2||⊢ (¬ φ → (φ → ψ))|
Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of implication and 'not'. In particular, φ ∨ ψ is not equivalent to ¬ φ → ψ, nor is φ → ψ equivalent to ¬ φ ∨ ψ, nor is φ ∧ ψ equivalent to ¬ (φ → ¬ ψ).
The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if φ implies a contradiction (such as its own negation), then one can conclude ¬ φ. By contrast, assuming ¬ φ and then deriving a contradiction only serves to prove ¬ ¬ φ, which in intuitionistic logic is not the same as φ.
The biconditional can be defined as the conjunction of two implications, as in dfbi2 and df-bi.
Predicate logic adds set variables (individual variables) and the ability to quantify them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃ cannot be defined in terms of ∀. As in classical logic, we also add = for equality (which is key to how we handle substitution in metamath) and ∈ (which for current purposes can just be thought of as an arbitrary predicate, but which will later come to mean set membership).
Our axioms are based on the classical set.mm predicate logic axioms, but adapted for intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing some aspects of how we handle negations.
|Axiom of Specialization||ax-4||⊢ (∀xφ → φ)|
|Axiom of Quantified Implication||ax-5||⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))|
|The converse of ax-5o||ax-i5r||⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ))|
|Axiom of Quantifier Commutation||ax-7||⊢ (∀x∀yφ → ∀y∀xφ)|
|Rule of Generalization||ax-gen||⊢ φ => ⊢ ∀xφ|
|x is bound in ∀xφ||ax-ial||⊢ (∀xφ → ∀x∀xφ)|
|x is bound in ∃xφ||ax-ie1||⊢ (∃xφ → ∀x∃xφ)|
|Define existential quantification||ax-ie2||⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ)))|
|Axiom of Equality||ax-8||⊢ (x = y → (x = z → y = z))|
|Axiom of Existence||ax-i9||⊢ ∃x x = y|
|Axiom of Quantifier Substitution||ax-10||⊢ (∀x x = y → ∀y y = x)|
|Axiom of Variable Substitution||ax-11||⊢ (x = y → (∀yφ → ∀x(x = y → φ)))|
|Axiom of Quantifier Introduction||ax-i12||⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))|
|Axiom of Bundling||ax-bnd||⊢ (∀z z = x ∨ (∀z z = y ∨ ∀x∀z(x = y → ∀z x = y)))|
|Left Membership Equality||ax-13||⊢ (x = y → (x ∈ z → y ∈ z))|
|Right Membership Equality||ax-14||⊢ (x = y → (z ∈ x → z ∈ y))|
|Distinctness||ax-17||⊢ (φ → ∀xφ), where x does not occur in φ|
Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. These axioms were developed in response to Russell's Paradox, a discovery that revolutionized the foundations of mathematics and logic.
In the IZF axioms that follow, all set variables are assumed to be distinct. If you click on their links you will see the explicit distinct variable conditions.
|Axiom of Extensionality||ax-ext||⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)|
|Axiom of Collection||ax-coll||⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀x ∈ 𝑎 ∃y ∈ 𝑏 φ)|
|Axiom of Separation||ax-sep||⊢ ∃y∀x(x ∈ y ↔ (x ∈ z ∧ φ))|
|Axiom of Power Sets||ax-pow||⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y)|
|Axiom of Pairing||ax-pr||⊢ ∃z∀w((w = x ∨ w = y) → w ∈ z)|
|Axiom of Union||ax-un||⊢ ∃y∀z(∃w(z ∈ w ∧ w ∈ x) → z ∈ y)|
|Axiom of Set Induction||ax-setind||⊢ (∀𝑎(∀y ∈ 𝑎 [y / 𝑎]φ → φ) → ∀𝑎φ)|
|Axiom of Infinity||ax-iinf||⊢ ∃x(∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x))|
We develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in metamath because the Axiom of Restricted Separation would require us to develop the ability to classify formulas as bounded formulas, similar to the machinery we have built up for asserting on whether variables are free in formulas.
Listed here are some examples of starting points for your journey through the maze. Some are stated just as they would be in a non-constructive context; others are here to highlight areas which look different constructively. You should study some simple proofs from propositional calculus until you get the hang of it. Then try some predicate calculus and finally set theory. The Theorem List shows the complete set of theorems in the database. You may also find the Bibliographic Cross-Reference useful.
|Propositional calculus Predicate calculus Set theory|
For the student or master of classical mathematics, constructive mathematics can be baffling. One can get over some of the intial hurdles of understanding how constructive mathematics works and why it might be interesting by reading [Bauer] but that work does little to explain in concrete terms how to write proofs in intuitionistic logic. Fortunately, metamath helps with one of the biggest hurdles: noticing when one is even using the law of the excluded middle or the axiom of choice. But suppose you have a classical proof from the Metamath Proof Explorer and it fails to verify when you copy it over to the Intuitionistic Logic Explorer. What then? Here are some rules of thumb in converting classical proofs to intuitionistic ones.
This is a list of theorems from the Metamath Proof Explorer (which assumes the law of the excluded middle throughout) which we do not have in the Intuitionistic Logic Explorer (generally because they are not provable without the law of the excluded middle, although some could be proved but aren't for a variety of reasons), together with the closest replacements.
|ax-3 , con4d , con34b , necon4bd||con3||The form of contraposition which removes negation does not hold in intutionistic logic.|
|pm2.18||pm2.01||See for example [Bauer] who uses the terminology "proof of negation" versus "proof by contradiction" to distinguish these.|
|notnotrd , notnotri , notnot2 , notnot||notnot1||Double negation introduction holds but not double negation elimination.|
|pm2.61 , pm2.61d , pm2.61d1 , pm2.61d2 , pm2.61i , pm2.61ii , pm2.61nii , pm2.61iii , pm2.61ian , pm2.61dan , pm2.61ddan , pm2.61dda , pm2.61ine , pm2.61ne , pm2.61dne , pm2.61dane , pm2.61da2ne , pm2.61da3ne , pm2.61iine||none|
|df-or , pm4.64 , pm2.54 , orri , orrd||pm2.53, ori, ord|
|ecase3d||none||This is a form of case elimination.|
|dfif4||none||Unused in set.mm|
|dfif5||none||Unused in set.mm|
|elimif , ifbothda , ifboth , ifid , eqif , ifval , elif , ifel , ifcl , ifcld , ifeqor , 2if2 , ifcomnan , csbif , csbifgOLD||none|
|dedth , dedth2h , dedth3h , dedth4h , dedth2v , dedth3v , dedth4v , elimhyp , elimhyp2v , elimhyp3v , elimhyp4v , elimel , elimdhyp , keephyp , keephyp2v , keephyp3v , keepel , ifex , ifexg||none||Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.|
|moabex||euabex||In general, most of the set.mm theorems still hold, but a decent number of the ones get caught up on "there are two cases: the set exists or it does not"|
|snex||snexg, snex||The iset.mm version of snex has an additional hypothesis|
|opex||opexg, opex||The iset.mm version of opex has additional hypotheses|
|df-so||df-iso||Although we define to describe a weakly linear order (such as real numbers), there are some orders which are also trichotomous, for example nntri3or, pitri3or, and nqtri3or.|
|sotric||sotricim||One direction, for any weak linear order.|
|sotritric||For a trichotomous order.|
|nntri2||For the specific order|
|pitric||For the specific order|
|nqtric||For the specific order|
|sotrieq||sotritrieq||For a trichotomous order|
|sotrieq2||see sotrieq and then apply ioran|
|issoi||issod, ispod||Many of the set.mm usages of issoi don't carry over, so there is less need for this convenience theorem.|
|isso2i||issod||Presumably this could be proved if we need it.|
|df-fr and all theorems using Fr||none||Because iset.mm assumes ax-setind without reluctance, all sets are well-founded. We could adopt a treatment more like set.mm if people want to investigate set theories which are constructive but which do not assume ax-setind.|
|df-we (and all theorems using We)||none||Ordering is moderately different in constructive logic, so if there is anything along these lines worth doing it will be different from set.mm.|
|ordsseleq||onelss, eqimss||Taken together, onelss and eqimss represent the reverse direction of the biconditional from ordsseleq|
|ordtri3or||nntri3or||Ordinal trichotomy implies the law of the excluded middle as shown in ordtriexmid.|
|ordtri2||nntri2||ordtri2 for all ordinals presumably implies excluded middle although we don't have a specific proof analogous to ordtriexmid.|
|ordtri3 , ordtri4 , ordtri2or2 , ordtri2or3 , dford2||none||Ordinal trichotomy implies the law of the excluded middle as shown in ordtriexmid. We don't have similar proofs for every variation of of trichotomy but most of them are presumably powerful enough to imply excluded middle.|
|ordtri1 , ontri1 , onssneli , onssnel2i||ssnel, nntri1||ssnel is a trichotomy-like theorem which does hold, although it is an implication whereas ordtri1 is a biconditional. nntri1 is biconditional, but just for natural numbers.|
|ordtr2 , ontr2||none||See also ordelpss in set.mm|
|ordtr3||none||This is weak linearity of ordinals, which presumably implies excluded middle by ordsoexmid.|
|ordtri2or||none||Implies excluded middle as shown at ordtri2orexmid.|
|ord0eln0 , on0eln0||ne0i, nn0eln0|
|ordequn||none||If you know which ordinal is larger, you can achieve a similar result via theorems such as oneluni or ssequn1.|
|fvex||funfvex||when evaluating a function within its domain|
|fvexg, fvex||when the function is a set and is evaluated at a set|
|relrnfvex||when evaluating a relation whose range is a set|
|1stexg, 2ndexg||for the functions and|
|ndmfv||ndmfvg||The case is fvprc.|
|csbriota , csbriotagOLD||csbriotag|
|riotaxfrd||none||Although it may be intuitionizable, it is lightly used in set.mm.|
|ovex||fnovex||Sometimes or fvexg or relrnfvex may work.|
|ov3||ovi3||Although set.mm's ov3 could be proved, it is only used a few places in set.mm, and in iset.mm those places need the modified form found in ovi3.|
|ndmovg , ndmov , ndmovcl , ndmovrcl , ndmovcom , ndmovass , ndmovdistr , ndmovord , and ndmovordi||none||Although these theorems are moderately widely used in set.mm, in many cases they are being used for case elimination and the set.mm proofs are not intuitionistic. We might bring back some of them later (which may have conditions that the sets exist, but aren't in the domain), but maybe we'll find a way to avoid them.|
|caov4||caov4d||Note that caov4d has a closure hypothesis.|
|caov411||caov411d||Note that caov411d has a closure hypothesis.|
|caov42||caov42d||Note that caov42d has a closure hypothesis.|
|caovdir||caovdird||caovdird adds some constraints about where the operations are evaluated.|
|ssonprc||none||not provable (we conjecture), but interesting enough to intuitionize anyway. is provable, and is provable. (Why isn't df-pss stated so that the set difference is inhabited? If so, you could prove .)|
|onint||none||Conjectured to not be provable without excluded middle. If you apply onint to a pair you can derive totality of the order.|
|onint0||none||Thought to be "trivially not intuitionistic", and it is not clear if there is an alternate way to state it that is true. If the empty set is in A then of course |^| A = (/), but the converse seems difficult. I don't know so much about the structure of the ordinals without linearity,|
|onssmin, onminesb, onminsb||none||Conjectured to not be provable without excluded middle, for the same reason as onint.|
|oninton||none yet||This one (with non-empty changed to inhabited) I think can still be salvaged though. From the fact that it is inhabited you get that it exists, and is a subset of an ordinal x. It is an intersection of transitive sets so it is transitive, and of course all its members are members of x so they are transitive too. And E. Fr A falls to subsets.|
|onintrab, onintrab2||none yet||The set.mm proof uses oninton.|
|oneqmin||none||Falls as written because it implies onint, but it might be useful to keep the reverse direction for subsets that do have a minimum.|
|onminex||none yet||falls as written because it implies onint, but it might be useful to keep the reverse direction for subsets that do have a minimum.|
|onmindif2||none||Conjectured to not be provable without excluded middle.|
|onmindif2||none||Conjectured to not be provable without excluded middle.|
|ordpwsuc||ordpwsucss||See the ordpwsucss comment for discussion of the succcessor-like properites of . Full ordpwsuc implies excluded middle as seen at ordpwsucexmid.|
|ordsucelsuc||onsucelsucr, nnsucelsuc||The converse of onsucelsucr implies excluded middle, as shown at onsucelsucexmid.|
|ordsucsssuc||onsucsssucr, nnsucsssuc||The converse of onsucsssucr implies excluded middle, as shown at onsucsssucexmid.|
|ordsucuniel||sucunielr||Full ordsucuniel implies excluded middle, as shown at ordsucunielexmid.|
|ordsucun||none yet||Conjectured to be provable in the reverse direction, but not the forward direction (implies some order totality).|
|ordunpr||none||Presumably not provable without excluded middle.|
|ordunel||none||Conjectured to not be provable (ordunel implies ordsucun).|
|onsucuni, ordsucuni||none||Conjectured to not be provable without excluded middle.|
|orduniorsuc||none||Presumably not provable.|
|ordunisuc||onunisuci, unisuc, unisucg|
|onsucuni2||none yet||Conjectured to be provable.|
|0elsuc||none yet||Conjectured to be provable.|
|onuniorsuci||none||Conjectured to not be provable without excluded middle.|
|onuninsuci, ordununsuc||none||Conjectured to be provable in the forward direction but not the reverse one.|
|onsucssi||none yet||Conjectured to be provable.|
|nlimsucg||none yet||Conjectured to be provable.|
The forward direction is conjectured to imply excluded middle. Here is a sketch of the proposed proof.
Let om' be the set of all finite iterations of suc' A = on . (We can formalize this proof but not until we have om and at least finite induction.) Then om' = U. om' because if x e. om' then x = suc'^n (/) for some n, and then x C_ suc'^n (/) implies x e. suc'^(n+1) (/) e. om' so x e. U. om'.
Now supposing the theorem, we know that A. x e. om' suc x e. om', so in particular 2o e. om', that is, 2o = suc'^n (/) for some n. (Note that 1o = suc' (/) - see pw0.) For n = 0 and n = 1 this is clearly false, and for n = m+3 we have 1o e. suc' suc' (/) , so 2o C_ suc' suc' (/), so 2o e. suc' suc' suc' (/) C_ suc' suc' suc' suc'^m (/) = 2o, contradicting ordirr.
Thus 2o = suc' suc' (/) = suc' 1o. Applying this to X = we have X C_ 1o implies X e. suc' 1o = 2o and hence X = (/) \/ X = 1o, and LEM follows (by ordtriexmidlem2 for and rabsnt as seen in the onsucsssucexmid proof for ).
|ordzsl, onzsl, dflim3, nlimon||none|
|dflim4||df-ilim||We conjecture that dflim4 is not equivalent to df-ilim.|
|limsuc||none yet||Conjectured to be provable.|
|limsssuc||none yet||Conjectured to be provable.|
|1stnpr||none||May be intuitionizable, but very lightly used in set.mm.|
|2ndnpr||none||May be intuitionizable, but very lightly used in set.mm.|
|pwuninel||pwuninel2||The set.mm proof of pwuninel uses case elimination.|
|tfr2b , recsfnon , recsval||none||These transfinite recursion theorems are lightly used in set.mm.|
|df-rdg||df-irdg||This definition combines the successor and limit cases (rather than specifying them as separate cases in a way which relies on excluded middle). In the words of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic", "we can still define many of the familiar set-theoretic operations by transfinite recursion on ordinals (see Aczel and Rathjen 2001, Section 4.2). This is fine as long as the definitions by transfinite recursion do not make case distinctions such as in the classical ordinal cases of successor and limit."|
|ondif1||dif1o||In set.mm, ondif1 is used for Cantor Normal Form|
|ondif2 , dif20el||none||The set.mm proof is not intuitionistic|
|brwitnlem||none||The set.mm proof is not intuitionistic|
|frfnom||frecfnom||frecfnom adopts the frec notation and adds conditions on the characteristic function and initial value.|
|fr0g||frec0g||frec0g adopts the frec notation and adds a condition on the characteristic function.|
|frsuc||frecsuc||frecsuc adopts the frec notation and adds conditions on the characteristic function and initial value.|
|oaword||oawordi||The other direction presumably could be proven but isn't yet.|
|omwordi||nnmword||The set.mm proof of omwordi relies on case elimination.|
|nnawordex||nnaordex||nnawordex is only used a few places in set.mm|
|swoso||none||Unused in set.mm.|
|erdisj, qsdisj, qsdisj2, uniinqs||none||These could presumably be restated to be provable, but they are lightly used in set.mm|
|brecop2||none||This is a form of reverse closure and uses excluded middle in its proof.|
|erov , erov2||none||Unused in set.mm.|
|eceqoveq||none||Unused in set.mm.|
|ax-reg , axreg2 , zfregcl||ax-setind||ax-reg implies excluded middle as seen at regexmid|
|indpi||finds||Although indpi presumably could be proved, it is lightly used in set.mm|
|df-erq||none||Not needed given df-nqqs|
|elpqn||none||Not needed given df-nqqs|
|addnqf||dmaddpq, addclnq||It should be possible to prove that is a function, but so far there hasn't been a need to do so.|
|elnp , elnpi||elinp|
|ltprord||ltprordil||There hasn't yet been a need to investigate versions which are biconditional or which involve proper subsets.|
|ltexprlem1 , ltexprlem2 , ltexprlem3 , ltexprlem4||none||See the lemmas for ltexpri generally.|
|reclem4pr||recexprlemss1l, recexprlemss1u, recexprlemex|
|supexpr , suplem1pr , suplem2pr||none||The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle.|
|dmaddsr , dmmulsr||none||Although these presumably could be proved in a way similar to dmaddpq and dmmulpq (in fact dmaddpqlem would seem to be easily generalizable to anything of the form ), we haven't yet had a need to do so.|
|sqgt0sr||mulgt0sr||The case would also intuitionize (looks like this part of set.mm's sqgt0sr proof would intuitionize well), but for the theorem as a whole, "not equal to zero" would need to be changed to "apart from zero".|
|recexsr||recexsrlem||This would follow from sqgt0sr (as in the set.mm proof of recexsr), but as noted there "not equal to zero" would need to be changed to "apart from zero".|
|mappsrpr , ltpsrpr , map2psrpr||none||Although variants of these theorems could be intuitionized, in set.mm they are only used for supremum theorems, so we can consider this in more detail when we tackle what kind of supremum theorems to prove.|
|supsrlem , supsr||none||The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle.|
|axaddf , ax-addf , axmulf , ax-mulf||none||Because these are described as deprecated in set.mm, we haven't figured out what would be involved in proving them for iset.mm.|
|ax1ne0 , ax-1ne0||ax0lt1, ax-0lt1|
|axrrecex , ax-rrecex||axprecex, ax-precex|
|axpre-lttri , ax-pre-lttri||axpre-ltirr, axpre-ltwlin, ax-pre-ltirr, ax-pre-ltwlin|
|axpre-sup , ax-pre-sup , axsup||none yet||The Least Upper Bound property for sets of real numbers does not hold, in general, without excluded middle. If we want a set of axioms for real numbers which allows us to avoid construction-dependent theorems beyond this point, we'll need a modified Least Upper Bound property, a statement concerning Dedekind cuts or something similar, or some other axiom(s).|
|elimne0||none||Even in set.mm, the weak deduction theorem is discouraged in favor of theorems in deduction form.|
|ssxr||df-xr||Lightly used in set.mm|
|ltnle , ltnlei , ltnled||lenlt|
|lttri2 , lttri3 , lttri4 , letri3||none||Real number trichotomy is not provable.|
|leloe , eqlelt , leloei , leloed , eqleltd||none|
|leltne , leltned||none|
|lelttr , lelttri , lelttrd||none yet||This should be provable, as it is part of Definition 11.2.7(vi) of [HoTT], p. (varies). But the set.mm proof does not work as-is.|
|ltletr , ltletri , ltletrd||none yet||This should be provable, as it is part of Definition 11.2.7(vi) of [HoTT], p. (varies). But the set.mm proof does not work as-is.|
|letric , letrii , letrid||none||A form of real number trichotomy|
|ltlen , ltleni , ltlend||none|
|ltadd2 , ltadd2i , ltadd2d , ltadd2dd||none yet||This will be provable based on additive inverse once we have addcom and a few other prerequisites. We'll probably end up moving it down to the "Ordering on reals (cont.)" section.|
|ne0gt0||none||We presumably could prove this if we changed "not equal to zero" to "apart from zero".|
|lecasei , lelttric , ltlecasei||none||These are real number trichotomy|
|lttri2i||none||This can be read as "two real numbers are non-equal if and only if they are apart" which relies on excluded middle|
|lttri3i||none||The direction (which can also be read as "if two real numbers are not apart, then they are equal) is proved via trichotomy in set.mm|
|letri3i||none||Equivalent to lttri3i by lenlt|
|le2tri3i||none||This is an elaborated version of letri3i|
|ne0gt0d , lttrid , lttri2d , lttri3d , lttri4d , letri3d||none||These are real number trichotomy|
|dedekind , dedekindle||none|
|mul02lem1||none||The one use in set.mm is not needed in iset.mm.|
page was last updated on 15-Aug-2015.
Your comments are welcome: Norman Megill
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