6938:
6792:) will equal 0, so that IVT is valid automatically. Thus in classical analysis, which uses classical logic, in order to prove the full IVT, it is sufficient to prove the constructive version. From this perspective, the full IVT fails in constructive analysis simply because constructive analysis does not accept classical logic. Conversely, one may argue that the true meaning of IVT, even in classical mathematics, is the constructive version involving the
1400:
1814:
47:, which in this context means analysis done according to the more common principles of classical mathematics. However, there are various schools of thought and many different formalizations of constructive analysis. Whether classical or constructive in some fashion, any such framework of analysis axiomatizes the
1244:
proves the non-contradiction claim for any proposition, and that the negation of excluded middle for any given proposition would be absurd. This also means there is no consistent theory (even if anti-classical) rejecting the excluded middle disjunction for any given proposition. Indeed, it holds that
366:
being provable would be absurd, and so the latter cannot also be provable in a consistent theory. The double-negated existence claim is a logically negative statement and implied by, but generally not equivalent to the existence claim itself. Much of the intricacies of constructive analysis can be
1819:
even if these disjunction claims here do not carry any information. In the absence of further axioms breaking the meta-logical properties, constructive entailment instead generally reflects provability. Taboo statements that ought not be decidable (if the aim is to respect the provability
3777:. The constant sequences correspond to rational numbers. Algebraic operations such as addition and multiplication can be defined component-wise, together with a systematic reindexing for speedup. The definition in terms of sequences furthermore enables the definition of a strict order "
3170:
and zero are related, with respect to the ordering predicate on the reals, while a demonstration of the latter shows how negation of such conditions would imply to a contradiction. In turn, there is then also a strong and a looser notion of, e.g., being a third-order polynomial.
6876:.) Again, this is classically equivalent to the full least upper bound principle, since every set is located in classical mathematics. And again, while the definition of located set is complicated, nevertheless it is satisfied by many commonly studied sets, including all
5755:. An upshot is that the set theory will prove properties for the reals, i.e. for this class of sets, expressed using the logical equality. Constructive reals in the presence of appropriate choice axioms will be Cauchy-complete but not automatically order-complete.
6796:
condition, with the full IVT following by "pure logic" afterwards. Some logicians, while accepting that classical mathematics is correct, still believe that the constructive approach gives a better insight into the true meaning of theorems, in much this way.
1251:
3688:
has better completeness properties as a collection, but a more intricate theory of its order-relation and, in turn, worse locatedness properties. While less commonly employed, also this construction simplifies to the classical real numbers when assuming
1669:
6069:
are always uncountable also over a constructive framework. One variant of the diagonal construction relevant for the present context may be formulated as follows, proven using countable choice and for reals as sequences of rationals:
493:
1409:
to the non-existence claim of a sequence for which the excluded middle disjunction about equality-to-zero would be disprovable. No sequence with that disjunction being rejected can be exhibited. Assume the theories at hand are
6636:
However, constructive analysis provides several alternative formulations of IVT, all of which are equivalent to the usual form in classical analysis, but not in constructive analysis. For example, under the same conditions on
4975:, and likewise for the equality of two such reals. And also the sequences of rationals may be required to carry a modulus of convergence. Positivity properties may defined as being eventually forever apart by some rational.
927:
2897:
5234:
4442:
1620:
6397:
Through the optional adoption of further consistent axioms, the negation of decidability may be provable. For example, equality-to-zero is rejected to be decidable when adopting
Brouwerian continuity principles or
1109:
5389:
1558:
5321:
1044:
4201:
5751:. In that language, regular rational sequences are degraded to a mere representative of a Cauchy real. Equality of those reals is then given by the equality of sets, which is governed by the set theoretical
3553:
4519:
4354:
2749:
5152:
735:
6237:
5451:
2156:
790:
4356:, where the relation on the right hand side is in terms of rational numbers. Formally, a positive real in this language is a regular sequence together with a natural witnessing positivity. Further,
2946:. In words: A demonstration that a number is somehow apart from zero is also a demonstration that this number is non-zero. But constructively it does not follow that the doubly negative statement
2667:
2238:
5861:
4973:
3965:
560:
1455:
2414:
2310:
6891:
are constructively validâor from another point of view, there are several different concepts that are classically equivalent but not constructively equivalent. Indeed, if the interval were
3863:, eventually has an index beyond which all its elements are invertible. Various implications between the relations, as well as between sequences with various properties, may then be proven.
2476:
5505:
498:
While a constructive theory proves fewer theorems than its classical counter-part in its classical presentation, it may exhibit attractive meta-logical properties. For example, if a theory
323:
6005:
5930:
5743:
5053:
5012:
3775:
2602:
661:
153:
5972:
4575:
4710:
1395:{\displaystyle {\mathsf {T}}\,\,\,\vdash \,\,\,\forall (x\in {\mathbb {Q} }^{\mathbb {N} }).\,\neg \neg {\big (}(x\cong _{\mathrm {ext} }0)\lor \neg (x\cong _{\mathrm {ext} }0){\big )}}
1661:
620:
590:
364:
6325:
6516:. Notably, there are two anti-classical schools as incompatible with one-another. This article discusses principles compatible with the classical theory and choice is made explicit.
6372:
3967:. Such a modulus may be considered for a sequence of reals, but it may also be considered for all the reals themselves, in which case one is really dealing with a sequence of pairs.
2944:
2808:
1997:
6510:
6472:
6434:
4577:
are deemed equivalent to the zero sequence. Such definitions are of course compatible with classical investigations and variations thereof were well studied also before. One has
3594:
3445:
3074:
3044:
6280:
is adopted in the
Russian school of recursive mathematics. This principle strengthens the impact of proven negation of strict equality. A so-called analytical form of it grants
3407:
3910:
3480:
1809:{\displaystyle {\mathsf {T}}+{\mathrm {PEM} }\,\,\,\vdash \,\,\,\forall (x\in {\mathbb {Q} }^{\mathbb {N} }).\,(x\cong _{\mathrm {ext} }0)\lor \neg (x\cong _{\mathrm {ext} }0)}
6039:
5819:
3717:
3682:
2345:
275:
5690:
5659:
4822:
4793:
4289:
4260:
2979:
391:
199:
6067:
5789:
5627:
5529:
5077:
1167:
226:
3317:
1945:
1479:
1234:
520:
3930:
1199:
6177:
4639:
5576:
4764:
3224:
109:
4665:
4022:
3620:
3369:
3343:
3278:
6098:
5710:
5600:
4892:
4601:
4231:
3244:
2690:
2525:
2091:
2053:
2027:
1910:
1884:
1838:
83:
6125:
5893:
4042:
3642:
3168:
3121:
411:
4086:
4066:
3861:
3795:
3198:
3005:
2839:
2176:
3727:
In the commutative ring of real numbers, a provably non-invertible element equals zero. This and the most basic locality structure is abstracted in the theory of
7254:
6145:
4862:
4842:
4730:
4117:
3996:
3835:
3815:
3141:
3094:
2496:
2073:
As explained, various predicates will fail to be decidable in a constructive formulation, such as these formed from order-theoretical relations. This includes "
1499:
1129:
967:
947:
830:
810:
6399:
6966:
4291:
apart and so one may compute natural numbers exceeding any real. For the regular sequences, one defines the logically positive loose positivity property as
416:
115:. While this term is thus overloaded in the subject, all the frameworks share a broad common core of results that are also theorems of classical analysis.
6749:
can be constructed. This is a complicated condition, but there are several other conditions that imply it and that are commonly met; for example, every
6250:, "a historical misrepresentation has been perpetuated that associates diagonalization with non-constructivity" and a constructive component of the
1131:" is used in several contexts here. For any theory capturing arithmetic, there are many yet undecided and even provenly independent such statements
6821:(or supremum), possibly infinite. However, as with the intermediate value theorem, an alternative version survives; in constructive analysis, any
835:
229:
2846:
5157:
4359:
1563:
7540:
7015:
1049:
1999:, which constitutes a strict total order in its constructive formulation (also called linear order or, to be explicit about the context, a
5334:
1504:
5239:
972:
4126:
622:. Already in classical arithmetic, this is violated for the most basic propositions about sequences of numbers - as demonstrated next.
3496:
4447:
3685:
7247:
7223:
7204:
6956:
5899:
existence. As a result, the constructive theory of cardinal order can diverge substantially from the classical one. Here, sets like
5324:
4294:
2698:
2062:
theory is relevant as the structures discussed below are model thereof. However, this section thus does not concern aspects akin to
5082:
670:
6895:
in constructive analysis, then the classical IVT would follow from the first constructive version in the example; one could find
5764:
5747:
3448:
2316:
6182:
5394:
2108:
4262:, and in the latter case non-zero indices ought to be used. No two of the rational entries in a regular sequence are more than
3797:" fulfilling the desired axioms. Other relations discussed above may then be defined in terms of it. In particular, any number
1840:" in formalizations below as well. For implications of disjunctions of yet not proven or disproven propositions, one speaks of
740:
3007:. Consequently, many classically equivalent statements bifurcate into distinct statement. For example, for a fixed polynomial
7535:
3975:
Given such a model then enables the definition of more set theoretic notions. For any subset of reals, one may speak of an
2619:
5940:
2184:
6251:
5832:
4897:
3935:
529:
7240:
1856:
may be axiomatized such that all the non-logical axioms are in accordance with constructive principles. This concerns a
1421:
245:
2351:
7401:
6806:
2259:
3143:
is apart from zero is stronger than the mere statement that it is non-zero. A demonstration of former explicates how
2419:
5460:
4099:
of constructive analysis, modeling the order properties described above, proves theorems for sequences of rationals
284:
7294:
7153:
6614:
6549:
5977:
5902:
5715:
5025:
4984:
3747:
2549:
2528:
633:
125:
5946:
6271:
5535:
to begin with, the encoding also means these objects are in scope of the quantifiers in second-order arithmetic.
4528:
6805:
Another difference between classical and constructive analysis is that constructive analysis does not prove the
4670:
5532:
1625:
595:
565:
328:
32:
6283:
6330:
2902:
7299:
5752:
5630:
2760:
1950:
168:
156:
6532:. Generally speaking, theorem formulation in constructive analysis mirrors the classical theory closest in
7088:
6877:
3976:
3883:
6481:
6439:
6405:
3886:
is often employed in the constructive study of Cauchy sequences of reals, meaning the association of any
3561:
3412:
3049:
3010:
1415:
7545:
7504:
6382:
3374:
2102:
164:
28:
6937:
6277:
3889:
3452:
2093:", which will be rendered equivalent to a negation. Crucial disjunctions are now discussed explicitly.
6014:
5794:
3692:
3657:
2320:
250:
59:
definable from an asymmetric order structure. Center stage takes a positivity predicate, here denoted
7434:
7375:
7337:
7332:
7279:
6892:
6525:
6008:
5665:
5634:
4798:
4769:
4522:
4265:
4236:
3654:
Demanding good order properties as above but strong completeness properties at the same time implies
2949:
1406:
523:
370:
241:
174:
6048:
5770:
5608:
5510:
5058:
1134:
204:
7514:
7347:
7263:
6951:
6862:
6553:
4978:
3872:
3487:
3287:
1915:
1460:
1237:
1215:
1202:
501:
5331:
as well, or extension encoding tags or metadata. For an example using this encoding, the sequence
3915:
3226:. However, see the discussion of possible further axiomatic principles regarding the strength of "
2501:
The theory shall validate further axioms concerning the relation between the positivity predicate
1172:
7509:
7424:
7289:
7168:
6943:
6262:
All these considerations may also be undertaken in a topos or appropriate dependent type theory.
6150:
5896:
3875:
on a finite set of rationals is decidable, an absolute value map on the reals may be defined and
2609:
1841:
119:
56:
4606:
3622:, but the other direction is also not provable in general. In a constructive real closed field,
1820:
interpretation of constructive claims) can be designed for definitions of a custom equivalence "
7499:
7464:
7452:
7429:
7416:
7406:
7393:
7219:
7200:
7011:
6904:
6818:
6750:
5555:
5454:
4743:
3203:
2248:
2059:
1853:
1206:
969:. Here, for the sake of demonstration, defining an extensional equality to the zero sequence
664:
88:
4644:
4001:
3599:
3348:
3322:
3257:
7457:
7148:
7096:
6475:
6378:
6247:
6077:
6042:
5695:
5585:
5328:
4867:
4580:
4206:
3409:, which as noted will not be provable in general. But neither will the totality disjunction
3229:
2692:", rendering it a negation. All negations are stable in intuitionistic logic, and therefore
2675:
2504:
2416:. Likewise, the asymmetric order in the theory ought to fulfill the weak linearity property
2076:
2032:
2006:
1889:
1863:
1857:
1823:
62:
48:
7025:
6613:. In constructive analysis, this does not hold, because the constructive interpretation of
6103:
5878:
5629:
and validates the existence of even classically uncountable function spaces (and certainly
4027:
3627:
3146:
3099:
396:
7444:
7360:
7355:
7317:
7021:
6773:
6633:
hovers near zero during a stretch along its domain, then this cannot necessarily be done.
6626:
6591:
6579:
6560:
6533:
6529:
6513:
5933:
5872:
4071:
4051:
3876:
3840:
3780:
3177:
2984:
2818:
2161:
630:
A common strategy of formalization of real numbers is in terms of sequences or rationals,
2531:
for polynomial. In this theory, between any two separated numbers, other numbers exist.
663:
and so we draw motivation and examples in terms of those. So to define terms, consider a
6658:
6642:
6130:
4847:
4827:
4715:
4102:
3981:
3820:
3800:
3126:
3079:
2481:
1484:
1114:
952:
932:
815:
795:
5763:
In this context it may also be possible to model a theory or real numbers in terms of
3490:, the conjunction of such statements is also rendered a negation of apartness, and so
488:{\displaystyle {\big (}\exists x.\theta (x){\big )}\to \neg \forall x.\neg \theta (x)}
7529:
7192:
6961:
6900:
6888:
6537:
6386:
4096:
4048:
is an upper bound given through a sequence of reals, positively characterized using "
3728:
2067:
1241:
7101:
7365:
7271:
6971:
2000:
160:
7079:
5055:
can be coded rather compactly, as they each may be mapped to a unique subclass of
7042:, Mathematical Logic Quarterly, Volume 48, Issue 2, Pages: 163-320, February 2002
6887:
Closely related to this, in constructive mathematics, fewer characterisations of
3744:
A common approach is to identify the real numbers with non-volatile sequences in
2348:. Other disjunctions are however implied based on other positivity results, e.g.
7383:
7309:
6881:
6869:
6689:
Alternatively, we can keep the same conclusion as in the classical IVTâa single
6678:. That is, we can get as close to zero as we like, even if we can't construct a
6595:
2527:
and the algebraic operations including multiplicative inversion, as well as the
1411:
278:
20:
6243:
Formulations of the reals aided by explicit moduli permit separate treatments.
367:
framed in terms of the weakness of propositions of the logically negative form
7322:
6933:
6648:(no matter how large), there exists (that is, we can construct) a real number
6610:
5582:. Note, however, that there are properties which can distinguish between two
7476:
7327:
6564:
5552:
is given to such well-behaved sequences or rationals, and relations such as
4795:. Other formalizations directly take as definition that for any fixed bound
52:
4864:
must eventually be forever at least as close. Exponentially falling bounds
1663:
again amounts to the proposition of universally quantified form. Trivially
922:{\displaystyle q_{n}\,:=\,{\textstyle \sum }_{k=0}^{n}\chi _{Q}(n)/2^{n+1}}
2892:{\displaystyle (\phi \to \neg \psi )\leftrightarrow (\psi \to \neg \phi )}
7055:; Studies in Logic and the Foundations of Mathematics; (2023) pp. 201-207
5863:, holding pairs representing inhabited, pairwise intersecting intervals.
5229:{\displaystyle \langle i,n_{i},d_{i},s_{i}\rangle \in {\mathbb {N} }^{4}}
4732:, but then shown to be equivalent of the logical negation of the former.
4437:{\displaystyle x\cong y\,:=\,\forall n.|x_{n}-y_{n}|\leq {\tfrac {2}{n}}}
4045:
2063:
1615:{\displaystyle {\mathsf {T}}\,\nvdash \,\neg (g\cong _{\mathrm {ext} }0)}
6625:(in the sense that it can be approximated to any desired precision by a
2243:
and indeed at most one of the three can hold at once. But the stronger,
7469:
7116:
3932:) is required in the form of an explicit, strictly increasing function
1104:{\displaystyle q\cong _{\mathrm {ext} }0\leftrightarrow \forall n.Q(n)}
7232:
6994:; Studies in Logic and the Foundations of Mathematics; Springer, 1988;
667:
predicate on the naturals, which in the constructive vernacular means
7151:, "The Mathematical Development of Set Theory from Cantor to Cohen",
6810:
6524:
Many classical theorems can only be proven in a formulation that is
5384:{\displaystyle i\mapsto {\textstyle \sum }_{k=0}^{i}{\tfrac {1}{k}}}
4068:". If a subset with an upper bound is well-behaved with respect to "
3912:
to an appropriate index (beyond which the sequences are closer than
1553:{\displaystyle {\mathsf {T}}\,\nvdash \,(g\cong _{\mathrm {ext} }0)}
7173:
6402:
in recursive mathematics. The weak continuity principle as well as
5829:
Another approach is to define a real number as a certain subset of
3644:" is a negation and is not equivalent to the disjunction in general
7051:
Bridges D., Ishihara H., Rathjen M., Schwichtenberg H. (Editors),
5316:{\displaystyle 2^{i}\cdot 3^{n_{i}}\cdot 5^{d_{i}}\cdot 7^{s_{i}}}
1039:{\displaystyle (q\cong _{\mathrm {ext} }0)\,:=\,\forall n.q_{n}=0}
4196:{\displaystyle |x_{n}-x_{m}|\leq {\tfrac {1}{n}}+{\tfrac {1}{m}}}
2613:. With it, the substitute of the principles above give tightness
929:
is monotone, with values non-strictly growing between the bounds
7481:
3548:{\displaystyle (x\geq y\land y\geq x)\leftrightarrow (x\cong y)}
7236:
6772:
For another way to view this example, notice that according to
4514:{\displaystyle \neg \exists n.|x_{n}-y_{n}|>{\tfrac {2}{n}}}
4349:{\displaystyle x>0\,:=\,\exists n.x_{n}>{\tfrac {1}{n}}}
2744:{\displaystyle \neg \neg (x\cong y)\leftrightarrow (x\cong y)}
118:
Constructive frameworks for its formulation are extensions of
111:. The members of the collection are generally just called the
325:
is provable, this exactly means that the non-existence claim
7005:
5952:
5147:{\displaystyle i\mapsto {\tfrac {n_{i}}{d_{i}}}(-1)^{s_{i}}}
4667:
may be defined from a numerical non-negativity property, as
730:{\displaystyle \forall n.{\big (}Q(n)\lor \neg Q(n){\big )}}
7167:
Diener, Hannes (2020). "Constructive
Reverse Mathematics".
7118:
6780:
condition fails, then it must fail at some specific point
6552:(IVT). In classical analysis, IVT implies that, given any
6232:{\displaystyle \forall (n\in {\mathbb {N} }).x_{n}\,\#\,z}
5446:{\displaystyle 1,2,{\tfrac {5}{2}},{\tfrac {8}{3}},\dots }
3879:
and limits of sequences of reals can be defined as usual.
2151:{\displaystyle (\phi \lor \psi )\to (\neg \phi \to \psi )}
5531:. While this example, an explicit sequence of sums, is a
785:{\displaystyle \chi _{Q}\colon {\mathbb {N} }\to \{0,1\}}
6753:
is locally non-zero (assuming that it already satisfies
6701:) is exactly zeroâwhile strengthening the conditions on
5745:
may be collected into a set and then this is called the
4525:. Via this predicate, the regular sequences in the band
6825:
subset of the real line has a supremum. (Here a subset
4024:. One may speak of least upper bounds with respect to "
1481:
proves the zero-sequence to be a good approximation to
7040:
5821:
or dependent choice, these structures are isomorphic.
5605:
In contrast, in a set theory that models the naturals
5426:
5411:
5370:
5346:
5093:
4803:
4774:
4691:
4556:
4500:
4423:
4335:
4270:
4241:
4182:
4167:
3371:. Decidability of positivity may thus be expressed as
2672:
Thus, apartness can also function as a definition of "
856:
6484:
6442:
6408:
6333:
6286:
6185:
6153:
6133:
6106:
6080:
6051:
6017:
5980:
5949:
5905:
5881:
5835:
5797:
5773:
5718:
5698:
5668:
5637:
5611:
5588:
5558:
5513:
5463:
5397:
5337:
5242:
5160:
5085:
5061:
5028:
4987:
4900:
4870:
4850:
4830:
4801:
4772:
4746:
4718:
4673:
4647:
4609:
4583:
4531:
4450:
4362:
4297:
4268:
4239:
4209:
4129:
4105:
4074:
4054:
4030:
4004:
3984:
3938:
3918:
3892:
3843:
3823:
3803:
3783:
3750:
3695:
3660:
3630:
3602:
3564:
3499:
3455:
3415:
3377:
3351:
3325:
3290:
3260:
3232:
3206:
3180:
3149:
3129:
3102:
3082:
3052:
3013:
2987:
2952:
2905:
2849:
2821:
2763:
2754:
The elusive trichotomy disjunction itself then reads
2701:
2678:
2662:{\displaystyle \neg (x\#0)\leftrightarrow (x\cong 0)}
2622:
2552:
2507:
2484:
2422:
2354:
2323:
2262:
2187:
2164:
2111:
2079:
2035:
2009:
1953:
1918:
1892:
1866:
1826:
1672:
1628:
1566:
1507:
1501:, but it can also meta-logically be established that
1487:
1463:
1424:
1254:
1218:
1175:
1137:
1117:
1052:
975:
955:
935:
838:
818:
798:
743:
673:
636:
598:
568:
532:
504:
419:
399:
373:
331:
287:
253:
207:
177:
128:
91:
65:
7130:
Bauer, A., Hanson, J. A. "The countable reals", 2022
2233:{\displaystyle \neg (x>0\lor 0>x)\to x\cong 0}
1886:, with a positive unit and non-positive zero, i.e.,
7492:
7443:
7415:
7392:
7374:
7346:
7308:
7270:
6536:. Some theorems can only be formulated in terms of
5856:{\displaystyle {\mathbb {Q} }\times {\mathbb {Q} }}
4968:{\displaystyle \forall n.|x_{n}-x_{n+1}|<2^{-n}}
4894:are also used, also say in a real number condition
3960:{\displaystyle \varepsilon \mapsto N(\varepsilon )}
555:{\displaystyle {\mathsf {T}}\vdash \phi \lor \psi }
6504:
6466:
6428:
6366:
6319:
6231:
6171:
6139:
6119:
6092:
6061:
6033:
5999:
5966:
5924:
5887:
5855:
5813:
5783:
5737:
5704:
5684:
5653:
5621:
5594:
5570:
5523:
5499:
5457:and with the above coding it maps to the subclass
5445:
5383:
5315:
5236:. In turn, this can be encoded as unique naturals
5228:
5146:
5071:
5047:
5006:
4967:
4886:
4856:
4836:
4816:
4787:
4758:
4724:
4704:
4659:
4633:
4595:
4569:
4513:
4436:
4348:
4283:
4254:
4225:
4195:
4111:
4080:
4060:
4036:
4016:
3990:
3959:
3924:
3904:
3855:
3829:
3809:
3789:
3769:
3711:
3676:
3636:
3614:
3588:
3547:
3474:
3439:
3401:
3363:
3337:
3311:
3272:
3238:
3218:
3192:
3162:
3135:
3115:
3088:
3068:
3038:
2999:
2973:
2938:
2891:
2833:
2802:
2743:
2684:
2661:
2596:
2519:
2490:
2470:
2408:
2339:
2304:
2232:
2170:
2150:
2085:
2047:
2021:
1991:
1939:
1904:
1878:
1832:
1808:
1655:
1614:
1552:
1493:
1473:
1450:{\displaystyle g\in {\mathbb {Q} }^{\mathbb {N} }}
1449:
1394:
1228:
1193:
1161:
1123:
1103:
1038:
961:
941:
921:
824:
804:
784:
729:
655:
614:
584:
554:
514:
487:
405:
385:
358:
317:
269:
220:
193:
147:
103:
77:
6845:are real numbers, either there exists an element
5895:" in set theory is the primary notion defined as
2409:{\displaystyle (x+y>0)\to (x>0\lor y>0)}
6992:Constructivism in mathematics: an introduction 1
6801:The least-upper-bound principle and compact sets
6721:, there exists (we can construct) a real number
4444:, which is logically equivalent to the negation
2607:may be independently defined and constitutes an
2305:{\displaystyle (x>0\lor 0>x)\lor x\cong 0}
792:be the characteristic function defined to equal
7010:. Cambridge, U.K.: Cambridge University Press.
5932:or some models of the reals can be taken to be
2471:{\displaystyle (y>x)\to (y>t\lor t>x)}
5500:{\displaystyle \{15,90,24300,6561000,\dots \}}
3280:may be defined by or proven equivalent to the
2253:, i.e. it is not provable that for all reals,
2066:and relevant arithmetic substructures are not
318:{\displaystyle \neg \neg \exists x.\theta (x)}
7248:
6000:{\displaystyle {\mathbb {Q} }^{\mathbb {N} }}
5925:{\displaystyle {\mathbb {Q} }^{\mathbb {N} }}
5738:{\displaystyle {\mathbb {Q} }^{\mathbb {N} }}
5048:{\displaystyle {\mathbb {Q} }^{\mathbb {N} }}
5007:{\displaystyle {\mathbb {N} }^{\mathbb {N} }}
4521:. This is provably transitive and in turn an
3770:{\displaystyle {\mathbb {Q} }^{\mathbb {N} }}
2597:{\displaystyle x\#y\,:=\,(x>y\lor y>x)}
1387:
1314:
722:
685:
656:{\displaystyle {\mathbb {Q} }^{\mathbb {N} }}
450:
422:
148:{\displaystyle {\mathbb {N} }^{\mathbb {N} }}
8:
6967:Indecomposability (constructive mathematics)
6745:)| > 0. In this case, the desired number
6617:("there exists") requires one to be able to
5967:{\displaystyle {\mathcal {P}}{\mathbb {N} }}
5494:
5464:
5206:
5161:
5014:or stronger principles aid such frameworks.
779:
767:
7157:/ Volume 2 / Issue 01 / March 1996, pp 1-71
7139:See, e.g., Theorem 1 in Bishop, 1967, p. 25
4570:{\displaystyle |x_{n}|\leq {\tfrac {2}{n}}}
1860:with postulates for a positivity predicate
240:The base logic of constructive analysis is
7255:
7241:
7233:
5692:) the numbers equivalent with respect to "
4705:{\displaystyle x_{n}\geq -{\tfrac {1}{n}}}
2539:In the context of analysis, the auxiliary
51:by some means, a collection extending the
7172:
7100:
6495:
6487:
6485:
6483:
6441:
6419:
6411:
6409:
6407:
6332:
6285:
6225:
6221:
6215:
6200:
6199:
6198:
6184:
6152:
6132:
6111:
6105:
6079:
6054:
6053:
6052:
6050:
6019:
6018:
6016:
5991:
5990:
5989:
5984:
5983:
5982:
5979:
5959:
5958:
5957:
5951:
5950:
5948:
5943:proving uncountability of powersets like
5916:
5915:
5914:
5909:
5908:
5907:
5904:
5880:
5848:
5847:
5846:
5838:
5837:
5836:
5834:
5799:
5798:
5796:
5776:
5775:
5774:
5772:
5729:
5728:
5727:
5722:
5721:
5720:
5717:
5697:
5670:
5669:
5667:
5639:
5638:
5636:
5614:
5613:
5612:
5610:
5587:
5557:
5548:In some frameworks of analysis, the name
5516:
5515:
5514:
5512:
5462:
5425:
5410:
5396:
5369:
5363:
5352:
5345:
5336:
5305:
5300:
5285:
5280:
5265:
5260:
5247:
5241:
5220:
5215:
5214:
5213:
5200:
5187:
5174:
5159:
5136:
5131:
5109:
5099:
5092:
5084:
5064:
5063:
5062:
5060:
5039:
5038:
5037:
5032:
5031:
5030:
5027:
4998:
4997:
4996:
4991:
4990:
4989:
4986:
4956:
4944:
4932:
4919:
4910:
4899:
4875:
4869:
4849:
4829:
4802:
4800:
4773:
4771:
4745:
4717:
4690:
4678:
4672:
4646:
4608:
4582:
4555:
4547:
4541:
4532:
4530:
4499:
4491:
4485:
4472:
4463:
4449:
4422:
4414:
4408:
4395:
4386:
4376:
4372:
4361:
4334:
4325:
4311:
4307:
4296:
4269:
4267:
4240:
4238:
4214:
4208:
4181:
4166:
4158:
4152:
4139:
4130:
4128:
4104:
4073:
4053:
4029:
4003:
3983:
3937:
3917:
3891:
3842:
3822:
3802:
3782:
3761:
3760:
3759:
3754:
3753:
3752:
3749:
3697:
3696:
3694:
3662:
3661:
3659:
3629:
3601:
3563:
3498:
3457:
3456:
3454:
3414:
3376:
3350:
3324:
3289:
3259:
3231:
3205:
3179:
3154:
3148:
3128:
3107:
3101:
3081:
3061:
3060:
3059:
3051:
3022:
3021:
3020:
3012:
2986:
2951:
2904:
2848:
2820:
2762:
2700:
2677:
2621:
2566:
2562:
2551:
2506:
2483:
2421:
2353:
2325:
2324:
2322:
2261:
2186:
2163:
2110:
2078:
2034:
2008:
1967:
1963:
1952:
1917:
1891:
1865:
1825:
1787:
1786:
1751:
1750:
1739:
1727:
1726:
1725:
1720:
1719:
1718:
1704:
1703:
1702:
1698:
1697:
1696:
1684:
1683:
1674:
1673:
1671:
1656:{\displaystyle g\cong _{\mathrm {ext} }0}
1637:
1636:
1627:
1593:
1592:
1578:
1574:
1568:
1567:
1565:
1531:
1530:
1519:
1515:
1509:
1508:
1506:
1486:
1465:
1464:
1462:
1441:
1440:
1439:
1434:
1433:
1432:
1423:
1386:
1385:
1366:
1365:
1330:
1329:
1313:
1312:
1305:
1293:
1292:
1291:
1286:
1285:
1284:
1270:
1269:
1268:
1264:
1263:
1262:
1256:
1255:
1253:
1220:
1219:
1217:
1185:
1180:
1174:
1136:
1116:
1061:
1060:
1051:
1024:
1010:
1006:
987:
986:
974:
954:
934:
907:
898:
883:
873:
862:
855:
853:
849:
843:
837:
817:
797:
759:
758:
757:
748:
742:
721:
720:
684:
683:
672:
647:
646:
645:
640:
639:
638:
635:
615:{\displaystyle {\mathsf {T}}\vdash \psi }
600:
599:
597:
585:{\displaystyle {\mathsf {T}}\vdash \phi }
570:
569:
567:
534:
533:
531:
506:
505:
503:
449:
448:
421:
420:
418:
398:
372:
359:{\displaystyle \neg \exists x.\theta (x)}
330:
286:
255:
254:
252:
209:
208:
206:
179:
178:
176:
139:
138:
137:
132:
131:
130:
127:
90:
64:
6717:in the interval and any natural number
6320:{\displaystyle \neg (x\leq 0)\to x>0}
4088:" (discussed below), it has a supremum.
1418:mean that there is an explicit sequence
6983:
6641:as in the classical theorem, given any
6367:{\displaystyle \neg (x\cong 0)\to x\#0}
2939:{\displaystyle x\#0\to \neg (x\cong 0)}
2498:, related to locatedness of the reals.
2178:-direction. In a pseudo-order, one has
277:is not automatically assumed for every
43:The name of the subject contrasts with
7216:Real Analysis: A Constructive Approach
5677:
5674:
5671:
5646:
5643:
5640:
4203:. An alternative is using the tighter
2803:{\displaystyle (x\#0)\lor \neg (x\#0)}
1992:{\displaystyle y>x\,:=\,(y-x>0)}
1675:
1569:
1510:
1466:
1257:
1221:
601:
571:
535:
507:
213:
210:
186:
183:
180:
5022:It is worth noting that sequences in
1240:, rational-valued sequences. Already
31:done according to some principles of
7:
7197:Foundations of Constructive Analysis
7082:Foundations of Constructive Analysis
7066:Foundations of Constructive Analysis
7053:Handbook of Constructive Mathematics
5154:may be encoded as set of quadruples
2251:disjunction does not hold in general
1457:such that, for any fixed precision,
85:, which governs an equality-to-zero
7007:An introduction to Gödel's Theorems
6548:For a simple example, consider the
6505:{\displaystyle {\mathrm {CT} _{0}}}
6467:{\displaystyle x\geq 0\lor 0\geq x}
6429:{\displaystyle {\mathrm {CT} _{0}}}
6254:already appeared in Cantor's work.
3589:{\displaystyle x>y\lor x\cong y}
3440:{\displaystyle x\geq 0\lor 0\geq x}
3069:{\displaystyle k\in {\mathbb {N} }}
3039:{\displaystyle p\in {\mathbb {R} }}
1947:. In any such ring, one may define
6491:
6488:
6415:
6412:
6374:. Weaker forms may be formulated.
6358:
6334:
6287:
6222:
6186:
6026:
6023:
6020:
5806:
5803:
5800:
4901:
4454:
4451:
4377:
4312:
3847:
3704:
3701:
3698:
3669:
3666:
3663:
3467:
3464:
3461:
3458:
3402:{\displaystyle x>0\lor 0\geq x}
3291:
3200:is apriori stronger than that for
3184:
2991:
2953:
2918:
2909:
2880:
2859:
2843:, in both senses of the word. Via
2825:
2791:
2782:
2770:
2705:
2702:
2632:
2623:
2556:
2332:
2329:
2326:
2188:
2158:generally really only goes in the
2133:
1919:
1794:
1791:
1788:
1773:
1758:
1755:
1752:
1705:
1691:
1688:
1685:
1644:
1641:
1638:
1600:
1597:
1594:
1579:
1538:
1535:
1532:
1373:
1370:
1367:
1352:
1337:
1334:
1331:
1309:
1306:
1271:
1177:
1138:
1080:
1068:
1065:
1062:
1011:
994:
991:
988:
832:is true. The associated sequence
705:
674:
526:, then if it proves a disjunction
470:
461:
458:
427:
377:
374:
335:
332:
294:
291:
288:
262:
259:
256:
14:
6957:Constructive nonstandard analysis
6385:and adopts the classically valid
5325:fundamental theorem of arithmetic
3998:, negatively characterized using
3905:{\displaystyle \varepsilon >0}
3475:{\displaystyle {\mathrm {LLPO} }}
393:, which is generally weaker than
201:, a constructive counter-part of
6990:Troelstra, A. S., van Dalen D.,
6936:
6034:{\displaystyle {\mathrm {PEM} }}
5814:{\displaystyle {\mathrm {PEM} }}
3712:{\displaystyle {\mathrm {PEM} }}
3677:{\displaystyle {\mathrm {PEM} }}
2340:{\displaystyle {\mathrm {LPO} }}
270:{\displaystyle {\mathrm {PEM} }}
7102:10.1090/s0002-9904-1970-12455-7
6713:, meaning that given any point
6512:. Such phenomena also occur in
6274:is adopted in various schools.
6270:For practical mathematics, the
5974:and plain function spaces like
5685:{\displaystyle {\mathsf {ZFC}}}
5654:{\displaystyle {\mathsf {CZF}}}
4817:{\displaystyle {\tfrac {2}{N}}}
4788:{\displaystyle {\tfrac {2}{n}}}
4284:{\displaystyle {\tfrac {3}{2}}}
4255:{\displaystyle {\tfrac {1}{n}}}
2974:{\displaystyle \neg (x\cong 0)}
1842:weak Brouwerian counterexamples
495:can generally be not reversed.
413:. In turn, also an implication
386:{\displaystyle \neg \neg \phi }
194:{\displaystyle {\mathsf {CZF}}}
6657:in the interval such that the
6544:The intermediate value theorem
6352:
6349:
6337:
6305:
6302:
6290:
6205:
6189:
6062:{\displaystyle {\mathbb {R} }}
5784:{\displaystyle {\mathbb {Q} }}
5622:{\displaystyle {\mathbb {N} }}
5524:{\displaystyle {\mathbb {N} }}
5341:
5128:
5118:
5089:
5072:{\displaystyle {\mathbb {N} }}
4945:
4911:
4622:
4610:
4548:
4533:
4492:
4464:
4415:
4387:
4159:
4131:
3954:
3948:
3942:
3542:
3530:
3527:
3524:
3500:
3306:
3294:
3033:
3027:
2968:
2956:
2933:
2921:
2915:
2886:
2877:
2871:
2868:
2865:
2856:
2850:
2797:
2785:
2776:
2764:
2738:
2726:
2723:
2720:
2708:
2656:
2644:
2641:
2638:
2626:
2591:
2567:
2465:
2441:
2438:
2435:
2423:
2403:
2379:
2376:
2373:
2355:
2287:
2263:
2218:
2215:
2191:
2165:
2145:
2139:
2130:
2127:
2124:
2112:
1986:
1968:
1934:
1922:
1803:
1776:
1767:
1740:
1733:
1708:
1609:
1582:
1547:
1520:
1414:and arithmetically sound. Now
1382:
1355:
1346:
1319:
1299:
1274:
1236:with quantifiers ranging over
1162:{\displaystyle \forall n.Q(n)}
1156:
1150:
1098:
1092:
1077:
1003:
976:
895:
889:
764:
717:
711:
699:
693:
482:
476:
455:
445:
439:
353:
347:
312:
306:
221:{\displaystyle {\mathsf {ZF}}}
1:
7078:Stolzenberg, Gabriel (1970).
5941:Cantors diagonal construction
4091:
3312:{\displaystyle \neg (x>0)}
1940:{\displaystyle \neg (0>0)}
1474:{\displaystyle {\mathsf {T}}}
1229:{\displaystyle {\mathsf {T}}}
515:{\displaystyle {\mathsf {T}}}
7541:Constructivism (mathematics)
5871:Recall that the preorder on
3925:{\displaystyle \varepsilon }
2841:carries positive information
2101:In intuitonistic logic, the
1194:{\displaystyle \Pi _{1}^{0}}
246:principle of excluded middle
7402:Ontology (computer science)
6807:least-upper-bound principle
6725:in the interval such that |
6381:school reasons in terms of
6172:{\displaystyle a<z<b}
3174:So the excluded middle for
7562:
7295:Intuitionistic type theory
7154:Bulletin of Symbolic Logic
6615:existential quantification
6601:in the interval such that
6550:intermediate value theorem
6100:and any sequence of reals
6074:For any two pair of reals
5327:. There are more economic
4634:{\displaystyle (y-x)>0}
2529:intermediate value theorem
6272:axiom of dependent choice
5791:. At least when assuming
5453:, may be used to compute
3076:, the statement that the
2815:proof of the disjunction
169:constructive set theories
7080:"Review: Errett Bishop,
6258:Category and type theory
5580:equality or real numbers
5571:{\displaystyle x\cong y}
5533:total recursive function
4759:{\displaystyle x\cong y}
4740:The above definition of
3250:Non-strict partial order
3219:{\displaystyle x\cong 0}
1622:. Here this proposition
1111:. Note that the symbol "
232:may be studied as well.
104:{\displaystyle x\cong 0}
33:constructive mathematics
7300:Constructive set theory
5753:axiom of extensionality
5079:. A sequence rationals
4660:{\displaystyle x\geq 0}
4017:{\displaystyle x\leq b}
3615:{\displaystyle x\geq y}
3364:{\displaystyle 0\geq x}
3338:{\displaystyle x\leq 0}
3273:{\displaystyle 0\geq x}
244:, which means that the
157:second-order arithmetic
7214:Bridger, Mark (2007).
7089:Bull. Amer. Math. Soc.
6594:, then there exists a
6506:
6468:
6430:
6393:Anti-classical schools
6368:
6321:
6233:
6173:
6141:
6127:, there exists a real
6121:
6094:
6093:{\displaystyle a<b}
6063:
6035:
6001:
5968:
5926:
5889:
5857:
5815:
5785:
5739:
5706:
5705:{\displaystyle \cong }
5686:
5655:
5623:
5596:
5595:{\displaystyle \cong }
5572:
5525:
5501:
5447:
5385:
5317:
5230:
5148:
5073:
5049:
5008:
4969:
4888:
4887:{\displaystyle 2^{-n}}
4858:
4838:
4818:
4789:
4760:
4726:
4706:
4661:
4635:
4597:
4596:{\displaystyle y>x}
4571:
4515:
4438:
4350:
4285:
4256:
4227:
4226:{\displaystyle 2^{-n}}
4197:
4113:
4092:Bishop's formalization
4082:
4062:
4038:
4018:
3992:
3961:
3926:
3906:
3884:modulus of convergence
3857:
3831:
3811:
3791:
3771:
3713:
3678:
3638:
3616:
3590:
3549:
3476:
3441:
3403:
3365:
3339:
3313:
3274:
3240:
3239:{\displaystyle \cong }
3220:
3194:
3164:
3137:
3117:
3090:
3070:
3040:
3001:
2975:
2940:
2893:
2835:
2804:
2745:
2686:
2685:{\displaystyle \cong }
2663:
2598:
2521:
2520:{\displaystyle x>0}
2492:
2472:
2410:
2341:
2306:
2234:
2172:
2152:
2087:
2086:{\displaystyle \cong }
2049:
2048:{\displaystyle 0>x}
2023:
2022:{\displaystyle x<0}
1993:
1941:
1906:
1905:{\displaystyle 1>0}
1880:
1879:{\displaystyle x>0}
1848:Order vs. disjunctions
1834:
1833:{\displaystyle \cong }
1810:
1657:
1616:
1554:
1495:
1475:
1451:
1396:
1230:
1195:
1163:
1125:
1105:
1040:
963:
943:
923:
826:
806:
786:
731:
657:
626:Undecidable predicates
616:
586:
556:
516:
489:
407:
387:
360:
319:
271:
222:
195:
149:
105:
79:
78:{\displaystyle x>0}
7536:Mathematical analysis
7285:Constructive analysis
7004:Smith, Peter (2007).
6507:
6474:. The existence of a
6469:
6431:
6369:
6322:
6234:
6174:
6142:
6122:
6120:{\displaystyle x_{n}}
6095:
6064:
6041:or alternatively the
6036:
6002:
5969:
5927:
5890:
5888:{\displaystyle \leq }
5858:
5816:
5786:
5740:
5707:
5687:
5656:
5624:
5597:
5573:
5526:
5502:
5448:
5386:
5318:
5231:
5149:
5074:
5050:
5009:
4970:
4889:
4859:
4839:
4819:
4790:
4761:
4727:
4707:
4662:
4636:
4598:
4572:
4516:
4439:
4351:
4286:
4257:
4228:
4198:
4114:
4083:
4063:
4039:
4037:{\displaystyle \leq }
4019:
3993:
3962:
3927:
3907:
3858:
3832:
3812:
3792:
3772:
3714:
3679:
3639:
3637:{\displaystyle \geq }
3617:
3591:
3550:
3477:
3442:
3404:
3366:
3340:
3314:
3275:
3254:Lastly, the relation
3241:
3221:
3195:
3165:
3163:{\displaystyle a_{k}}
3138:
3118:
3116:{\displaystyle a_{k}}
3091:
3071:
3041:
3002:
2976:
2941:
2899:it also follows that
2894:
2836:
2805:
2746:
2687:
2664:
2599:
2522:
2493:
2473:
2411:
2342:
2307:
2235:
2173:
2153:
2103:disjunctive syllogism
2088:
2050:
2024:
1994:
1942:
1907:
1881:
1835:
1811:
1658:
1617:
1555:
1496:
1476:
1452:
1397:
1231:
1196:
1164:
1126:
1106:
1041:
964:
944:
924:
827:
807:
787:
737:is provable, and let
732:
658:
617:
587:
557:
517:
490:
408:
406:{\displaystyle \phi }
388:
361:
320:
272:
236:Logical preliminaries
230:direct axiomatization
223:
196:
150:
106:
80:
29:mathematical analysis
25:constructive analysis
16:Mathematical analysis
7338:Fuzzy set operations
7333:Fuzzy finite element
7280:Intuitionistic logic
7115:Robert S. Lubarsky,
6893:sequentially compact
6526:logically equivalent
6482:
6440:
6406:
6331:
6284:
6183:
6151:
6131:
6104:
6078:
6049:
6015:
5978:
5947:
5903:
5879:
5833:
5795:
5771:
5716:
5696:
5666:
5635:
5609:
5586:
5556:
5511:
5461:
5395:
5335:
5240:
5158:
5083:
5059:
5026:
4985:
4898:
4868:
4848:
4828:
4799:
4770:
4766:uses a common bound
4744:
4716:
4671:
4645:
4607:
4581:
4529:
4523:equivalence relation
4448:
4360:
4295:
4266:
4237:
4207:
4127:
4103:
4081:{\displaystyle <}
4072:
4061:{\displaystyle <}
4052:
4028:
4002:
3982:
3936:
3916:
3890:
3856:{\displaystyle x\#0}
3841:
3821:
3801:
3790:{\displaystyle >}
3781:
3748:
3693:
3686:MacNeille completion
3658:
3628:
3600:
3562:
3497:
3453:
3413:
3375:
3349:
3323:
3288:
3258:
3230:
3204:
3193:{\displaystyle x\#0}
3178:
3147:
3127:
3100:
3080:
3050:
3011:
3000:{\displaystyle x\#0}
2985:
2950:
2903:
2847:
2834:{\displaystyle x\#y}
2819:
2761:
2699:
2676:
2620:
2550:
2505:
2482:
2420:
2352:
2321:
2260:
2185:
2171:{\displaystyle \to }
2162:
2109:
2077:
2033:
2007:
1951:
1916:
1890:
1864:
1824:
1670:
1626:
1564:
1505:
1485:
1461:
1422:
1407:logically equivalent
1252:
1216:
1212:Consider any theory
1173:
1135:
1115:
1050:
973:
953:
933:
836:
816:
796:
741:
671:
634:
596:
566:
530:
524:disjunction property
502:
417:
397:
371:
329:
285:
251:
242:intuitionistic logic
205:
175:
126:
89:
63:
7515:Non-monotonic logic
7264:Non-classical logic
6952:Computable analysis
6554:continuous function
6514:realizability topoi
5825:Interval arithmetic
5368:
1238:primitive recursive
1203:Goldbach conjecture
1190:
878:
281:. If a proposition
159:, or strong enough
122:by types including
7510:Intermediate logic
7290:Heyting arithmetic
7218:. Hoboken: Wiley.
6944:Mathematics portal
6705:. We require that
6502:
6464:
6426:
6364:
6317:
6278:Markov's principle
6229:
6169:
6137:
6117:
6090:
6059:
6031:
6009:intuitionistically
5997:
5964:
5922:
5885:
5853:
5811:
5781:
5748:Cauchy real number
5735:
5702:
5682:
5651:
5619:
5592:
5568:
5521:
5497:
5443:
5435:
5420:
5381:
5379:
5350:
5344:
5313:
5226:
5144:
5116:
5069:
5045:
5004:
4965:
4884:
4854:
4834:
4814:
4812:
4785:
4783:
4756:
4722:
4702:
4700:
4657:
4631:
4593:
4567:
4565:
4511:
4509:
4434:
4432:
4346:
4344:
4281:
4279:
4252:
4250:
4223:
4193:
4191:
4176:
4109:
4078:
4058:
4034:
4014:
3988:
3971:Bounds and suprema
3957:
3922:
3902:
3877:Cauchy convergence
3853:
3827:
3807:
3787:
3767:
3740:Rational sequences
3709:
3674:
3634:
3612:
3586:
3545:
3472:
3437:
3399:
3361:
3335:
3309:
3282:logically negative
3270:
3236:
3216:
3190:
3160:
3133:
3113:
3086:
3066:
3036:
2997:
2971:
2936:
2889:
2831:
2800:
2741:
2682:
2659:
2610:apartness relation
2594:
2541:logically positive
2517:
2488:
2468:
2406:
2337:
2302:
2245:logically positive
2230:
2168:
2148:
2083:
2045:
2019:
1989:
1937:
1902:
1876:
1852:The theory of the
1830:
1806:
1653:
1612:
1550:
1491:
1471:
1447:
1392:
1226:
1201:-examples are the
1191:
1176:
1159:
1121:
1101:
1046:, it follows that
1036:
959:
939:
919:
860:
854:
822:
802:
782:
727:
653:
612:
582:
552:
512:
485:
403:
383:
356:
315:
267:
218:
191:
145:
120:Heyting arithmetic
101:
75:
57:apartness relation
45:classical analysis
7523:
7522:
7505:Inquisitive logic
7500:Dynamic semantics
7453:Three-state logic
7407:Ontology language
7017:978-0-521-67453-9
6905:infinite sequence
6819:least upper bound
6813:of the real line
6751:analytic function
6674:) is less than 1/
6252:diagonal argument
6140:{\displaystyle z}
6045:axiom, models of
5434:
5419:
5378:
5329:pairing functions
5115:
4857:{\displaystyle y}
4837:{\displaystyle x}
4811:
4782:
4725:{\displaystyle n}
4699:
4564:
4508:
4431:
4343:
4278:
4249:
4190:
4175:
4112:{\displaystyle x}
4097:One formalization
3991:{\displaystyle b}
3830:{\displaystyle 0}
3810:{\displaystyle x}
3136:{\displaystyle p}
3089:{\displaystyle k}
2491:{\displaystyle t}
2249:law of trichotomy
1854:real closed field
1494:{\displaystyle g}
1124:{\displaystyle 0}
962:{\displaystyle 1}
942:{\displaystyle 0}
825:{\displaystyle Q}
805:{\displaystyle 0}
7553:
7458:Tri-state buffer
7257:
7250:
7243:
7234:
7229:
7210:
7179:
7178:
7176:
7164:
7158:
7149:Akihiro Kanamori
7146:
7140:
7137:
7131:
7128:
7122:
7113:
7107:
7106:
7104:
7075:
7069:
7062:
7056:
7049:
7043:
7036:
7030:
7029:
7001:
6995:
6988:
6946:
6941:
6940:
6809:, i.e. that any
6794:locally non-zero
6778:locally non-zero
6711:locally non-zero
6621:the real number
6534:separable spaces
6511:
6509:
6508:
6503:
6501:
6500:
6499:
6494:
6476:Specker sequence
6473:
6471:
6470:
6465:
6435:
6433:
6432:
6427:
6425:
6424:
6423:
6418:
6373:
6371:
6370:
6365:
6326:
6324:
6323:
6318:
6238:
6236:
6235:
6230:
6220:
6219:
6204:
6203:
6178:
6176:
6175:
6170:
6146:
6144:
6143:
6138:
6126:
6124:
6123:
6118:
6116:
6115:
6099:
6097:
6096:
6091:
6068:
6066:
6065:
6060:
6058:
6057:
6043:countable choice
6040:
6038:
6037:
6032:
6030:
6029:
6011:valid. Assuming
6006:
6004:
6003:
5998:
5996:
5995:
5994:
5988:
5987:
5973:
5971:
5970:
5965:
5963:
5962:
5956:
5955:
5931:
5929:
5928:
5923:
5921:
5920:
5919:
5913:
5912:
5894:
5892:
5891:
5886:
5862:
5860:
5859:
5854:
5852:
5851:
5842:
5841:
5820:
5818:
5817:
5812:
5810:
5809:
5790:
5788:
5787:
5782:
5780:
5779:
5744:
5742:
5741:
5736:
5734:
5733:
5732:
5726:
5725:
5711:
5709:
5708:
5703:
5691:
5689:
5688:
5683:
5681:
5680:
5660:
5658:
5657:
5652:
5650:
5649:
5628:
5626:
5625:
5620:
5618:
5617:
5602:-related reals.
5601:
5599:
5598:
5593:
5577:
5575:
5574:
5569:
5530:
5528:
5527:
5522:
5520:
5519:
5506:
5504:
5503:
5498:
5452:
5450:
5449:
5444:
5436:
5427:
5421:
5412:
5390:
5388:
5387:
5382:
5380:
5371:
5367:
5362:
5351:
5322:
5320:
5319:
5314:
5312:
5311:
5310:
5309:
5292:
5291:
5290:
5289:
5272:
5271:
5270:
5269:
5252:
5251:
5235:
5233:
5232:
5227:
5225:
5224:
5219:
5218:
5205:
5204:
5192:
5191:
5179:
5178:
5153:
5151:
5150:
5145:
5143:
5142:
5141:
5140:
5117:
5114:
5113:
5104:
5103:
5094:
5078:
5076:
5075:
5070:
5068:
5067:
5054:
5052:
5051:
5046:
5044:
5043:
5042:
5036:
5035:
5013:
5011:
5010:
5005:
5003:
5002:
5001:
4995:
4994:
4974:
4972:
4971:
4966:
4964:
4963:
4948:
4943:
4942:
4924:
4923:
4914:
4893:
4891:
4890:
4885:
4883:
4882:
4863:
4861:
4860:
4855:
4843:
4841:
4840:
4835:
4823:
4821:
4820:
4815:
4813:
4804:
4794:
4792:
4791:
4786:
4784:
4775:
4765:
4763:
4762:
4757:
4731:
4729:
4728:
4723:
4711:
4709:
4708:
4703:
4701:
4692:
4683:
4682:
4666:
4664:
4663:
4658:
4640:
4638:
4637:
4632:
4602:
4600:
4599:
4594:
4576:
4574:
4573:
4568:
4566:
4557:
4551:
4546:
4545:
4536:
4520:
4518:
4517:
4512:
4510:
4501:
4495:
4490:
4489:
4477:
4476:
4467:
4443:
4441:
4440:
4435:
4433:
4424:
4418:
4413:
4412:
4400:
4399:
4390:
4355:
4353:
4352:
4347:
4345:
4336:
4330:
4329:
4290:
4288:
4287:
4282:
4280:
4271:
4261:
4259:
4258:
4253:
4251:
4242:
4232:
4230:
4229:
4224:
4222:
4221:
4202:
4200:
4199:
4194:
4192:
4183:
4177:
4168:
4162:
4157:
4156:
4144:
4143:
4134:
4118:
4116:
4115:
4110:
4087:
4085:
4084:
4079:
4067:
4065:
4064:
4059:
4043:
4041:
4040:
4035:
4023:
4021:
4020:
4015:
3997:
3995:
3994:
3989:
3966:
3964:
3963:
3958:
3931:
3929:
3928:
3923:
3911:
3909:
3908:
3903:
3862:
3860:
3859:
3854:
3836:
3834:
3833:
3828:
3816:
3814:
3813:
3808:
3796:
3794:
3793:
3788:
3776:
3774:
3773:
3768:
3766:
3765:
3764:
3758:
3757:
3718:
3716:
3715:
3710:
3708:
3707:
3683:
3681:
3680:
3675:
3673:
3672:
3643:
3641:
3640:
3635:
3621:
3619:
3618:
3613:
3595:
3593:
3592:
3587:
3558:The disjunction
3554:
3552:
3551:
3546:
3481:
3479:
3478:
3473:
3471:
3470:
3446:
3444:
3443:
3438:
3408:
3406:
3405:
3400:
3370:
3368:
3367:
3362:
3344:
3342:
3341:
3336:
3318:
3316:
3315:
3310:
3279:
3277:
3276:
3271:
3245:
3243:
3242:
3237:
3225:
3223:
3222:
3217:
3199:
3197:
3196:
3191:
3169:
3167:
3166:
3161:
3159:
3158:
3142:
3140:
3139:
3134:
3122:
3120:
3119:
3114:
3112:
3111:
3096:'th coefficient
3095:
3093:
3092:
3087:
3075:
3073:
3072:
3067:
3065:
3064:
3045:
3043:
3042:
3037:
3026:
3025:
3006:
3004:
3003:
2998:
2980:
2978:
2977:
2972:
2945:
2943:
2942:
2937:
2898:
2896:
2895:
2890:
2840:
2838:
2837:
2832:
2809:
2807:
2806:
2801:
2750:
2748:
2747:
2742:
2691:
2689:
2688:
2683:
2668:
2666:
2665:
2660:
2603:
2601:
2600:
2595:
2526:
2524:
2523:
2518:
2497:
2495:
2494:
2489:
2477:
2475:
2474:
2469:
2415:
2413:
2412:
2407:
2346:
2344:
2343:
2338:
2336:
2335:
2311:
2309:
2308:
2303:
2239:
2237:
2236:
2231:
2177:
2175:
2174:
2169:
2157:
2155:
2154:
2149:
2092:
2090:
2089:
2084:
2054:
2052:
2051:
2046:
2028:
2026:
2025:
2020:
2003:). As is usual,
1998:
1996:
1995:
1990:
1946:
1944:
1943:
1938:
1911:
1909:
1908:
1903:
1885:
1883:
1882:
1877:
1858:commutative ring
1839:
1837:
1836:
1831:
1815:
1813:
1812:
1807:
1799:
1798:
1797:
1763:
1762:
1761:
1732:
1731:
1730:
1724:
1723:
1695:
1694:
1679:
1678:
1662:
1660:
1659:
1654:
1649:
1648:
1647:
1621:
1619:
1618:
1613:
1605:
1604:
1603:
1573:
1572:
1559:
1557:
1556:
1551:
1543:
1542:
1541:
1514:
1513:
1500:
1498:
1497:
1492:
1480:
1478:
1477:
1472:
1470:
1469:
1456:
1454:
1453:
1448:
1446:
1445:
1444:
1438:
1437:
1416:Gödel's theorems
1405:This theorem is
1401:
1399:
1398:
1393:
1391:
1390:
1378:
1377:
1376:
1342:
1341:
1340:
1318:
1317:
1298:
1297:
1296:
1290:
1289:
1261:
1260:
1235:
1233:
1232:
1227:
1225:
1224:
1200:
1198:
1197:
1192:
1189:
1184:
1168:
1166:
1165:
1160:
1130:
1128:
1127:
1122:
1110:
1108:
1107:
1102:
1073:
1072:
1071:
1045:
1043:
1042:
1037:
1029:
1028:
999:
998:
997:
968:
966:
965:
960:
948:
946:
945:
940:
928:
926:
925:
920:
918:
917:
902:
888:
887:
877:
872:
861:
848:
847:
831:
829:
828:
823:
811:
809:
808:
803:
791:
789:
788:
783:
763:
762:
753:
752:
736:
734:
733:
728:
726:
725:
689:
688:
662:
660:
659:
654:
652:
651:
650:
644:
643:
621:
619:
618:
613:
605:
604:
591:
589:
588:
583:
575:
574:
561:
559:
558:
553:
539:
538:
521:
519:
518:
513:
511:
510:
494:
492:
491:
486:
454:
453:
426:
425:
412:
410:
409:
404:
392:
390:
389:
384:
365:
363:
362:
357:
324:
322:
321:
316:
276:
274:
273:
268:
266:
265:
227:
225:
224:
219:
217:
216:
200:
198:
197:
192:
190:
189:
154:
152:
151:
146:
144:
143:
142:
136:
135:
110:
108:
107:
102:
84:
82:
81:
76:
49:real number line
7561:
7560:
7556:
7555:
7554:
7552:
7551:
7550:
7526:
7525:
7524:
7519:
7488:
7439:
7411:
7388:
7370:
7361:Relevance logic
7356:Structural rule
7342:
7318:Degree of truth
7304:
7266:
7261:
7226:
7213:
7207:
7191:
7188:
7186:Further reading
7183:
7182:
7166:
7165:
7161:
7147:
7143:
7138:
7134:
7129:
7125:
7114:
7110:
7077:
7076:
7072:
7064:Errett Bishop,
7063:
7059:
7050:
7046:
7038:Erik Palmgren,
7037:
7033:
7018:
7003:
7002:
6998:
6989:
6985:
6980:
6942:
6935:
6932:
6925:
6915:
6803:
6774:classical logic
6673:
6656:
6627:rational number
6561:closed interval
6546:
6530:classical logic
6522:
6486:
6480:
6479:
6478:is proven from
6438:
6437:
6410:
6404:
6403:
6400:Church's thesis
6395:
6329:
6328:
6282:
6281:
6268:
6260:
6211:
6181:
6180:
6149:
6148:
6129:
6128:
6107:
6102:
6101:
6076:
6075:
6047:
6046:
6013:
6012:
5981:
5976:
5975:
5945:
5944:
5906:
5901:
5900:
5877:
5876:
5869:
5831:
5830:
5827:
5793:
5792:
5769:
5768:
5761:
5719:
5714:
5713:
5694:
5693:
5664:
5663:
5633:
5632:
5607:
5606:
5584:
5583:
5578:are called the
5554:
5553:
5546:
5541:
5509:
5508:
5459:
5458:
5393:
5392:
5333:
5332:
5301:
5296:
5281:
5276:
5261:
5256:
5243:
5238:
5237:
5212:
5196:
5183:
5170:
5156:
5155:
5132:
5127:
5105:
5095:
5081:
5080:
5057:
5056:
5029:
5024:
5023:
5020:
4988:
4983:
4982:
4979:Function choice
4952:
4928:
4915:
4896:
4895:
4871:
4866:
4865:
4846:
4845:
4826:
4825:
4797:
4796:
4768:
4767:
4742:
4741:
4738:
4714:
4713:
4674:
4669:
4668:
4643:
4642:
4605:
4604:
4579:
4578:
4537:
4527:
4526:
4481:
4468:
4446:
4445:
4404:
4391:
4358:
4357:
4321:
4293:
4292:
4264:
4263:
4235:
4234:
4210:
4205:
4204:
4148:
4135:
4125:
4124:
4119:fulfilling the
4101:
4100:
4094:
4070:
4069:
4050:
4049:
4026:
4025:
4000:
3999:
3980:
3979:
3973:
3934:
3933:
3914:
3913:
3888:
3887:
3869:
3839:
3838:
3819:
3818:
3799:
3798:
3779:
3778:
3751:
3746:
3745:
3742:
3737:
3725:
3691:
3690:
3684:. Notably, the
3656:
3655:
3652:
3626:
3625:
3598:
3597:
3560:
3559:
3495:
3494:
3488:De Morgan's law
3451:
3450:
3411:
3410:
3373:
3372:
3347:
3346:
3321:
3320:
3286:
3285:
3256:
3255:
3252:
3228:
3227:
3202:
3201:
3176:
3175:
3150:
3145:
3144:
3125:
3124:
3103:
3098:
3097:
3078:
3077:
3048:
3047:
3009:
3008:
2983:
2982:
2948:
2947:
2901:
2900:
2845:
2844:
2817:
2816:
2813:Importantly, a
2759:
2758:
2697:
2696:
2674:
2673:
2618:
2617:
2548:
2547:
2537:
2503:
2502:
2480:
2479:
2418:
2417:
2350:
2349:
2319:
2318:
2258:
2257:
2183:
2182:
2160:
2159:
2107:
2106:
2099:
2075:
2074:
2031:
2030:
2005:
2004:
1949:
1948:
1914:
1913:
1888:
1887:
1862:
1861:
1850:
1822:
1821:
1782:
1746:
1717:
1668:
1667:
1632:
1624:
1623:
1588:
1562:
1561:
1526:
1503:
1502:
1483:
1482:
1459:
1458:
1431:
1420:
1419:
1361:
1325:
1283:
1250:
1249:
1214:
1213:
1207:Rosser sentence
1171:
1170:
1133:
1132:
1113:
1112:
1056:
1048:
1047:
1020:
982:
971:
970:
951:
950:
931:
930:
903:
879:
839:
834:
833:
814:
813:
794:
793:
744:
739:
738:
669:
668:
637:
632:
631:
628:
594:
593:
564:
563:
528:
527:
500:
499:
415:
414:
395:
394:
369:
368:
327:
326:
283:
282:
249:
248:
238:
228:. Of course, a
203:
202:
173:
172:
155:, constructive
129:
124:
123:
87:
86:
61:
60:
41:
17:
12:
11:
5:
7559:
7557:
7549:
7548:
7543:
7538:
7528:
7527:
7521:
7520:
7518:
7517:
7512:
7507:
7502:
7496:
7494:
7490:
7489:
7487:
7486:
7485:
7484:
7474:
7473:
7472:
7462:
7461:
7460:
7449:
7447:
7441:
7440:
7438:
7437:
7432:
7427:
7421:
7419:
7413:
7412:
7410:
7409:
7404:
7398:
7396:
7390:
7389:
7387:
7386:
7380:
7378:
7376:Paraconsistent
7372:
7371:
7369:
7368:
7363:
7358:
7352:
7350:
7344:
7343:
7341:
7340:
7335:
7330:
7325:
7320:
7314:
7312:
7306:
7305:
7303:
7302:
7297:
7292:
7287:
7282:
7276:
7274:
7272:Intuitionistic
7268:
7267:
7262:
7260:
7259:
7252:
7245:
7237:
7231:
7230:
7224:
7211:
7205:
7193:Bishop, Errett
7187:
7184:
7181:
7180:
7159:
7141:
7132:
7123:
7108:
7095:(2): 301â323.
7070:
7057:
7044:
7031:
7016:
6996:
6982:
6981:
6979:
6976:
6975:
6974:
6969:
6964:
6959:
6954:
6948:
6947:
6931:
6928:
6917:
6911:
6889:compact spaces
6802:
6799:
6682:that gives us
6669:
6659:absolute value
6652:
6643:natural number
6545:
6542:
6538:approximations
6521:
6518:
6498:
6493:
6490:
6463:
6460:
6457:
6454:
6451:
6448:
6445:
6422:
6417:
6414:
6394:
6391:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6339:
6336:
6316:
6313:
6310:
6307:
6304:
6301:
6298:
6295:
6292:
6289:
6267:
6264:
6259:
6256:
6241:
6240:
6228:
6224:
6218:
6214:
6210:
6207:
6202:
6197:
6194:
6191:
6188:
6168:
6165:
6162:
6159:
6156:
6136:
6114:
6110:
6089:
6086:
6083:
6056:
6028:
6025:
6022:
5993:
5986:
5961:
5954:
5918:
5911:
5884:
5868:
5867:Uncountability
5865:
5850:
5845:
5840:
5826:
5823:
5808:
5805:
5802:
5778:
5760:
5759:Dedekind reals
5757:
5731:
5724:
5701:
5679:
5676:
5673:
5648:
5645:
5642:
5616:
5591:
5567:
5564:
5561:
5545:
5542:
5540:
5537:
5518:
5496:
5493:
5490:
5487:
5484:
5481:
5478:
5475:
5472:
5469:
5466:
5455:Euler's number
5442:
5439:
5433:
5430:
5424:
5418:
5415:
5409:
5406:
5403:
5400:
5377:
5374:
5366:
5361:
5358:
5355:
5349:
5343:
5340:
5308:
5304:
5299:
5295:
5288:
5284:
5279:
5275:
5268:
5264:
5259:
5255:
5250:
5246:
5223:
5217:
5211:
5208:
5203:
5199:
5195:
5190:
5186:
5182:
5177:
5173:
5169:
5166:
5163:
5139:
5135:
5130:
5126:
5123:
5120:
5112:
5108:
5102:
5098:
5091:
5088:
5066:
5041:
5034:
5019:
5016:
5000:
4993:
4962:
4959:
4955:
4951:
4947:
4941:
4938:
4935:
4931:
4927:
4922:
4918:
4913:
4909:
4906:
4903:
4881:
4878:
4874:
4853:
4833:
4824:, the numbers
4810:
4807:
4781:
4778:
4755:
4752:
4749:
4737:
4734:
4721:
4698:
4695:
4689:
4686:
4681:
4677:
4656:
4653:
4650:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4592:
4589:
4586:
4563:
4560:
4554:
4550:
4544:
4540:
4535:
4507:
4504:
4498:
4494:
4488:
4484:
4480:
4475:
4471:
4466:
4462:
4459:
4456:
4453:
4430:
4427:
4421:
4417:
4411:
4407:
4403:
4398:
4394:
4389:
4385:
4382:
4379:
4375:
4371:
4368:
4365:
4342:
4339:
4333:
4328:
4324:
4320:
4317:
4314:
4310:
4306:
4303:
4300:
4277:
4274:
4248:
4245:
4220:
4217:
4213:
4189:
4186:
4180:
4174:
4171:
4165:
4161:
4155:
4151:
4147:
4142:
4138:
4133:
4108:
4093:
4090:
4077:
4057:
4033:
4013:
4010:
4007:
3987:
3972:
3969:
3956:
3953:
3950:
3947:
3944:
3941:
3921:
3901:
3898:
3895:
3868:
3865:
3852:
3849:
3846:
3826:
3806:
3786:
3763:
3756:
3741:
3738:
3736:
3733:
3729:Heyting fields
3724:
3721:
3706:
3703:
3700:
3671:
3668:
3665:
3651:
3648:
3633:
3624:the relation "
3611:
3608:
3605:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3556:
3555:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3469:
3466:
3463:
3460:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3360:
3357:
3354:
3345:is defined as
3334:
3331:
3328:
3308:
3305:
3302:
3299:
3296:
3293:
3269:
3266:
3263:
3251:
3248:
3235:
3215:
3212:
3209:
3189:
3186:
3183:
3157:
3153:
3132:
3110:
3106:
3085:
3063:
3058:
3055:
3035:
3032:
3029:
3024:
3019:
3016:
2996:
2993:
2990:
2970:
2967:
2964:
2961:
2958:
2955:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2830:
2827:
2824:
2811:
2810:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2752:
2751:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2681:
2670:
2669:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2605:
2604:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2565:
2561:
2558:
2555:
2536:
2533:
2516:
2513:
2510:
2487:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2334:
2331:
2328:
2313:
2312:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2241:
2240:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2167:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2098:
2095:
2082:
2044:
2041:
2038:
2029:is defined as
2018:
2015:
2012:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1966:
1962:
1959:
1956:
1936:
1933:
1930:
1927:
1924:
1921:
1901:
1898:
1895:
1875:
1872:
1869:
1849:
1846:
1829:
1817:
1816:
1805:
1802:
1796:
1793:
1790:
1785:
1781:
1778:
1775:
1772:
1769:
1766:
1760:
1757:
1754:
1749:
1745:
1742:
1738:
1735:
1729:
1722:
1716:
1713:
1710:
1707:
1701:
1693:
1690:
1687:
1682:
1677:
1652:
1646:
1643:
1640:
1635:
1631:
1611:
1608:
1602:
1599:
1596:
1591:
1587:
1584:
1581:
1577:
1571:
1549:
1546:
1540:
1537:
1534:
1529:
1525:
1522:
1518:
1512:
1490:
1468:
1443:
1436:
1430:
1427:
1403:
1402:
1389:
1384:
1381:
1375:
1372:
1369:
1364:
1360:
1357:
1354:
1351:
1348:
1345:
1339:
1336:
1333:
1328:
1324:
1321:
1316:
1311:
1308:
1304:
1301:
1295:
1288:
1282:
1279:
1276:
1273:
1267:
1259:
1223:
1209:of a theory.
1188:
1183:
1179:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1120:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1070:
1067:
1064:
1059:
1055:
1035:
1032:
1027:
1023:
1019:
1016:
1013:
1009:
1005:
1002:
996:
993:
990:
985:
981:
978:
958:
938:
916:
913:
910:
906:
901:
897:
894:
891:
886:
882:
876:
871:
868:
865:
859:
852:
846:
842:
821:
812:exactly where
801:
781:
778:
775:
772:
769:
766:
761:
756:
751:
747:
724:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
687:
682:
679:
676:
649:
642:
627:
624:
611:
608:
603:
581:
578:
573:
551:
548:
545:
542:
537:
509:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
452:
447:
444:
441:
438:
435:
432:
429:
424:
402:
382:
379:
376:
355:
352:
349:
346:
343:
340:
337:
334:
314:
311:
308:
305:
302:
299:
296:
293:
290:
264:
261:
258:
237:
234:
215:
212:
188:
185:
182:
141:
134:
100:
97:
94:
74:
71:
68:
40:
37:
15:
13:
10:
9:
6:
4:
3:
2:
7558:
7547:
7544:
7542:
7539:
7537:
7534:
7533:
7531:
7516:
7513:
7511:
7508:
7506:
7503:
7501:
7498:
7497:
7495:
7491:
7483:
7480:
7479:
7478:
7475:
7471:
7468:
7467:
7466:
7463:
7459:
7456:
7455:
7454:
7451:
7450:
7448:
7446:
7445:Digital logic
7442:
7436:
7433:
7431:
7428:
7426:
7423:
7422:
7420:
7418:
7414:
7408:
7405:
7403:
7400:
7399:
7397:
7395:
7391:
7385:
7382:
7381:
7379:
7377:
7373:
7367:
7364:
7362:
7359:
7357:
7354:
7353:
7351:
7349:
7348:Substructural
7345:
7339:
7336:
7334:
7331:
7329:
7326:
7324:
7321:
7319:
7316:
7315:
7313:
7311:
7307:
7301:
7298:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7277:
7275:
7273:
7269:
7265:
7258:
7253:
7251:
7246:
7244:
7239:
7238:
7235:
7227:
7225:0-471-79230-6
7221:
7217:
7212:
7208:
7206:4-87187-714-0
7202:
7198:
7194:
7190:
7189:
7185:
7175:
7170:
7163:
7160:
7156:
7155:
7150:
7145:
7142:
7136:
7133:
7127:
7124:
7120:
7119:
7112:
7109:
7103:
7098:
7094:
7091:
7090:
7085:
7083:
7074:
7071:
7067:
7061:
7058:
7054:
7048:
7045:
7041:
7035:
7032:
7027:
7023:
7019:
7013:
7009:
7008:
7000:
6997:
6993:
6987:
6984:
6977:
6973:
6970:
6968:
6965:
6963:
6962:Heyting field
6960:
6958:
6955:
6953:
6950:
6949:
6945:
6939:
6934:
6929:
6927:
6924:
6920:
6914:
6910:
6906:
6902:
6901:cluster point
6898:
6894:
6890:
6885:
6883:
6879:
6875:
6871:
6867:
6864:
6860:
6856:
6852:
6848:
6844:
6840:
6837:if, whenever
6836:
6832:
6828:
6824:
6820:
6817:would have a
6816:
6812:
6808:
6800:
6798:
6795:
6791:
6787:
6783:
6779:
6775:
6770:
6768:
6764:
6761:) < 0 and
6760:
6756:
6752:
6748:
6744:
6740:
6736:
6732:
6728:
6724:
6720:
6716:
6712:
6708:
6704:
6700:
6696:
6692:
6687:
6685:
6681:
6677:
6672:
6668:
6664:
6660:
6655:
6651:
6647:
6644:
6640:
6634:
6632:
6628:
6624:
6620:
6616:
6612:
6609:) is exactly
6608:
6604:
6600:
6597:
6593:
6589:
6585:
6581:
6577:
6573:
6569:
6566:
6562:
6558:
6555:
6551:
6543:
6541:
6539:
6535:
6531:
6527:
6519:
6517:
6515:
6496:
6477:
6461:
6458:
6455:
6452:
6449:
6446:
6443:
6420:
6401:
6392:
6390:
6388:
6387:bar induction
6384:
6380:
6375:
6361:
6355:
6346:
6343:
6340:
6314:
6311:
6308:
6299:
6296:
6293:
6279:
6275:
6273:
6265:
6263:
6257:
6255:
6253:
6249:
6246:According to
6244:
6226:
6216:
6212:
6208:
6195:
6192:
6166:
6163:
6160:
6157:
6154:
6134:
6112:
6108:
6087:
6084:
6081:
6073:
6072:
6071:
6044:
6010:
5942:
5937:
5935:
5898:
5882:
5874:
5866:
5864:
5843:
5824:
5822:
5766:
5765:Dedekind cuts
5758:
5756:
5754:
5750:
5749:
5699:
5661:
5603:
5589:
5581:
5565:
5562:
5559:
5551:
5543:
5538:
5536:
5534:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5456:
5440:
5437:
5431:
5428:
5422:
5416:
5413:
5407:
5404:
5401:
5398:
5375:
5372:
5364:
5359:
5356:
5353:
5347:
5338:
5330:
5326:
5306:
5302:
5297:
5293:
5286:
5282:
5277:
5273:
5266:
5262:
5257:
5253:
5248:
5244:
5221:
5209:
5201:
5197:
5193:
5188:
5184:
5180:
5175:
5171:
5167:
5164:
5137:
5133:
5124:
5121:
5110:
5106:
5100:
5096:
5086:
5017:
5015:
4980:
4976:
4960:
4957:
4953:
4949:
4939:
4936:
4933:
4929:
4925:
4920:
4916:
4907:
4904:
4879:
4876:
4872:
4851:
4831:
4808:
4805:
4779:
4776:
4753:
4750:
4747:
4735:
4733:
4719:
4696:
4693:
4687:
4684:
4679:
4675:
4654:
4651:
4648:
4628:
4625:
4619:
4616:
4613:
4590:
4587:
4584:
4561:
4558:
4552:
4542:
4538:
4524:
4505:
4502:
4496:
4486:
4482:
4478:
4473:
4469:
4460:
4457:
4428:
4425:
4419:
4409:
4405:
4401:
4396:
4392:
4383:
4380:
4373:
4369:
4366:
4363:
4340:
4337:
4331:
4326:
4322:
4318:
4315:
4308:
4304:
4301:
4298:
4275:
4272:
4246:
4243:
4218:
4215:
4211:
4187:
4184:
4178:
4172:
4169:
4163:
4153:
4149:
4145:
4140:
4136:
4122:
4106:
4098:
4089:
4075:
4055:
4047:
4031:
4011:
4008:
4005:
3985:
3978:
3970:
3968:
3951:
3945:
3939:
3919:
3899:
3896:
3893:
3885:
3880:
3878:
3874:
3866:
3864:
3850:
3844:
3824:
3804:
3784:
3739:
3735:Formalization
3734:
3732:
3730:
3723:Invertibility
3722:
3720:
3687:
3649:
3647:
3645:
3631:
3609:
3606:
3603:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3539:
3536:
3533:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3493:
3492:
3491:
3489:
3484:
3482:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3358:
3355:
3352:
3332:
3329:
3326:
3303:
3300:
3297:
3283:
3267:
3264:
3261:
3249:
3247:
3233:
3213:
3210:
3207:
3187:
3181:
3172:
3155:
3151:
3130:
3108:
3104:
3083:
3056:
3053:
3030:
3017:
3014:
2994:
2988:
2965:
2962:
2959:
2930:
2927:
2924:
2912:
2906:
2883:
2874:
2862:
2853:
2842:
2828:
2822:
2794:
2788:
2779:
2773:
2767:
2757:
2756:
2755:
2735:
2732:
2729:
2717:
2714:
2711:
2695:
2694:
2693:
2679:
2653:
2650:
2647:
2635:
2629:
2616:
2615:
2614:
2612:
2611:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2563:
2559:
2553:
2546:
2545:
2544:
2542:
2534:
2532:
2530:
2514:
2511:
2508:
2499:
2485:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2432:
2429:
2426:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2370:
2367:
2364:
2361:
2358:
2347:
2299:
2296:
2293:
2290:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2256:
2255:
2254:
2252:
2250:
2246:
2227:
2224:
2221:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2181:
2180:
2179:
2142:
2136:
2121:
2118:
2115:
2104:
2096:
2094:
2080:
2071:
2069:
2065:
2061:
2056:
2042:
2039:
2036:
2016:
2013:
2010:
2002:
1983:
1980:
1977:
1974:
1971:
1964:
1960:
1957:
1954:
1931:
1928:
1925:
1899:
1896:
1893:
1873:
1870:
1867:
1859:
1855:
1847:
1845:
1843:
1827:
1800:
1783:
1779:
1770:
1764:
1747:
1743:
1736:
1714:
1711:
1699:
1680:
1666:
1665:
1664:
1650:
1633:
1629:
1606:
1589:
1585:
1575:
1544:
1527:
1523:
1516:
1488:
1428:
1425:
1417:
1413:
1408:
1379:
1362:
1358:
1349:
1343:
1326:
1322:
1302:
1280:
1277:
1265:
1248:
1247:
1246:
1243:
1242:minimal logic
1239:
1210:
1208:
1204:
1186:
1181:
1153:
1147:
1144:
1141:
1118:
1095:
1089:
1086:
1083:
1074:
1057:
1053:
1033:
1030:
1025:
1021:
1017:
1014:
1007:
1000:
983:
979:
956:
936:
914:
911:
908:
904:
899:
892:
884:
880:
874:
869:
866:
863:
857:
850:
844:
840:
819:
799:
776:
773:
770:
754:
749:
745:
714:
708:
702:
696:
690:
680:
677:
666:
625:
623:
609:
606:
579:
576:
549:
546:
543:
540:
525:
522:exhibits the
496:
479:
473:
467:
464:
442:
436:
433:
430:
400:
380:
350:
344:
341:
338:
309:
303:
300:
297:
280:
247:
243:
235:
233:
231:
170:
166:
162:
158:
121:
116:
114:
98:
95:
92:
72:
69:
66:
58:
54:
50:
46:
38:
36:
34:
30:
26:
22:
7546:Intuitionism
7425:Three-valued
7366:Linear logic
7284:
7215:
7196:
7162:
7152:
7144:
7135:
7126:
7117:
7111:
7092:
7087:
7081:
7073:
7065:
7060:
7052:
7047:
7039:
7034:
7006:
6999:
6991:
6986:
6972:Pseudo-order
6922:
6918:
6912:
6908:
6896:
6886:
6882:compact sets
6873:
6865:
6858:
6854:
6850:
6846:
6842:
6838:
6834:
6830:
6826:
6822:
6814:
6804:
6793:
6789:
6785:
6781:
6777:
6771:
6766:
6762:
6758:
6754:
6746:
6742:
6738:
6734:
6730:
6726:
6722:
6718:
6714:
6710:
6706:
6702:
6698:
6694:
6690:
6688:
6683:
6679:
6675:
6670:
6666:
6662:
6653:
6649:
6645:
6638:
6635:
6630:
6622:
6618:
6606:
6602:
6598:
6587:
6583:
6575:
6571:
6567:
6556:
6547:
6523:
6436:even refute
6396:
6376:
6276:
6269:
6261:
6245:
6242:
5938:
5934:subcountable
5870:
5828:
5762:
5746:
5604:
5579:
5550:real numbers
5549:
5547:
5544:Cauchy reals
5021:
4977:
4739:
4120:
4095:
3974:
3881:
3870:
3743:
3726:
3653:
3623:
3557:
3485:
3281:
3253:
3173:
2981:would imply
2814:
2812:
2753:
2671:
2608:
2606:
2540:
2538:
2500:
2314:
2247:
2244:
2242:
2105:in the form
2100:
2072:
2057:
2001:pseudo-order
1851:
1818:
1404:
1211:
629:
497:
239:
117:
113:real numbers
112:
55:and with an
44:
42:
39:Introduction
24:
18:
7465:Four-valued
7435:Ćukasiewicz
7430:Four-valued
7417:Many-valued
7394:Description
7384:Dialetheism
7121:, July 2015
7068:, July 1967
6870:upper bound
6784:; and then
6769:) > 0).
6596:real number
5939:That said,
4233:instead of
3977:upper bound
3817:apart from
3486:By a valid
3449:analytical
3447:, see also
3319:, and then
2317:analytical
2060:first-order
1560:as well as
279:proposition
21:mathematics
7530:Categories
7323:Fuzzy rule
7174:1804.05495
6978:References
6853:such that
6693:such that
6629:). But if
6379:Brouwerian
6266:Principles
5539:Set theory
5323:using the
4736:Variations
4123:condition
4121:regularity
3650:Variations
3284:statement
3046:and fixed
2543:predicate
2097:Trichotomy
1412:consistent
562:then also
7477:IEEE 1164
7328:Fuzzy set
6878:intervals
6776:, if the
6733:| < 1/
6619:construct
6565:real line
6459:≥
6453:∨
6447:≥
6359:#
6353:→
6344:≅
6335:¬
6306:→
6297:≤
6288:¬
6223:#
6196:∈
6187:∀
5897:injection
5883:≤
5873:cardinals
5844:×
5700:≅
5590:≅
5563:≅
5492:…
5441:…
5348:∑
5342:↦
5294:⋅
5274:⋅
5254:⋅
5210:∈
5207:⟩
5162:⟨
5122:−
5090:↦
4958:−
4926:−
4902:∀
4877:−
4751:≅
4688:−
4685:≥
4652:≥
4617:−
4553:≤
4479:−
4455:∃
4452:¬
4420:≤
4402:−
4378:∀
4367:≅
4313:∃
4216:−
4164:≤
4146:−
4032:≤
4009:≤
3952:ε
3943:↦
3940:ε
3920:ε
3894:ε
3848:#
3632:≥
3607:≥
3581:≅
3575:∨
3537:≅
3528:↔
3519:≥
3513:∧
3507:≥
3432:≥
3426:∨
3420:≥
3394:≥
3388:∨
3356:≥
3330:≤
3292:¬
3265:≥
3246:" below.
3234:≅
3211:≅
3185:#
3057:∈
3018:∈
2992:#
2963:≅
2954:¬
2928:≅
2919:¬
2916:→
2910:#
2884:ϕ
2881:¬
2878:→
2875:ψ
2869:↔
2863:ψ
2860:¬
2857:→
2854:ϕ
2826:#
2792:#
2783:¬
2780:∨
2771:#
2733:≅
2724:↔
2715:≅
2706:¬
2703:¬
2680:≅
2651:≅
2642:↔
2633:#
2624:¬
2580:∨
2557:#
2535:Apartness
2454:∨
2439:→
2392:∨
2377:→
2297:≅
2291:∨
2276:∨
2225:≅
2219:→
2204:∨
2189:¬
2166:→
2143:ψ
2140:→
2137:ϕ
2134:¬
2128:→
2122:ψ
2119:∨
2116:ϕ
2081:≅
2070:therein.
2068:definable
1975:−
1920:¬
1828:≅
1784:≅
1774:¬
1771:∨
1748:≅
1715:∈
1706:∀
1700:⊢
1634:≅
1590:≅
1580:¬
1576:⊬
1528:≅
1517:⊬
1429:∈
1363:≅
1353:¬
1350:∨
1327:≅
1310:¬
1307:¬
1281:∈
1272:∀
1266:⊢
1178:Π
1139:∀
1081:∀
1078:↔
1058:≅
1012:∀
984:≅
881:χ
858:∑
765:→
755::
746:χ
706:¬
703:∨
675:∀
665:decidable
610:ψ
607:⊢
580:ϕ
577:⊢
550:ψ
547:∨
544:ϕ
541:⊢
474:θ
471:¬
462:∀
459:¬
456:→
437:θ
428:∃
401:ϕ
381:ϕ
378:¬
375:¬
345:θ
336:∃
333:¬
304:θ
295:∃
292:¬
289:¬
96:≅
53:rationals
7195:(1967).
6930:See also
6880:and all
6592:positive
6580:negative
6520:Theorems
6248:Kanamori
5662:or even
4712:for all
4641:. Also,
4046:supremum
3596:implies
2478:for all
2064:topology
1205:and the
171:such as
7470:Verilog
7026:2384958
6903:of the
6835:located
6823:located
6684:exactly
6563:to the
6559:from a
6528:, over
6383:spreads
5486:6561000
3873:maximum
3871:As the
3837:, i.e.
7493:Others
7222:
7203:
7024:
7014:
6868:is an
6811:subset
6686:zero.
6582:while
5018:Coding
3867:Moduli
1169:. Two
7310:Fuzzy
7169:arXiv
6899:as a
6857:<
6841:<
6737:and |
6590:) is
6578:) is
6570:, if
6147:with
5712:" in
5480:24300
5391:, or
4044:". A
2058:This
167:- or
161:topos
7482:VHDL
7220:ISBN
7201:ISBN
7012:ISBN
6611:zero
6377:The
6312:>
6179:and
6164:<
6158:<
6085:<
5631:say
4950:<
4844:and
4626:>
4588:>
4497:>
4332:>
4302:>
4076:<
4056:<
3897:>
3785:>
3569:>
3382:>
3301:>
2586:>
2574:>
2512:>
2460:>
2448:>
2430:>
2398:>
2386:>
2368:>
2315:See
2282:>
2270:>
2210:>
2198:>
2040:>
2014:<
1981:>
1958:>
1929:>
1912:and
1897:>
1871:>
949:and
165:type
70:>
7097:doi
6872:of
6849:of
6833:is
6829:of
6709:be
6661:of
6327:or
6007:is
5767:of
5507:of
4981:in
4603:as
3123:of
592:or
163:-,
27:is
19:In
7532::
7199:.
7093:76
7086:.
7022:MR
7020:.
6926:.
6884:.
6863:or
6861:,
6729:-
6540:.
6389:.
5936:.
5474:90
5468:15
4374::=
4309::=
3882:A
3731:.
3719:.
3646:.
3483:.
2564::=
2055:.
1965::=
1844:.
1008::=
851::=
35:.
23:,
7256:e
7249:t
7242:v
7228:.
7209:.
7177:.
7171::
7105:.
7099::
7084:"
7028:.
6923:N
6921:â
6919:n
6916:)
6913:n
6909:c
6907:(
6897:c
6874:S
6866:y
6859:s
6855:x
6851:S
6847:s
6843:y
6839:x
6831:R
6827:S
6815:R
6790:x
6788:(
6786:f
6782:x
6767:b
6765:(
6763:f
6759:a
6757:(
6755:f
6747:c
6743:y
6741:(
6739:f
6735:m
6731:x
6727:y
6723:y
6719:m
6715:x
6707:f
6703:f
6699:c
6697:(
6695:f
6691:c
6680:c
6676:n
6671:n
6667:c
6665:(
6663:f
6654:n
6650:c
6646:n
6639:f
6631:f
6623:c
6607:c
6605:(
6603:f
6599:c
6588:b
6586:(
6584:f
6576:a
6574:(
6572:f
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