1199:
295:
5441:
3249:
700:", denote the same term and correspond to the same tree, viz. the left tree in the above picture. Separating the tree structure of a term from its graphical representation on paper, it is also easy to account for parentheses (being only representation, not structure) and invisible multiplication operators (existing only in structure, not in representation).
1771:
indicated by blue variables immediately left to their black substitutes. Intuitively, the pattern, except for its variables, must be contained in the subject; if a variable occurs multiple times in the pattern, equal subterms are required at the respective positions of the
1347:, can be assigned, that is, a string of natural numbers indicating the node's place in the hierarchy. The empty string, commonly denoted by ε, is assigned to the root node. Position strings within the black term are indicated in red in the picture.
1630:
by some authors) can be assigned, i.e. its distance (number of edges) from the root. In this setting, the depth of a node always equals the length of its position string. In the picture, depth levels in the black term are indicated in
1638:
of a term commonly refers to the number of its nodes, or, equivalently, to the length of the term's written representation, counting symbols without parentheses. The black and the blue term in the picture has the size 15 and 5,
1041:
2885:. The general sum operator Σ can then be considered as a ternary function symbol taking a lower bound value, an upper bound value and a function to be summed-up. Due to its latter argument, the Σ operator is called a
2259:
2839:
2769:
2098:
1291:
2044:
2157:
2627:
1609:
732:. While structural equality can be checked without any knowledge about the meaning of the symbols, semantic equality cannot. If the function / is e.g. interpreted not as rational but as
1539:
1186:. An atomic formula is called ground if it is built entirely from ground terms; all ground atomic formulas composable from a given set of function and predicate symbols make up the
2533:
1968:
698:
259:
2373:
3820:
2333:
2294:
1172:
1138:
147:
4495:
1743:
1331:
199:
175:
4578:
3719:
632:
became popular in computer science, it turned out to be more convenient to think of a term as a tree. For example, several distinct character strings, like "
1615:
and embedding are converse to each other: while the former appends function symbols at the bottom of the term, the latter appends them at the top. The
1802:
When the domain of discourse contains elements of basically different kinds, it is useful to split the set of all terms accordingly. To this end, a
1059:-ary root function symbol. The sum has a finite value if there is only a finite number of constants and function symbols, which is usually the case.
961:
4892:
2936:
The rightmost column of the table indicates how each mathematical notation example can be represented by a lambda term, also converting common
5050:
3622:
3838:
2181:
4905:
4228:
4490:
4910:
4900:
4637:
3843:
1810:) is assigned to each variable and each constant symbol, and a declaration of domain sorts and range sort to each function symbol. A
4388:
3834:
787:. In many contexts, the particular variable names in a term don't matter, e.g. the commutativity axiom for addition can be stated as
5046:
46:
denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a
523:
and higher-arity functions are possible but uncommon in practice. Many authors consider constant symbols as 0-ary function symbols
5143:
4887:
3712:
3482:) formulas can be renamed in a similar way as terms. In fact, some authors consider a quantifier-free formula as a term (of type
4448:
4141:
3882:
3478:
Since atomic formulas can be viewed as trees, too, and renaming is essentially a concept on trees, atomic (and, more generally,
5475:
294:
5404:
5106:
4869:
4864:
4689:
4110:
3794:
3676:
1775:
740:=2 the left and right term evaluates to 3 and 2, respectively. Structural equal terms need to agree in their variable names.
2844:
All these operators can be viewed as taking a function rather than a value term as one of their arguments. For example, the
2778:
2708:
1198:
5399:
5182:
5099:
4812:
4743:
4620:
3862:
720:
equal if they correspond to the same tree. For example, the left and the right tree in the above picture are structurally
4470:
5324:
5150:
4836:
4069:
2053:
1205:
43:
4475:
2848:
operator is applied to a sequence, i.e. to a mapping from positive integer to e.g. real numbers. As another example, a
1987:
5465:
4807:
4546:
3804:
3705:
5202:
5197:
2852:
function to implement the second example from the table, Σ, would have a function pointer argument (see box below).
2105:
5470:
5131:
4721:
4115:
4083:
3774:
2573:
3848:
1551:
5421:
5370:
5267:
4765:
4726:
4203:
5262:
3877:
1477:
5192:
4731:
4583:
4566:
4289:
3769:
3508:
3459:
2849:
5094:
5071:
5032:
4918:
4859:
4505:
4425:
4269:
4213:
3826:
3479:
70:
5384:
5111:
5089:
5056:
4949:
4795:
4780:
4753:
4704:
4588:
4523:
4348:
4314:
4309:
4183:
4014:
3991:
2490:
1616:
94:
90:
66:
51:
2693:
Mathematical notations as shown in the table do not fit into the scheme of a first-order term as defined
1931:
659:
5314:
5167:
4959:
4677:
4413:
4319:
4178:
4163:
4044:
4019:
629:
608:
by the first, second, and third term building rule, respectively. The latter term is usually written as
5440:
208:
2340:
5287:
5249:
5126:
4930:
4770:
4694:
4672:
4500:
4458:
4357:
4324:
4188:
3976:
3887:
1612:
768:
351:
62:
5416:
5307:
5292:
5272:
5229:
5116:
5066:
4992:
4937:
4874:
4667:
4662:
4610:
4378:
4367:
4039:
3939:
3867:
3858:
3854:
3789:
3784:
1881:
819:; in such cases the whole formula may be renamed, while an arbitrary subterm usually may not, e.g.
534:
39:
2303:
2264:
1155:
1121:
5445:
5214:
5177:
5162:
5155:
5138:
4924:
4790:
4716:
4699:
4652:
4465:
4374:
4208:
4193:
4153:
4105:
4090:
4078:
4034:
4009:
3779:
3728:
2862:
1387:
1115:
1111:
102:
31:
4942:
4398:
5380:
5187:
4997:
4987:
4879:
4760:
4595:
4571:
4352:
4336:
4241:
4218:
4095:
4064:
4029:
3924:
3759:
3682:
3672:
3645:
3618:
3589:
2385:
1797:
1187:
1107:
889:
520:
485:
442:
279:
74:
58:
1710:
1298:
5394:
5389:
5282:
5239:
5061:
5022:
5017:
5002:
4828:
4785:
4682:
4480:
4430:
4004:
3966:
3641:
733:
283:
1982:, and the vector addition, the scalar multiplication, and the inner product is declared as
5375:
5365:
5319:
5302:
5257:
5219:
5121:
5041:
4848:
4775:
4748:
4736:
4642:
4556:
4530:
4485:
4453:
4254:
4056:
3999:
3949:
3914:
3872:
3504:
2941:
903:
729:
5360:
5339:
5297:
5277:
5172:
5027:
4625:
4615:
4605:
4600:
4534:
4408:
4284:
4173:
4168:
4146:
3747:
2937:
1780:
1183:
882:
759:
if the latter resulted from consistently renaming all variables of the former, i.e. if
613:
184:
160:
78:
3248:
5459:
5334:
5012:
4519:
4304:
4294:
4264:
4249:
3919:
3610:
1383:
86:
5234:
5081:
4982:
4974:
4854:
4711:
4647:
4630:
4561:
4420:
4279:
3981:
3764:
3606:
3123:// implements anonymous function (lambda i. i*i); however, C requires a name for it
1877:
897:
3652:. Studies in Logic and the Foundation of Mathematics. Vol. 73. North Holland.
5344:
5224:
4403:
4393:
4340:
4024:
3944:
3929:
3809:
3754:
3664:
3283:
of variable symbols, the set of lambda terms is defined recursively as follows:
2856:
2683:
1141:
894:
857:
264:
47:
17:
4274:
4129:
4100:
3906:
3686:
1747:
matches the black subject term at position 1, with the matching substitution
1431:
can also be viewed as resulting from a generalized concatenation of the term
1374:. For example, at position 122 of the black term in the picture, the subterm
1036:{\displaystyle \theta _{h+1}=\sum _{i=0}^{\infty }f_{i}\cdot \theta _{h}^{i}}
861:; a term that doesn't contain multiple occurrences of a variable is called a
342:-ary function symbols, also called operator symbols, for each natural number
5426:
5329:
4382:
4299:
4259:
4223:
4159:
3971:
3961:
3934:
1339:
Since a term has the structure of a tree hierarchy, to each of its nodes a
881:+1) is a non-linear term. These properties are important in, for example,
5411:
5209:
4657:
4362:
3956:
3454:
2914:
1085:≥ 1, an (unsorted first-order) atomic formula is obtained by applying an
5007:
3799:
1182:+1) ≥ 0 is a mathematical formula evaluating to true in the algebra of
2877:
from the C program below can be written anonymously as a lambda term λ
2771:
doesn't make sense. In contrast, the other variables, referred to as
2254:{\displaystyle \langle ({\vec {v}}+{\vec {0}})*a,{\vec {w}}*b\rangle }
3697:
2377:
is required. Function symbols having several declarations are called
2933:
operator takes such a sequence and returns its limit (if defined).
4551:
3897:
3742:
2705:, variable that may not appear outside the notation's scope, e.g.
1197:
865:. For example, 2+2 is a ground term and hence also a linear term,
628:
adhering to certain building rules. However, since the concept of
558:
497:
are inhabited. Well-known examples are the unary function symbols
275:
82:
3360:, etc. which are, however, not admitted in pure lambda calculus.
1928:
the set of vector and number constants, respectively. Then e.g.
3701:
3243:
1910:
be the set of vector and number variables, respectively, and
779:, too, since a renaming substitution σ has an inverse σ, and
3352:
The above motivating examples also used some constants like
892:
for the function symbols, the set of all terms forms the
1892:
denote the sort of vectors and numbers, respectively, let
488:
of the term language describes which function symbol sets
956:, since a ground term of height 0 can only be a constant,
855:). A term that doesn't contain any variables is called a
2775:, behave like ordinary first-order term variables, e.g.
1541:. The latter term also results from embedding the term
3260:
1619:
relates a term and any result of appends on both sides.
616:
and the more common operator symbol + for convenience.
3617:. Cambridge University Press. pp. 1–2 and 34–35.
2834:{\displaystyle k\cdot \int _{a}^{b}\sin(k\cdot t)\;dt}
2764:{\displaystyle t\cdot \int _{a}^{b}\sin(k\cdot t)\;dt}
1705:, respectively. In the picture, the blue pattern term
1093:
terms. As for function symbols, a relation symbol set
354:
to be the smallest set with the following properties:
3503:
Renaming of the commutativity axiom can be viewed as
2781:
2711:
2576:
2493:
2343:
2306:
2267:
2184:
2108:
2056:
1990:
1934:
1713:
1554:
1480:
1301:
1208:
1158:
1124:
964:
662:
211:
187:
163:
105:
941:
can be computed by the following recursion formula:
541:
denotes a named object from that domain, a variable
530:, thus needing no special syntactic class for them.
5353:
5248:
5080:
4973:
4825:
4518:
4441:
4335:
4239:
4128:
4055:
3990:
3905:
3896:
3818:
3735:
835:is not a valid version of the commutativity axiom.
2833:
2763:
2621:
2527:
2388:for more information, including extensions of the
2367:
2327:
2288:
2253:
2151:
2093:{\displaystyle \langle .,.\rangle :W\times W\to N}
2092:
2038:
1962:
1737:
1603:
1533:
1325:
1286:{\displaystyle {\frac {a*((a+1)*(a+2))}{1*(2*3)}}}
1285:
1166:
1132:
1035:
692:
253:
193:
169:
151:is a term built from the constant 1, the variable
141:
2039:{\displaystyle +:W\times W\to W,*:W\times N\to W}
1390:since each term is trivially a subterm of itself.
724:equal terms, although they might be considered "
3422:denotes the constant function always returning
2913:/3, ..., respectively, that is, it denotes the
728:" as they always evaluate to the same value in
545:ranges over the objects in that domain, and an
512:, and the binary function symbols +, −, ⋅, / ∈
77:to an appropriate number of terms is called an
27:Components of a mathematical or logical formula
2335:a well-sorted term, an additional declaration
2296:is not (since + doesn't accept a term of sort
2152:{\displaystyle {\vec {v}},{\vec {w}}\in V_{W}}
1081:-ary relation symbols for each natural number
533:A term denotes a mathematical object from the
3713:
3392:) denotes the result of calling the function
2901:denotes a function that maps 1, 2, 3, ... to
2622:{\displaystyle \int _{a}^{b}\sin(k\cdot t)dt}
624:Originally, logicians defined a term to be a
346:≥ 1, the set of (unsorted first-order) terms
8:
2248:
2185:
2069:
2057:
1604:{\displaystyle {\frac {a*(\;.\;)}{1*(2*3)}}}
937:of distinct ground terms of a height up to
4539:
4134:
3902:
3720:
3706:
3698:
2824:
2754:
2102:, respectively. Assuming variable symbols
1611:. In an informal sense, the operations of
1571:
1567:
1534:{\displaystyle {\frac {a*(b+1)}{1*(2*3)}}}
783:= uσ. Both terms are then also said to be
3592:section of the Signature (logic) article.
3222:// applies sum operator to sum up squares
2797:
2792:
2780:
2727:
2722:
2710:
2586:
2581:
2575:
2519:
2509:
2498:
2492:
2342:
2314:
2313:
2305:
2269:
2268:
2266:
2231:
2230:
2207:
2206:
2192:
2191:
2183:
2143:
2125:
2124:
2110:
2109:
2107:
2055:
1989:
1954:
1936:
1935:
1933:
1712:
1555:
1553:
1481:
1479:
1300:
1209:
1207:
1160:
1159:
1157:
1126:
1125:
1123:
1027:
1022:
1009:
999:
988:
969:
963:
663:
661:
210:
186:
162:
104:
2399:
911:Abbreviating the number of constants as
901:. The set of all ground terms forms the
445:notation, this is sometimes written as:
293:
3633:
3471:
2889:. As another example, the lambda term λ
1853:th subterm's sort matches the declared
1833:) may be composed from sorted subterms
1685:and some substitution σ. In this case,
1465:is the black term in the picture, then
561:of objects to objects. For example, if
3414:denotes the identity function, while λ
3371:denotes a unary function that returns
3348:) is also a lambda term (application).
2694:
1866:
1449:(indicated by "."; its position being
1106:. In mathematical logic, more complex
1043:, since a ground term of height up to
1362:starts, which is commonly denoted by
1202:Tree structure of black example term
1110:are built from atomic formulas using
73:. An expression formed by applying a
7:
3318:is also a lambda term (abstraction);
2528:{\displaystyle \sum _{i=1}^{n}i^{2}}
1102:is usually non-empty only for small
1047:+1 can be obtained by composing any
3441:and returns the result of applying
2300:as 2nd argument). In order to make
1963:{\displaystyle {\vec {0}}\in C_{W}}
1865:; any other term (i.e. obeying the
693:{\displaystyle {\frac {n(n+1)}{2}}}
203:; it is part of the atomic formula
3669:Introduction to Mathematical Logic
2994:// implements general sum operator
1000:
584:is a binary function symbol, then
155:, and the binary function symbols
25:
3576:
3491:
1657:structurally equals a subterm of
620:Term structure vs. representation
369:every constant symbol is a term:
358:every variable symbol is a term:
263:which evaluates to true for each
254:{\displaystyle (x+1)*(x+1)\geq 0}
5439:
3406:. For example, the abstraction λ
3247:
2368:{\displaystyle *:N\times W\to W}
1653:, if a substitution instance of
278:, terms play important roles in
50:refers to an object and a whole
2866:to be supplied as arguments to
2697:, as they all introduce an own
843:The set of variables of a term
3363:Intuitively, the abstraction λ
2821:
2809:
2751:
2739:
2610:
2598:
2359:
2319:
2274:
2236:
2218:
2212:
2197:
2188:
2130:
2115:
2084:
2030:
2006:
1941:
1732:
1720:
1595:
1583:
1572:
1564:
1525:
1513:
1502:
1490:
1378:+2 has its root. The relation
1320:
1308:
1277:
1265:
1254:
1251:
1239:
1233:
1221:
1218:
681:
669:
242:
230:
224:
212:
136:
124:
118:
106:
1:
5400:History of mathematical logic
1861:. Such a term is also called
1051:ground terms of height up to
329:of constant symbols and sets
5325:Primitive recursive function
3551:", which is synonymous to "∀
2887:second-order function symbol
2328:{\displaystyle a*{\vec {v}}}
2289:{\displaystyle {\vec {v}}+a}
1622:To each node of a term, its
1167:{\displaystyle \mathbb {R} }
1133:{\displaystyle \mathbb {R} }
1064:Building formulas from terms
301:tree structure of the term (
3615:Term Rewriting and All That
3588:I.e., "symbol type" in the
2401:Terms with bound variables
1386:on the set of terms; it is
441:Using an intuitive, pseudo-
325:of variable symbols, a set
142:{\displaystyle (x+1)*(x+1)}
5492:
4389:Schröder–Bernstein theorem
4116:Monadic predicate calculus
3775:Foundations of mathematics
3379:, while the application (
2873:For example, the function
2681:
1795:
1401:the subterm at a position
573:is a constant symbol, and
569:is a variable symbol, 1 ∈
5435:
5422:Philosophy of mathematics
5371:Automated theorem proving
4542:
4496:Von Neumann–Bernays–Gödel
4137:
3335:are lambda terms, then (
3306:is a variable symbol and
1880:comes with an associated
1441:; the latter is called a
922:-ary function symbols as
708:Two terms are said to be
3460:Expression (mathematics)
3310:is a lambda term, then λ
2946:
1435:with a term-like object
1089:-ary relation symbol to
5072:Self-verifying theories
4893:Tarski's axiomatization
3844:Tarski's undefinability
3839:incompleteness theorems
1884:of scalar numbers. Let
1806:(sometimes also called
1738:{\displaystyle x*(y*z)}
1693:, and σ are called the
1409:is commonly denoted by
1326:{\displaystyle x*(y*z)}
1190:for these symbol sets.
1118:. For example, letting
839:Ground and linear terms
65:from constant symbols,
63:recursively constructed
5476:Elementary mathematics
5446:Mathematics portal
5057:Proof of impossibility
4705:propositional variable
4015:Propositional calculus
3590:Many-sorted signatures
3287:every variable symbol
2860:can be used to denote
2835:
2765:
2623:
2529:
2514:
2369:
2329:
2290:
2261:is well-sorted, while
2255:
2153:
2094:
2040:
1964:
1739:
1617:encompassment ordering
1605:
1535:
1335:
1327:
1287:
1168:
1134:
1037:
1004:
873:+1) is a linear term,
694:
318:
255:
195:
171:
143:
5315:Kolmogorov complexity
5268:Computably enumerable
5168:Model complete theory
4960:Principia Mathematica
4020:Propositional formula
3849:Banach–Tarski paradox
2836:
2766:
2624:
2530:
2494:
2390:many-sorted framework
2370:
2330:
2291:
2256:
2154:
2095:
2041:
1965:
1740:
1703:matching substitution
1606:
1536:
1393:The term obtained by
1328:
1288:
1201:
1194:Operations with terms
1169:
1135:
1038:
984:
785:equal modulo renaming
769:renaming substitution
695:
404:-ary function symbol
297:
256:
196:
172:
144:
81:, which evaluates to
5263:Church–Turing thesis
5250:Computability theory
4459:continuum hypothesis
3977:Square of opposition
3835:Gödel's completeness
2779:
2709:
2574:
2491:
2341:
2304:
2265:
2182:
2106:
2054:
1988:
1932:
1711:
1552:
1478:
1474:results in the term
1299:
1206:
1156:
1122:
962:
918:, and the number of
743:In contrast, a term
660:
209:
185:
161:
103:
5417:Mathematical object
5308:P versus NP problem
5273:Computable function
5067:Reverse mathematics
4993:Logical consequence
4870:primitive recursive
4865:elementary function
4638:Free/bound variable
4491:Tarski–Grothendieck
4010:Logical connectives
3940:Logical equivalence
3790:Logical consequence
3691:; here: Sect.II.1.3
3671:. Springer London.
3527:" actually means "∀
3437:) takes a function
3426:. The lambda term λ
2863:anonymous functions
2802:
2732:
2591:
2402:
1112:logical connectives
1032:
730:rational arithmetic
704:Structural equality
535:domain of discourse
352:recursively defined
40:mathematical object
5466:Mathematical logic
5215:Transfer principle
5178:Semantics of logic
5163:Categorical theory
5139:Non-standard model
4653:Logical connective
3780:Information theory
3729:Mathematical logic
3259:. You can help by
2831:
2788:
2761:
2718:
2619:
2577:
2525:
2400:
2365:
2325:
2286:
2251:
2149:
2090:
2036:
1960:
1857:th domain sort of
1735:
1677:for some position
1661:, or formally, if
1601:
1531:
1461:. For example, if
1336:
1323:
1293:, with blue redex
1283:
1164:
1140:denote the set of
1130:
1033:
1018:
736:division, then at
734:truncating integer
726:semantically equal
690:
521:Ternary operations
319:
251:
191:
167:
139:
54:refers to a fact.
32:mathematical logic
5471:Rewriting systems
5453:
5452:
5385:Abstract category
5188:Theories of truth
4998:Rule of inference
4988:Natural deduction
4969:
4968:
4514:
4513:
4219:Cartesian product
4124:
4123:
4030:Many-valued logic
4005:Boolean functions
3888:Russell's paradox
3863:diagonal argument
3760:First-order logic
3646:H. Jerome Keisler
3624:978-0-521-77920-3
3509:universal closure
3486:rather than e.g.
3295:is a lambda term;
3277:
3276:
2841:does make sense.
2680:
2679:
2386:many-sorted logic
2322:
2277:
2239:
2215:
2200:
2133:
2118:
1944:
1798:Many-sorted logic
1599:
1548:into the context
1529:
1380:"is a subterm of"
1350:At each position
1281:
775:is a renaming of
771:σ. In that case,
688:
290:Formal definition
284:rewriting systems
280:universal algebra
194:{\displaystyle *}
170:{\displaystyle +}
16:(Redirected from
5483:
5444:
5443:
5395:History of logic
5390:Category of sets
5283:Decision problem
5062:Ordinal analysis
5003:Sequent calculus
4901:Boolean algebras
4841:
4840:
4815:
4786:logical/constant
4540:
4526:
4449:Zermelo–Fraenkel
4200:Set operations:
4135:
4072:
3903:
3883:Löwenheim–Skolem
3770:Formal semantics
3722:
3715:
3708:
3699:
3692:
3690:
3661:
3655:
3654:; here: Sect.1.3
3653:
3638:
3628:
3593:
3586:
3580:
3505:alpha-conversion
3501:
3495:
3476:
3272:
3269:
3251:
3244:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2840:
2838:
2837:
2832:
2801:
2796:
2770:
2768:
2767:
2762:
2731:
2726:
2628:
2626:
2625:
2620:
2590:
2585:
2534:
2532:
2531:
2526:
2524:
2523:
2513:
2508:
2451:
2443:
2442:
2441:
2440:
2433:
2403:
2392:described here.
2376:
2374:
2372:
2371:
2366:
2334:
2332:
2331:
2326:
2324:
2323:
2315:
2295:
2293:
2292:
2287:
2279:
2278:
2270:
2260:
2258:
2257:
2252:
2241:
2240:
2232:
2217:
2216:
2208:
2202:
2201:
2193:
2177:
2158:
2156:
2155:
2150:
2148:
2147:
2135:
2134:
2126:
2120:
2119:
2111:
2101:
2099:
2097:
2096:
2091:
2047:
2045:
2043:
2042:
2037:
1981:
1969:
1967:
1966:
1961:
1959:
1958:
1946:
1945:
1937:
1869:only) is called
1856:
1852:
1787:Related concepts
1770:
1746:
1744:
1742:
1741:
1736:
1676:
1610:
1608:
1607:
1602:
1600:
1598:
1575:
1556:
1547:
1540:
1538:
1537:
1532:
1530:
1528:
1505:
1482:
1473:
1447:term with a hole
1440:
1430:
1419:
1373:
1334:
1332:
1330:
1329:
1324:
1292:
1290:
1289:
1284:
1282:
1280:
1257:
1210:
1173:
1171:
1170:
1165:
1163:
1139:
1137:
1136:
1131:
1129:
1042:
1040:
1039:
1034:
1031:
1026:
1014:
1013:
1003:
998:
980:
979:
699:
697:
696:
691:
689:
684:
664:
655:
643:
626:character string
417:, a larger term
270:
262:
260:
258:
257:
252:
202:
200:
198:
197:
192:
178:
176:
174:
173:
168:
154:
150:
148:
146:
145:
140:
75:predicate symbol
71:function symbols
21:
5491:
5490:
5486:
5485:
5484:
5482:
5481:
5480:
5456:
5455:
5454:
5449:
5438:
5431:
5376:Category theory
5366:Algebraic logic
5349:
5320:Lambda calculus
5258:Church encoding
5244:
5220:Truth predicate
5076:
5042:Complete theory
4965:
4834:
4830:
4826:
4821:
4813:
4533: and
4529:
4524:
4510:
4486:New Foundations
4454:axiom of choice
4437:
4399:Gödel numbering
4339: and
4331:
4235:
4120:
4070:
4051:
4000:Boolean algebra
3986:
3950:Equiconsistency
3915:Classical logic
3892:
3873:Halting problem
3861: and
3837: and
3825: and
3824:
3819:Theorems (
3814:
3731:
3726:
3696:
3695:
3679:
3663:
3662:
3658:
3640:
3639:
3635:
3625:
3605:
3602:
3597:
3596:
3587:
3583:
3511:of the axiom: "
3502:
3498:
3480:quantifier-free
3477:
3473:
3468:
3451:
3405:
3399:with the input
3398:
3391:
3385:
3347:
3341:
3334:
3327:
3273:
3267:
3264:
3257:needs expansion
3242:
3237:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3165:" %d"
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3129:<stdio.h>
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2940:operators into
2777:
2776:
2707:
2706:
2691:
2686:
2572:
2571:
2515:
2489:
2488:
2435:
2434:
2431:
2430:
2429:
2428:
2422:
2417:
2412:
2407:
2398:
2339:
2338:
2336:
2302:
2301:
2263:
2262:
2180:
2179:
2176:
2160:
2139:
2104:
2103:
2052:
2051:
2049:
1986:
1985:
1983:
1980:
1971:
1950:
1930:
1929:
1927:
1918:
1909:
1900:
1876:For example, a
1854:
1850:
1848:
1839:
1832:
1823:
1800:
1794:
1789:
1748:
1709:
1708:
1706:
1675:
1662:
1576:
1557:
1550:
1549:
1542:
1506:
1483:
1476:
1475:
1472:
1466:
1436:
1429:
1421:
1418:
1410:
1372:
1363:
1297:
1296:
1294:
1258:
1211:
1204:
1203:
1196:
1184:complex numbers
1154:
1153:
1120:
1119:
1101:
1076:
1066:
1005:
965:
960:
959:
955:
948:
936:
930:
917:
841:
706:
665:
658:
657:
645:
633:
622:
583:
529:
518:
511:
496:
479:
470:
437:) can be built.
436:
427:
416:
399:
390:
337:
292:
268:
207:
206:
204:
183:
182:
180:
159:
158:
156:
152:
101:
100:
98:
97:. For example,
91:bivalent logics
28:
23:
22:
18:Variant (logic)
15:
12:
11:
5:
5489:
5487:
5479:
5478:
5473:
5468:
5458:
5457:
5451:
5450:
5436:
5433:
5432:
5430:
5429:
5424:
5419:
5414:
5409:
5408:
5407:
5397:
5392:
5387:
5378:
5373:
5368:
5363:
5361:Abstract logic
5357:
5355:
5351:
5350:
5348:
5347:
5342:
5340:Turing machine
5337:
5332:
5327:
5322:
5317:
5312:
5311:
5310:
5305:
5300:
5295:
5290:
5280:
5278:Computable set
5275:
5270:
5265:
5260:
5254:
5252:
5246:
5245:
5243:
5242:
5237:
5232:
5227:
5222:
5217:
5212:
5207:
5206:
5205:
5200:
5195:
5185:
5180:
5175:
5173:Satisfiability
5170:
5165:
5160:
5159:
5158:
5148:
5147:
5146:
5136:
5135:
5134:
5129:
5124:
5119:
5114:
5104:
5103:
5102:
5097:
5090:Interpretation
5086:
5084:
5078:
5077:
5075:
5074:
5069:
5064:
5059:
5054:
5044:
5039:
5038:
5037:
5036:
5035:
5025:
5020:
5010:
5005:
5000:
4995:
4990:
4985:
4979:
4977:
4971:
4970:
4967:
4966:
4964:
4963:
4955:
4954:
4953:
4952:
4947:
4946:
4945:
4940:
4935:
4915:
4914:
4913:
4911:minimal axioms
4908:
4897:
4896:
4895:
4884:
4883:
4882:
4877:
4872:
4867:
4862:
4857:
4844:
4842:
4823:
4822:
4820:
4819:
4818:
4817:
4805:
4800:
4799:
4798:
4793:
4788:
4783:
4773:
4768:
4763:
4758:
4757:
4756:
4751:
4741:
4740:
4739:
4734:
4729:
4724:
4714:
4709:
4708:
4707:
4702:
4697:
4687:
4686:
4685:
4680:
4675:
4670:
4665:
4660:
4650:
4645:
4640:
4635:
4634:
4633:
4628:
4623:
4618:
4608:
4603:
4601:Formation rule
4598:
4593:
4592:
4591:
4586:
4576:
4575:
4574:
4564:
4559:
4554:
4549:
4543:
4537:
4520:Formal systems
4516:
4515:
4512:
4511:
4509:
4508:
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4468:
4463:
4462:
4461:
4456:
4445:
4443:
4439:
4438:
4436:
4435:
4434:
4433:
4423:
4418:
4417:
4416:
4409:Large cardinal
4406:
4401:
4396:
4391:
4386:
4372:
4371:
4370:
4365:
4360:
4345:
4343:
4333:
4332:
4330:
4329:
4328:
4327:
4322:
4317:
4307:
4302:
4297:
4292:
4287:
4282:
4277:
4272:
4267:
4262:
4257:
4252:
4246:
4244:
4237:
4236:
4234:
4233:
4232:
4231:
4226:
4221:
4216:
4211:
4206:
4198:
4197:
4196:
4191:
4181:
4176:
4174:Extensionality
4171:
4169:Ordinal number
4166:
4156:
4151:
4150:
4149:
4138:
4132:
4126:
4125:
4122:
4121:
4119:
4118:
4113:
4108:
4103:
4098:
4093:
4088:
4087:
4086:
4076:
4075:
4074:
4061:
4059:
4053:
4052:
4050:
4049:
4048:
4047:
4042:
4037:
4027:
4022:
4017:
4012:
4007:
4002:
3996:
3994:
3988:
3987:
3985:
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3953:
3952:
3942:
3937:
3932:
3927:
3922:
3917:
3911:
3909:
3900:
3894:
3893:
3891:
3890:
3885:
3880:
3875:
3870:
3865:
3853:Cantor's
3851:
3846:
3841:
3831:
3829:
3816:
3815:
3813:
3812:
3807:
3802:
3797:
3792:
3787:
3782:
3777:
3772:
3767:
3762:
3757:
3752:
3751:
3750:
3739:
3737:
3733:
3732:
3727:
3725:
3724:
3717:
3710:
3702:
3694:
3693:
3677:
3656:
3632:
3631:
3630:
3629:
3623:
3601:
3598:
3595:
3594:
3581:
3496:
3470:
3469:
3467:
3464:
3463:
3462:
3457:
3450:
3447:
3403:
3396:
3389:
3383:
3350:
3349:
3345:
3339:
3332:
3325:
3319:
3296:
3275:
3274:
3254:
3252:
3241:
3238:
2947:
2929:/3, ...). The
2830:
2827:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2800:
2795:
2791:
2787:
2784:
2760:
2757:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2730:
2725:
2721:
2717:
2714:
2690:
2687:
2682:Main article:
2678:
2677:
2647:
2634:
2629:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2589:
2584:
2580:
2568:
2567:
2545:
2540:
2535:
2522:
2518:
2512:
2507:
2504:
2501:
2497:
2485:
2484:
2462:
2457:
2452:
2425:
2424:
2419:
2414:
2409:
2397:
2394:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2321:
2318:
2312:
2309:
2285:
2282:
2276:
2273:
2250:
2247:
2244:
2238:
2235:
2229:
2226:
2223:
2220:
2214:
2211:
2205:
2199:
2196:
2190:
2187:
2172:
2146:
2142:
2138:
2132:
2129:
2123:
2117:
2114:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1976:
1957:
1953:
1949:
1943:
1940:
1923:
1914:
1905:
1896:
1867:unsorted rules
1844:
1837:
1828:
1821:
1796:Main article:
1793:
1790:
1788:
1785:
1784:
1783:
1781:term rewriting
1778:
1776:unifying terms
1773:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1671:
1640:
1632:
1620:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1574:
1570:
1566:
1563:
1560:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1470:
1457:is said to be
1425:
1414:
1405:by a new term
1391:
1368:
1348:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1195:
1192:
1162:
1128:
1097:
1072:
1065:
1062:
1061:
1060:
1030:
1025:
1021:
1017:
1012:
1008:
1002:
997:
994:
991:
987:
983:
978:
975:
972:
968:
957:
953:
946:
932:
931:, the number θ
926:
915:
883:term rewriting
847:is denoted by
840:
837:
705:
702:
687:
683:
680:
677:
674:
671:
668:
621:
618:
614:infix notation
596:, and (hence)
581:
549:-ary function
527:
516:
509:
492:
482:
481:
475:
468:
439:
438:
432:
425:
412:
395:
388:
378:
367:
333:
299:Left to right:
291:
288:
250:
247:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
190:
166:
138:
135:
132:
129:
126:
123:
120:
117:
114:
111:
108:
95:interpretation
79:atomic formula
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5488:
5477:
5474:
5472:
5469:
5467:
5464:
5463:
5461:
5448:
5447:
5442:
5434:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5406:
5403:
5402:
5401:
5398:
5396:
5393:
5391:
5388:
5386:
5382:
5379:
5377:
5374:
5372:
5369:
5367:
5364:
5362:
5359:
5358:
5356:
5352:
5346:
5343:
5341:
5338:
5336:
5335:Recursive set
5333:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5309:
5306:
5304:
5301:
5299:
5296:
5294:
5291:
5289:
5286:
5285:
5284:
5281:
5279:
5276:
5274:
5271:
5269:
5266:
5264:
5261:
5259:
5256:
5255:
5253:
5251:
5247:
5241:
5238:
5236:
5233:
5231:
5228:
5226:
5223:
5221:
5218:
5216:
5213:
5211:
5208:
5204:
5201:
5199:
5196:
5194:
5191:
5190:
5189:
5186:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5166:
5164:
5161:
5157:
5154:
5153:
5152:
5149:
5145:
5144:of arithmetic
5142:
5141:
5140:
5137:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5109:
5108:
5105:
5101:
5098:
5096:
5093:
5092:
5091:
5088:
5087:
5085:
5083:
5079:
5073:
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5052:
5051:from ZFC
5048:
5045:
5043:
5040:
5034:
5031:
5030:
5029:
5026:
5024:
5021:
5019:
5016:
5015:
5014:
5011:
5009:
5006:
5004:
5001:
4999:
4996:
4994:
4991:
4989:
4986:
4984:
4981:
4980:
4978:
4976:
4972:
4962:
4961:
4957:
4956:
4951:
4950:non-Euclidean
4948:
4944:
4941:
4939:
4936:
4934:
4933:
4929:
4928:
4926:
4923:
4922:
4920:
4916:
4912:
4909:
4907:
4904:
4903:
4902:
4898:
4894:
4891:
4890:
4889:
4885:
4881:
4878:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4856:
4853:
4852:
4850:
4846:
4845:
4843:
4838:
4832:
4827:Example
4824:
4816:
4811:
4810:
4809:
4806:
4804:
4801:
4797:
4794:
4792:
4789:
4787:
4784:
4782:
4779:
4778:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4755:
4752:
4750:
4747:
4746:
4745:
4742:
4738:
4735:
4733:
4730:
4728:
4725:
4723:
4720:
4719:
4718:
4715:
4713:
4710:
4706:
4703:
4701:
4698:
4696:
4693:
4692:
4691:
4688:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4655:
4654:
4651:
4649:
4646:
4644:
4641:
4639:
4636:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4614:
4613:
4612:
4609:
4607:
4604:
4602:
4599:
4597:
4594:
4590:
4587:
4585:
4584:by definition
4582:
4581:
4580:
4577:
4573:
4570:
4569:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4544:
4541:
4538:
4536:
4532:
4527:
4521:
4517:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4479:
4477:
4474:
4472:
4471:Kripke–Platek
4469:
4467:
4464:
4460:
4457:
4455:
4452:
4451:
4450:
4447:
4446:
4444:
4440:
4432:
4429:
4428:
4427:
4424:
4422:
4419:
4415:
4412:
4411:
4410:
4407:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4384:
4380:
4376:
4373:
4369:
4366:
4364:
4361:
4359:
4356:
4355:
4354:
4350:
4347:
4346:
4344:
4342:
4338:
4334:
4326:
4323:
4321:
4318:
4316:
4315:constructible
4313:
4312:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4281:
4278:
4276:
4273:
4271:
4268:
4266:
4263:
4261:
4258:
4256:
4253:
4251:
4248:
4247:
4245:
4243:
4238:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4210:
4207:
4205:
4202:
4201:
4199:
4195:
4192:
4190:
4187:
4186:
4185:
4182:
4180:
4177:
4175:
4172:
4170:
4167:
4165:
4161:
4157:
4155:
4152:
4148:
4145:
4144:
4143:
4140:
4139:
4136:
4133:
4131:
4127:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4085:
4082:
4081:
4080:
4077:
4073:
4068:
4067:
4066:
4063:
4062:
4060:
4058:
4054:
4046:
4043:
4041:
4038:
4036:
4033:
4032:
4031:
4028:
4026:
4023:
4021:
4018:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3997:
3995:
3993:
3992:Propositional
3989:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3951:
3948:
3947:
3946:
3943:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3923:
3921:
3920:Logical truth
3918:
3916:
3913:
3912:
3910:
3908:
3904:
3901:
3899:
3895:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3864:
3860:
3856:
3852:
3850:
3847:
3845:
3842:
3840:
3836:
3833:
3832:
3830:
3828:
3822:
3817:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3776:
3773:
3771:
3768:
3766:
3763:
3761:
3758:
3756:
3753:
3749:
3746:
3745:
3744:
3741:
3740:
3738:
3734:
3730:
3723:
3718:
3716:
3711:
3709:
3704:
3703:
3700:
3688:
3684:
3680:
3674:
3670:
3666:
3660:
3657:
3651:
3647:
3643:
3637:
3634:
3626:
3620:
3616:
3612:
3611:Tobias Nipkow
3608:
3604:
3603:
3599:
3591:
3585:
3582:
3578:
3577:#Lambda terms
3574:
3570:
3566:
3562:
3558:
3554:
3550:
3546:
3542:
3538:
3534:
3530:
3526:
3522:
3518:
3514:
3510:
3506:
3500:
3497:
3493:
3492:#Sorted terms
3489:
3485:
3481:
3475:
3472:
3465:
3461:
3458:
3456:
3453:
3452:
3448:
3446:
3444:
3440:
3436:
3433:
3429:
3425:
3421:
3417:
3413:
3409:
3402:
3395:
3388:
3382:
3378:
3374:
3370:
3366:
3361:
3359:
3355:
3344:
3338:
3331:
3324:
3320:
3317:
3313:
3309:
3305:
3301:
3297:
3294:
3290:
3286:
3285:
3284:
3282:
3271:
3262:
3258:
3255:This section
3253:
3250:
3246:
3245:
3239:
2945:
2943:
2939:
2934:
2932:
2928:
2924:
2920:
2916:
2912:
2908:
2904:
2900:
2896:
2892:
2888:
2884:
2880:
2876:
2871:
2870:, Σ, ∫, etc.
2869:
2865:
2864:
2859:
2858:
2853:
2851:
2847:
2842:
2828:
2825:
2818:
2815:
2812:
2806:
2803:
2798:
2793:
2789:
2785:
2782:
2774:
2758:
2755:
2748:
2745:
2742:
2736:
2733:
2728:
2723:
2719:
2715:
2712:
2704:
2700:
2696:
2688:
2685:
2675:
2671:
2667:
2663:
2659:
2655:
2651:
2648:
2646:
2642:
2638:
2635:
2633:
2630:
2616:
2613:
2607:
2604:
2601:
2595:
2592:
2587:
2582:
2578:
2570:
2569:
2565:
2561:
2557:
2553:
2549:
2546:
2544:
2541:
2539:
2536:
2520:
2516:
2510:
2505:
2502:
2499:
2495:
2487:
2486:
2482:
2478:
2474:
2470:
2466:
2463:
2461:
2458:
2456:
2453:
2450:
2446:
2438:
2427:
2426:
2420:
2415:
2410:
2405:
2404:
2395:
2393:
2391:
2387:
2382:
2380:
2362:
2356:
2353:
2350:
2347:
2344:
2316:
2310:
2307:
2299:
2283:
2280:
2271:
2245:
2242:
2233:
2227:
2224:
2221:
2209:
2203:
2194:
2175:
2171:
2167:
2163:
2144:
2140:
2136:
2127:
2121:
2112:
2087:
2081:
2078:
2075:
2072:
2066:
2063:
2060:
2033:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2003:
2000:
1997:
1994:
1991:
1979:
1975:
1955:
1951:
1947:
1938:
1926:
1922:
1917:
1913:
1908:
1904:
1899:
1895:
1891:
1887:
1883:
1879:
1874:
1872:
1868:
1864:
1860:
1847:
1843:
1836:
1831:
1827:
1820:
1816:
1813:
1809:
1805:
1799:
1791:
1786:
1782:
1779:
1777:
1774:
1768:
1764:
1760:
1756:
1752:
1729:
1726:
1723:
1717:
1714:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1674:
1669:
1665:
1660:
1656:
1652:
1648:
1645:
1641:
1639:respectively.
1637:
1633:
1629:
1625:
1621:
1618:
1614:
1613:instantiating
1592:
1589:
1586:
1580:
1577:
1568:
1561:
1558:
1545:
1522:
1519:
1516:
1510:
1507:
1499:
1496:
1493:
1487:
1484:
1469:
1464:
1460:
1456:
1452:
1448:
1444:
1439:
1434:
1428:
1424:
1417:
1413:
1408:
1404:
1400:
1396:
1392:
1389:
1385:
1384:partial order
1381:
1377:
1371:
1366:
1361:
1357:
1353:
1349:
1346:
1342:
1338:
1337:
1317:
1314:
1311:
1305:
1302:
1274:
1271:
1268:
1262:
1259:
1248:
1245:
1242:
1236:
1230:
1227:
1224:
1215:
1212:
1200:
1193:
1191:
1189:
1188:Herbrand base
1185:
1181:
1177:
1151:
1147:
1143:
1117:
1113:
1109:
1105:
1100:
1096:
1092:
1088:
1084:
1080:
1075:
1071:
1063:
1058:
1054:
1050:
1046:
1028:
1023:
1019:
1015:
1010:
1006:
995:
992:
989:
985:
981:
976:
973:
970:
966:
958:
952:
944:
943:
942:
940:
935:
929:
925:
921:
914:
909:
907:
905:
900:
899:
896:
891:
886:
884:
880:
876:
872:
868:
864:
860:
859:
854:
850:
846:
838:
836:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
782:
778:
774:
770:
766:
762:
758:
754:
750:
746:
741:
739:
735:
731:
727:
723:
719:
718:syntactically
715:
711:
703:
701:
685:
678:
675:
672:
666:
653:
649:
641:
637:
631:
627:
619:
617:
615:
611:
607:
603:
599:
595:
591:
587:
580:
576:
572:
568:
564:
560:
556:
552:
548:
544:
540:
537:. A constant
536:
531:
526:
522:
515:
508:
504:
500:
495:
491:
487:
478:
474:
467:
463:
459:
455:
451:
448:
447:
446:
444:
435:
431:
424:
420:
415:
411:
407:
403:
398:
394:
387:
383:
379:
376:
372:
368:
365:
361:
357:
356:
355:
353:
349:
345:
341:
336:
332:
328:
324:
316:
312:
308:
304:
300:
296:
289:
287:
285:
281:
277:
272:
266:
265:real-numbered
248:
245:
239:
236:
233:
227:
221:
218:
215:
188:
164:
133:
130:
127:
121:
115:
112:
109:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
55:
53:
49:
45:
41:
37:
33:
19:
5437:
5235:Ultraproduct
5082:Model theory
5047:Independence
4983:Formal proof
4975:Proof theory
4958:
4931:
4888:real numbers
4860:second-order
4802:
4771:Substitution
4648:Metalanguage
4589:conservative
4562:Axiom schema
4506:Constructive
4476:Morse–Kelley
4442:Set theories
4421:Aleph number
4414:inaccessible
4320:Grothendieck
4204:intersection
4091:Higher-order
4079:Second-order
4025:Truth tables
3982:Venn diagram
3765:Formal proof
3668:
3665:Hermes, Hans
3659:
3650:Model Theory
3649:
3636:
3614:
3607:Franz Baader
3584:
3575:"; see also
3572:
3568:
3564:
3560:
3556:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3524:
3520:
3516:
3512:
3499:
3487:
3483:
3474:
3442:
3438:
3434:
3431:
3427:
3423:
3419:
3415:
3411:
3407:
3400:
3393:
3386:
3380:
3376:
3372:
3368:
3364:
3362:
3357:
3353:
3351:
3342:
3336:
3329:
3322:
3315:
3311:
3307:
3303:
3299:
3292:
3288:
3280:
3279:Given a set
3278:
3265:
3261:adding to it
3256:
2935:
2930:
2926:
2922:
2918:
2910:
2906:
2902:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2872:
2867:
2861:
2857:Lambda terms
2855:
2854:
2845:
2843:
2772:
2702:
2698:
2692:
2673:
2669:
2665:
2661:
2657:
2653:
2649:
2644:
2640:
2636:
2631:
2563:
2559:
2555:
2551:
2547:
2542:
2537:
2480:
2476:
2472:
2468:
2464:
2459:
2454:
2448:
2444:
2436:
2423:lambda-term
2396:Lambda terms
2389:
2383:
2378:
2297:
2173:
2169:
2165:
2161:
1977:
1973:
1924:
1920:
1915:
1911:
1906:
1902:
1897:
1893:
1889:
1885:
1878:vector space
1875:
1870:
1862:
1858:
1849:only if the
1845:
1841:
1834:
1829:
1825:
1818:
1814:
1811:
1807:
1803:
1801:
1792:Sorted terms
1766:
1762:
1758:
1754:
1750:
1702:
1699:subject term
1698:
1695:pattern term
1694:
1690:
1686:
1682:
1678:
1672:
1667:
1663:
1658:
1654:
1650:
1646:
1643:
1635:
1627:
1623:
1543:
1467:
1462:
1458:
1454:
1453:), in which
1450:
1446:
1442:
1437:
1432:
1426:
1422:
1415:
1411:
1406:
1402:
1398:
1394:
1379:
1375:
1369:
1364:
1359:
1355:
1351:
1344:
1340:
1179:
1175:
1149:
1145:
1142:real numbers
1103:
1098:
1094:
1090:
1086:
1082:
1078:
1073:
1069:
1068:Given a set
1067:
1056:
1052:
1048:
1044:
950:
938:
933:
927:
923:
919:
912:
910:
906:term algebra
902:
898:term algebra
893:
887:
878:
874:
870:
866:
862:
856:
852:
848:
844:
842:
832:
828:
824:
820:
816:
812:
808:
804:
800:
796:
792:
788:
784:
780:
776:
772:
764:
760:
756:
755:, of a term
752:
748:
747:is called a
744:
742:
737:
725:
721:
717:
713:
710:structurally
709:
707:
651:
647:
639:
635:
625:
623:
609:
605:
601:
597:
593:
589:
585:
578:
574:
570:
566:
562:
554:
550:
546:
542:
538:
532:
524:
513:
506:
502:
498:
493:
489:
483:
476:
472:
465:
461:
457:
453:
449:
440:
433:
429:
422:
418:
413:
409:
405:
401:
400:, and every
396:
392:
385:
381:
374:
370:
363:
359:
347:
343:
339:
334:
330:
326:
322:
321:Given a set
320:
314:
310:
306:
302:
298:
273:
56:
35:
29:
5345:Type theory
5293:undecidable
5225:Truth value
5112:equivalence
4791:non-logical
4404:Enumeration
4394:Isomorphism
4341:cardinality
4325:Von Neumann
4290:Ultrafilter
4255:Uncountable
4189:equivalence
4106:Quantifiers
4096:Fixed-point
4065:First-order
3945:Consistency
3930:Proposition
3907:Traditional
3878:Lindström's
3868:Compactness
3810:Type theory
3755:Cardinality
3445:to itself.
3375:when given
2684:Lambda term
2421:Written as
2178:, the term
1863:well-sorted
1812:sorted term
1420:. The term
1358:, a unique
1116:quantifiers
1055:, using an
863:linear term
858:ground term
443:grammatical
380:from every
309:+1))/2 and
274:Besides in
93:, given an
59:first-order
48:noun phrase
5460:Categories
5156:elementary
4849:arithmetic
4717:Quantifier
4695:functional
4567:Expression
4285:Transitive
4229:identities
4214:complement
4147:hereditary
4130:Set theory
3678:3540058192
3642:C.C. Chang
3600:References
3268:April 2021
3240:Definition
2689:Motivation
2379:overloaded
1871:ill-sorted
1701:, and the
1397:in a term
1354:of a term
612:+1, using
452: ::=
38:denotes a
5427:Supertask
5330:Recursion
5288:decidable
5122:saturated
5100:of models
5023:deductive
5018:axiomatic
4938:Hilbert's
4925:Euclidean
4906:canonical
4829:axiomatic
4761:Signature
4690:Predicate
4579:Extension
4501:Ackermann
4426:Operation
4305:Universal
4295:Recursive
4270:Singleton
4265:Inhabited
4250:Countable
4240:Types of
4224:power set
4194:partition
4111:Predicate
4057:Predicate
3972:Syllogism
3962:Soundness
3935:Inference
3925:Tautology
3827:paradoxes
3687:1431-4657
2816:⋅
2807:
2790:∫
2786:⋅
2746:⋅
2737:
2720:∫
2716:⋅
2605:⋅
2596:
2579:∫
2496:∑
2418:variables
2413:variables
2406:Notation
2360:→
2354:×
2345:∗
2320:→
2311:∗
2275:→
2249:⟩
2243:∗
2237:→
2222:∗
2213:→
2198:→
2186:⟨
2137:∈
2131:→
2116:→
2085:→
2079:×
2070:⟩
2058:⟨
2031:→
2025:×
2016:∗
2007:→
2001:×
1948:∈
1942:→
1727:∗
1718:∗
1590:∗
1581:∗
1562:∗
1520:∗
1511:∗
1488:∗
1395:replacing
1388:reflexive
1315:∗
1306:∗
1272:∗
1263:∗
1237:∗
1216:∗
1020:θ
1016:⋅
1001:∞
986:∑
967:θ
890:signature
767:for some
714:literally
486:signature
267:value of
246:≥
228:∗
189:∗
122:∗
67:variables
5412:Logicism
5405:timeline
5381:Concrete
5240:Validity
5210:T-schema
5203:Kripke's
5198:Tarski's
5193:semantic
5183:Strength
5132:submodel
5127:spectrum
5095:function
4943:Tarski's
4932:Elements
4919:geometry
4875:Robinson
4796:variable
4781:function
4754:spectrum
4744:Sentence
4700:variable
4643:Language
4596:Relation
4557:Automata
4547:Alphabet
4531:language
4385:-jection
4363:codomain
4349:Function
4310:Universe
4280:Infinite
4184:Relation
3967:Validity
3957:Argument
3855:theorem,
3667:(1973).
3648:(1977).
3613:(1999).
3455:Equation
3449:See also
3186:"%d
3126:#include
2915:sequence
2650:integral
1772:subject.
1765:+1, z ↦
1626:(called
1459:embedded
1341:position
1108:formulas
888:Given a
749:renaming
656:", and "
61:term is
52:sentence
42:while a
5354:Related
5151:Diagram
5049: (
5028:Hilbert
5013:Systems
5008:Theorem
4886:of the
4831:systems
4611:Formula
4606:Grammar
4522: (
4466:General
4179:Forcing
4164:Element
4084:Monadic
3859:paradox
3800:Theorem
3736:General
3507:on the
3494:below).
2408:example
2375:
2337:
2100:
2050:
2046:
1984:
1745:
1707:
1649:a term
1647:matches
1642:A term
1445:, or a
1443:context
1360:subterm
1333:
1295:
904:initial
753:variant
751:, or a
654:+1)))/2
604:, 1) ∈
471:, ...,
428:, ...,
261:
205:
201:
181:
177:
157:
149:
99:
44:formula
5117:finite
4880:Skolem
4833:
4808:Theory
4776:Symbol
4766:String
4749:atomic
4626:ground
4621:closed
4616:atomic
4572:ground
4535:syntax
4431:binary
4358:domain
4275:Finite
4040:finite
3898:Logics
3857:
3805:Theory
3685:
3675:
3621:
3579:below.
3490:, cf.
3225:return
3216:square
3192:"
3180:printf
3105:return
3087:square
3072:return
2944:form.
2942:prefix
2875:square
2411:Bound
2048:, and
1697:, the
1631:green.
1628:height
803:or as
642:+1))/2
592:, 1 ∈
559:tuples
384:terms
317:+1)/2)
282:, and
5107:Model
4855:Peano
4712:Proof
4552:Arity
4481:Naive
4368:image
4300:Fuzzy
4260:Empty
4209:union
4154:Class
3795:Model
3785:Lemma
3743:Axiom
3466:Notes
3358:power
3171:&
3159:scanf
3036:<=
2938:infix
2703:bound
2701:, or
2699:local
2695:above
2566:,2))
2560:power
2465:limit
2416:Free
1882:field
1840:,...,
1824:,...,
1769:+2 }
1624:depth
1382:is a
1343:, or
1178:+1)⋅(
716:, or
644:", "
553:maps
391:,...,
276:logic
87:false
5230:Type
5033:list
4837:list
4814:list
4803:Term
4737:rank
4631:open
4525:list
4337:Maps
4242:sets
4101:Free
4071:list
3821:list
3748:list
3683:ISSN
3673:ISBN
3619:ISBN
3484:bool
3328:and
3141:void
3135:main
2925:/2,
2921:/1,
2909:/2,
2905:/1,
2773:free
2384:See
2159:and
1972:0 ∈
1970:and
1919:and
1901:and
1888:and
1808:type
1804:sort
1666:σ =
1636:size
1634:The
1345:path
1114:and
895:free
849:vars
630:tree
484:The
179:and
83:true
69:and
36:term
34:, a
4917:of
4899:of
4847:of
4379:Sur
4353:Map
4160:Ur-
4142:Set
3488:int
3354:div
3321:if
3298:if
3263:.
3219:));
3198:sum
3150:int
3132:int
3093:int
3084:int
3075:res
3060:fct
3054:res
3039:upb
3027:lwb
3018:int
3012:for
3000:res
2997:int
2985:int
2979:fct
2976:int
2970:upb
2967:int
2961:lwb
2958:int
2952:sum
2949:int
2931:lim
2868:lim
2846:lim
2804:sin
2734:sin
2676:))
2666:sin
2593:sin
2550:(1,
2548:sum
2483:))
2473:div
2432:lim
1681:in
1174:⇒ (
1144:, ∀
1077:of
598:add
575:add
503:cos
499:sin
350:is
338:of
313:⋅((
89:in
85:or
30:In
5462::
5303:NP
4927::
4921::
4851::
4528:),
4383:Bi
4375:In
3681:.
3644:;
3609:;
3559::
3535::
3430:.(
3356:,
3189:\n
3177:);
3069:);
3057:+=
3045:++
2988:))
2893:.
2881:.
2664:.
2660:,λ
2643:,
2639:,
2558:.
2554:,λ
2471:.
2467:(λ
2439:→∞
2381:.
2168:∈
1873:.
1761:↦
1757:,
1753:↦
1749:{
1689:,
1546:+1
1471:12
1152:∈
1148::
949:=
908:.
885:.
877:⋅(
869:⋅(
765:tσ
763:=
722:un
712:,
650:⋅(
646:((
638:⋅(
588:∈
577:∈
565:∈
519:.
505:∈
501:,
480:).
460:|
456:|
408:∈
373:⊆
362:⊆
305:⋅(
286:.
271:.
57:A
5383:/
5298:P
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4835:(
4732:∀
4727:!
4722:∃
4683:=
4678:↔
4673:→
4668:∧
4663:∨
4658:¬
4381:/
4377:/
4351:/
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4158:(
4045:∞
4035:3
3823:)
3721:e
3714:t
3707:v
3689:.
3627:.
3573:a
3571:+
3569:b
3567:=
3565:b
3563:+
3561:a
3557:b
3555:,
3553:a
3549:x
3547:+
3545:y
3543:=
3541:y
3539:+
3537:x
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3531:,
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3525:x
3523:+
3521:y
3519:=
3517:y
3515:+
3513:x
3443:x
3439:x
3435:x
3432:x
3428:x
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3420:y
3418:.
3416:x
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3410:.
3408:x
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3312:x
3308:t
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3302:∈
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3270:)
3266:(
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3231:;
3228:0
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3120:}
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3114:i
3111:*
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3090:(
3081:}
3078:;
3066:i
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3015:(
3009:;
3006:0
3003:=
2991:{
2982:(
2973:,
2964:,
2955:(
2927:x
2923:x
2919:x
2917:(
2911:x
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2652:(
2645:k
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2564:i
2562:(
2556:i
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2521:2
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2503:=
2500:i
2481:n
2479:,
2477:x
2475:(
2469:n
2460:x
2455:n
2449:n
2447:/
2445:x
2437:n
2363:W
2357:W
2351:N
2348::
2317:v
2308:a
2298:N
2284:a
2281:+
2272:v
2246:b
2234:w
2228:,
2225:a
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2210:0
2204:+
2195:v
2189:(
2174:N
2170:V
2166:b
2164:,
2162:a
2145:W
2141:V
2128:w
2122:,
2113:v
2088:N
2082:W
2076:W
2073::
2067:.
2064:,
2061:.
2034:W
2028:N
2022:W
2019::
2013:,
2010:W
2004:W
1998:W
1995::
1992:+
1978:N
1974:C
1956:W
1952:C
1939:0
1925:N
1921:C
1916:W
1912:C
1907:N
1903:V
1898:W
1894:V
1890:N
1886:W
1859:f
1855:i
1851:i
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1842:t
1838:1
1835:t
1830:n
1826:t
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1819:t
1817:(
1815:f
1767:a
1763:a
1759:y
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1730:z
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1721:(
1715:x
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1673:p
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1508:1
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1500:1
1497:+
1494:b
1491:(
1485:a
1468:t
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1455:u
1451:p
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1427:p
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1416:p
1412:t
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1403:p
1399:t
1376:a
1370:p
1367:|
1365:t
1356:t
1352:p
1321:)
1318:z
1312:y
1309:(
1303:x
1278:)
1275:3
1269:2
1266:(
1260:1
1255:)
1252:)
1249:2
1246:+
1243:a
1240:(
1234:)
1231:1
1228:+
1225:a
1222:(
1219:(
1213:a
1180:x
1176:x
1161:R
1150:x
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1127:R
1104:n
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1070:R
1057:i
1053:h
1049:i
1045:h
1029:i
1024:h
1011:i
1007:f
996:0
993:=
990:i
982:=
977:1
974:+
971:h
954:0
951:f
947:0
945:θ
939:h
934:h
928:i
924:f
920:i
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913:f
879:n
875:n
871:n
867:x
853:t
851:(
845:t
833:a
831:+
829:b
827:=
825:y
823:+
821:x
817:a
815:+
813:b
811:=
809:b
807:+
805:a
801:x
799:+
797:y
795:=
793:y
791:+
789:x
781:t
777:t
773:u
761:u
757:u
745:t
738:n
686:2
682:)
679:1
676:+
673:n
670:(
667:n
652:n
648:n
640:n
636:n
634:(
610:n
606:T
602:n
600:(
594:T
590:T
586:n
582:2
579:F
571:C
567:V
563:n
557:-
555:n
551:f
547:n
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528:0
525:F
517:2
514:F
510:1
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494:n
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469:1
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464:(
462:f
458:c
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419:f
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366:,
364:T
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344:n
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269:x
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