5663:
405:
introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement. Viète's convention was to use consonants for known values, and vowels for unknowns.
1476:, the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.
2182:
664:
1709:
2444:
720:
2024:
2265:
1136:
2187:
However, in an experiment, in order to determine the dependence of pressure on a single one of the independent variables, it is necessary to fix all but one of the variables, say
1623:
2115:
961:
1062:
It is common for variables to play different roles in the same mathematical formula, and names or qualifiers have been introduced to distinguish them. For example, the general
998:
for variables. Variables with similar roles or meanings are often assigned consecutive letters or the same letter with different subscripts. For example, the three axes in 3D
4042:
525:
In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a nowhere
2515:
200:
1832:
885:
4717:
2778:
2553:
133:
2605:
2359:
1859:
232:
2918:
3390:
2720:
2631:
2070:
1909:
2740:
2691:
2671:
2651:
2573:
2329:
2305:
2285:
2205:
2110:
2090:
2044:
1952:
1929:
1883:
322:
252:
4800:
3941:
578:
1348:
is a quantity (usually a number) which is a part of the input of a problem, and remains constant during the whole solution of this problem. For example, in
834:) are commonly used for unknowns and variables of functions. In printed mathematics, the norm is to set variables and constants in an italic typeface.
5114:
1490:
The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. For example, in the notation
3289:
5272:
3909:
3864:
3809:
3748:
3727:
3704:
3454:
2092:, are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here
334:. These two notions are used almost identically, therefore one usually must be told whether a given symbol denotes a variable or a constant.
4060:
5127:
4450:
3326:. A quantity or force which, throughout a mathematical calculation or investigation, is assumed to vary or be capable of varying in value."
2364:
1405:
4712:
437:". Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in an 1887
5132:
5122:
4859:
4065:
1368:
137:
A variable may represent a unspecified number that remain fix during the resolution of a problem; in which case, it is often called a
4610:
4056:
5268:
3379:
3267:
3236:
1957:
1287:, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.
3298:
5687:
5365:
5109:
3934:
3482:
2334:
This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard
354:
4670:
4363:
4104:
2210:
394:
5702:
1509:, the three variables may be all independent and the notation represents a function of three variables. On the other hand, if
5626:
5328:
5091:
5086:
4911:
4332:
4016:
3202:
3823:
2459:
Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for a
1645:
5621:
5404:
5321:
5034:
4965:
4842:
4084:
4692:
5546:
5372:
5058:
4291:
3429:
668:
93:. This is the motivation for the choice of the term. Also, variables are used for denoting values of functions, such as
4697:
2780:
corresponds to a different parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a
5712:
5029:
4768:
4026:
3927:
3189:
3064:
1206:
803:
31:
5424:
5419:
3184:
2466:
5353:
4943:
4337:
4305:
3996:
3335:
1781:
is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.
4070:
5643:
5592:
5489:
4987:
4948:
4425:
3311:
1792:
449:
287:
35:
5484:
4099:
3253:
5414:
4953:
4805:
4788:
4511:
3991:
1307:
277:
1885:, the number of particles, is a positive integer (and therefore a discrete variable), while the other three,
5707:
5316:
5293:
5254:
5140:
5081:
4727:
4647:
4491:
4435:
4048:
3819:
3030:
2983:
2177:{\displaystyle P:\mathbb {R} _{>0}\times \mathbb {N} \times \mathbb {R} _{>0}\rightarrow \mathbb {R} }
1271:
786:
526:
3315:
1071:
5606:
5333:
5311:
5278:
5171:
5002:
4975:
4926:
4810:
4745:
4570:
4536:
4531:
4405:
4236:
4213:
3163:
2888:
1633:
1562:
1441:
1336:
1320:
1195:
900:
515:
466:
453:
342:
299:
258:
82:
63:
3549:
5536:
5389:
5181:
4899:
4635:
4541:
4400:
4385:
4266:
4241:
3618:
3056:
914:
350:
142:
5662:
1834:
This equation would generally be interpreted to have four variables, and one constant. The constant is
3366:
728:
This static formulation led to the modern notion of variable, which is simply a symbol representing a
5509:
5471:
5348:
5152:
4992:
4916:
4894:
4722:
4680:
4579:
4546:
4410:
4198:
4109:
2871:
402:
385:
338:
90:
756:, which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in
372:
is either a symbol representing an unspecified constant of the theory, or a variable which is being
151:
5638:
5529:
5514:
5494:
5451:
5338:
5288:
5214:
5159:
5096:
4889:
4884:
4832:
4600:
4589:
4261:
4161:
4089:
4080:
4076:
4011:
4006:
3339:
3226:
3194:
3153:
3077:
2308:
1770:
1328:
1316:
1284:
822:
are commonly used for known values and parameters, and letters at the end of the alphabet such as (
729:
537:
529:
519:
438:
358:
331:
141:. A variable may denote an unknown number that has to be determined; in which case, it is called a
75:
59:
5667:
5436:
5399:
5384:
5377:
5360:
5146:
5012:
4938:
4921:
4874:
4687:
4596:
4430:
4415:
4375:
4327:
4312:
4300:
4256:
4231:
4001:
3950:
3838:
3778:
3350:
3259:
3127:
3060:
1862:
1381:
1205:
refers commonly to the arguments of the functions. This is typically the case in sentences like "
995:
838:
572:", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula
373:
365:
146:
5164:
4620:
3505:
844:
5697:
5602:
5409:
5219:
5209:
5101:
4982:
4817:
4793:
4574:
4558:
4463:
4440:
4317:
4286:
4251:
4146:
3981:
3905:
3875:
3860:
3805:
3790:
3744:
3723:
3700:
3486:
3450:
3284:
3263:
3232:
3198:
3158:
1262:
1194:, although this last terminology is incorrect for an equation, and should be reserved for the
1015:
904:
807:
733:
410:
346:
3602:
3533:
2745:
2520:
1450:. To simplify formulas, it is often useful to use the same symbol for the dependent variable
100:
5692:
5616:
5611:
5504:
5461:
5283:
5244:
5239:
5224:
5050:
5007:
4904:
4702:
4652:
4226:
4188:
3770:
3442:
1357:
999:
533:
327:
295:
2578:
2337:
1837:
1468:. For example, the state of a physical system depends on measurable quantities such as the
806:, some variable names in pure mathematics consist of several letters and digits. Following
205:
5597:
5587:
5541:
5524:
5479:
5441:
5343:
5263:
5070:
4997:
4970:
4958:
4864:
4778:
4752:
4707:
4675:
4476:
4278:
4221:
4171:
4136:
4094:
3437:
3148:
2894:
1376:
1324:
1039:
2699:
2610:
2049:
1888:
2046:, as a function of the other variables, is the dependent variable, while its arguments,
1483:
is a variable that is implicitly a function of another (or several other) variables. An
5582:
5561:
5519:
5499:
5394:
5249:
4847:
4837:
4827:
4822:
4756:
4630:
4506:
4395:
4390:
4368:
3969:
3716:
3693:
3681:
3504:
Shapiro, Stewart; Kouri Kissel, Teresa (2024), Zalta, Edward N.; Nodelman, Uri (eds.),
3474:
3073:
2947:
2725:
2676:
2656:
2636:
2558:
2314:
2290:
2270:
2190:
2095:
2075:
2029:
1937:
1914:
1868:
1777:
stands for the function argument. When studying the polynomial as an object in itself,
1393:
1276:
1063:
753:
749:
659:{\displaystyle (\forall \epsilon >0)(\exists \eta >0)(\forall x)\;|x-a|<\eta }
471:
307:
237:
50:
1339:
that they define, many authors consider indeterminates as a special kind of variables.
5681:
5556:
5234:
4741:
4526:
4516:
4486:
4471:
4141:
3102:
2454:
988:
457:
1018:
they describe, but various naming conventions exist. A convention often followed in
5456:
5303:
5204:
5196:
5076:
5024:
4933:
4869:
4852:
4783:
4642:
4501:
4203:
3986:
3794:
2996:
2951:
1332:
445:
262:
2607:
in the 2D plane satisfying this equation trace out the graph of a parabola. Here,
1718:
is a summation variable which designates in turn each of the integers 1, 2, ...,
3899:
3854:
3738:
3574:
1765:(they are assumed to be fixed, i.e., parameters of the problem considered) while
5566:
5446:
4625:
4615:
4562:
4246:
4166:
4151:
4031:
3976:
3758:
3470:
3009:
3000:
2817:
1762:
1734:
1553:
1473:
1279:
to obtain the other antiderivatives. Because of the strong relationship between
1186:
1019:
911:
is the variable of the function. A more explicit way to denote this function is
737:
389:
291:
54:
42:
392:
used different colours to represent the unknowns in algebraic equations in the
17:
4496:
4351:
4322:
4128:
3878:
3850:
3446:
3121:
1312:
1280:
1023:
474:
fixed the terminology of infinitesimal calculus, and introduced the notation
5648:
5551:
4604:
4521:
4481:
4445:
4381:
4193:
4183:
4156:
3774:
3026:
2982:
as a second choice) for a fixed integer, such as a count of objects or the
1389:
1349:
1344:
1246:" (meaning that the argument of the function is referred to by the variable
536:
introduced a new formalism consisting of replacing the intuitive notion of
388:, single letters refer to geometric points and shapes. In the 7th century,
3394:
3126:Σ (capital sigma) for a sum, or σ (lowercase sigma) in statistics for the
5633:
5431:
4879:
4584:
4178:
3013:
2987:
2943:
2863:
2460:
1469:
1411:
1299:
748:
Variables are generally denoted by a single letter, most often from the
5229:
4021:
1415:
3842:
3782:
257:
Sometimes the same symbol can be used to denote both a variable and a
3919:
3512:(Spring 2024 ed.), Metaphysics Research Lab, Stanford University
3022:
2969:
also for a line, or in number theory for a prime number not equal to
1424:, whose possible values depend on the value of another variable, say
1331:. However, because of the strong relationship between polynomials or
732:
that either is unknown, or may be replaced by any element of a given
398:. One section of this book is called "Equations of Several Colours".
1418:
and other sciences, it is rather common to consider a variable, say
2307:
are also regarded as constants. Mathematically, this constitutes a
1014:. In physics, the names of variables are largely determined by the
4773:
4119:
3964:
1931:, for pressure, volume and temperature, are continuous variables.
413:"invented the convention of representing unknowns in equations by
3650:
464:
induces a corresponding variation of another quantity which is a
3490:
3136:
3039:
1275:
is an arbitrary constant function that is added to a particular
1054:
for variables representing corresponding better-defined values.
994:
Specific branches and applications of mathematics have specific
540:
by a formal definition. The older notion of limit was "when the
3923:
1319:. Formally speaking, an indeterminate is not a variable, but a
261:, that is a well defined mathematical object. For example, the
3761:(1954). "On Variables in Mathematics and in Natural Science".
1726:
because its variation is over a discrete set of values) while
1364:
has a different meaning and denotes an argument of a function.
1159:, which are taken to be given numbers and the fifth variable,
810:(1596–1650), letters at the beginning of the alphabet such as
725:
in which none of the five variables is considered as varying.
2653:
are regarded as constants, which specify the parabola, while
1636:
of the function being defined, which can be any real number.
3699:(classics ed.). Upper Saddle River, NJ: Prentice Hall.
3114:
as a second choice) for an arbitrarily small positive number
2842:, ... for situations where distinct letters are inconvenient
1769:
is called a variable. When studying this polynomial for its
272:
81:
Originally, the term "variable" was used primarily for the
3795:"Variables in Natural Language: Where do they come from?"
1311:
is a symbol, commonly called variable, that appears in a
1531:) then the notation represents a function of the single
1255:
In the same context, variables that are independent of
66:, often numbers. One say colloqually that the variable
1730:
is a parameter (it does not vary within the formula).
917:
847:
3579:. Munn & Company. September 3, 1887. p. 148.
2897:
2748:
2728:
2702:
2679:
2659:
2639:
2613:
2581:
2561:
2523:
2469:
2367:
2340:
2317:
2293:
2273:
2213:
2193:
2118:
2098:
2078:
2052:
2032:
1960:
1940:
1917:
1891:
1871:
1840:
1795:
1789:
Consider the equation describing the ideal gas law,
1704:{\displaystyle \sum _{i=1}^{n}i={\frac {n^{2}+n}{2}}}
1648:
1565:
1074:
671:
581:
310:
240:
208:
154:
103:
3695:
2439:{\displaystyle P(V,N,T,k_{B})={\frac {Nk_{B}T}{V}}.}
5575:
5470:
5302:
5195:
5047:
4740:
4663:
4557:
4461:
4350:
4277:
4212:
4127:
4118:
4040:
3957:
3901:
Algebra: Sets, Symbols, and the
Language of Thought
3683:
An
Elementary Treatise on the Differential Calculus
2742:as variables, we observe that each set of 3-tuples
1737:, a polynomial of degree 2 is generally denoted as
3715:
3692:
2912:
2772:
2734:
2714:
2685:
2665:
2645:
2625:
2599:
2567:
2547:
2509:
2438:
2353:
2323:
2299:
2279:
2259:
2199:
2176:
2104:
2084:
2064:
2038:
2018:
1946:
1923:
1903:
1877:
1853:
1826:
1703:
1617:
1130:
963:, which clarifies the function-argument status of
955:
879:
715:{\displaystyle \Rightarrow |L-f(x)|<\epsilon ,}
714:
658:
316:
246:
226:
194:
127:
3831:Proceedings of the American Philosophical Society
3763:The British Journal for the Philosophy of Science
3283:Ely, Robert; Adams, Anne E. (February 22, 2012).
2575:are all considered to be real. The set of points
3740:The Britannica Guide to Algebra and Trigonometry
1198:defined by the left-hand side of this equation.
983:occurs in a term that is a constant function of
456:, which essentially consists of studying how an
3722:(4th ed.). United States: Addison-Wesley.
3285:"Unknown, placeholder, or variable: what is x?"
1141:is interpreted as having five variables: four,
470:of the first variable. Almost a century later,
3837:(3). American Philosophical Society: 343–347.
3391:International Organization for Standardization
2019:{\displaystyle P(V,N,T)={\frac {Nk_{B}T}{V}}.}
510:. Until the end of the 19th century, the word
3935:
3800:. In Böttner, Michael; Thümmel, Wolf (eds.).
62:, typically a letter, that holds a place for
8:
3769:(18). University of Chicago Press: 134–142.
1934:One could rearrange this equation to obtain
1392:nature, and the way of computing with them (
1388:All these denominations of variables are of
290:, but has been used to denote an unassigned
3479:Dictionary of symbols of mathematical logic
4761:
4356:
4124:
3942:
3928:
3920:
2260:{\displaystyle P(T)={\frac {Nk_{B}T}{V}},}
1352:the mass and the size of a solid body are
1170:number. To distinguish them, the variable
630:
337:Variables are often used for representing
3686:(2nd ed.). London: MacMillan and Co.
3387:Quantities and units, Part 2: Mathematics
2896:
2747:
2727:
2701:
2678:
2658:
2638:
2612:
2580:
2560:
2522:
2483:
2468:
2418:
2408:
2396:
2366:
2345:
2339:
2316:
2292:
2272:
2239:
2229:
2212:
2192:
2170:
2169:
2157:
2153:
2152:
2144:
2143:
2131:
2127:
2126:
2117:
2097:
2077:
2051:
2031:
1998:
1988:
1959:
1939:
1916:
1890:
1870:
1845:
1839:
1812:
1794:
1683:
1676:
1664:
1653:
1647:
1585:
1564:
1098:
1082:
1073:
952:
931:
916:
876:
855:
846:
698:
675:
670:
645:
631:
580:
309:
239:
207:
162:
153:
102:
27:Symbol representing a mathematical object
3614:
3545:
1290:Other specific names for variables are:
3510:The Stanford Encyclopedia of Philosophy
3255:An Introduction to Algebraic Structures
3175:
74:the object, and that the object is the
30:For variables in computer science, see
3561:
3290:Mathematics Education Research Journal
1954:as a function of the other variables,
1380:is a kind of variable that is used in
1201:In the context of functions, the term
3859:. New York: Oxford University Press.
3743:. Britannica Educational Publishing.
3598:
3529:
1487:is a variable that is not dependent.
1180:, and the other variables are called
1131:{\displaystyle ax^{3}+bx^{2}+cx+d=0,}
776:), a word or abbreviation of a word (
276:, but has also been used to denote a
7:
3856:Descartes: A Very Short Introduction
3804:. Osnabrück Secolo. pp. 46–65.
2963:for the length and width of a figure
2361:as a variable to obtain a function
1618:{\displaystyle f(x)=x^{2}+\sin(x+4)}
1406:Dependent and independent variables
1400:Dependent and independent variables
956:{\textstyle x\mapsto ax^{2}+bx+c\,}
514:referred almost exclusively to the
3718:A First Course in Abstract Algebra
1369:Free variables and bound variables
1356:for the study of its movement. In
621:
603:
585:
25:
5661:
1215:is the variable of the function
401:At the end of the 16th century,
270:generally represents the number
3881:. In Weisstein, Eric W. (ed.).
3737:Hosch, William L., ed. (2010).
3299:Springer Science+Business Media
1632:is a variable standing for the
2907:
2901:
2767:
2749:
2594:
2582:
2510:{\displaystyle y=ax^{2}+bx+c,}
2402:
2371:
2223:
2217:
2166:
1982:
1964:
1612:
1600:
1575:
1569:
1240:is a function of the variable
921:
699:
695:
689:
676:
672:
646:
632:
627:
618:
615:
600:
597:
582:
195:{\displaystyle ax^{2}+bx+c=0,}
119:
113:
85:, in which case its value can
1:
5622:History of mathematical logic
1430:. In mathematical terms, the
841:is conventionally written as
5547:Primitive recursive function
3369:member of a class of objects
3225:Beckenbach, Edwin F (1982).
3183:Sobolev, S.K. (originator).
899:are parameters (also called
452:independently developed the
3231:(5th ed.). Wadsworth.
3190:Encyclopedia of Mathematics
2788:Conventional variable names
1827:{\displaystyle PV=Nk_{B}T.}
1479:Therefore, in a formula, a
1302:which has to be solved for.
1207:function of a real variable
1058:Specific kinds of variables
967:and the constant status of
326:has been used to denote an
32:Variable (computer science)
5729:
4611:Schröder–Bernstein theorem
4338:Monadic predicate calculus
3997:Foundations of mathematics
3824:"Variables Explained Away"
3714:Fraleigh, John B. (1989).
3691:Foerster, Paul A. (2006).
3349:an expression that can be
3336:Collins English Dictionary
2452:
1785:Example: the ideal gas law
1548:If one defines a function
1440:represents the value of a
1403:
1002:are conventionally called
880:{\textstyle ax^{2}+bx+c\,}
802:). Under the influence of
29:
5657:
5644:Philosophy of mathematics
5593:Automated theorem proving
4764:
4718:Von Neumann–Bernays–Gödel
4359:
3447:10.1007/978-3-540-85108-0
3312:Oxford English Dictionary
2782:moduli space of parabolas
1456:and the function mapping
1265:and are therefore called
532:. To solve this problem,
450:Gottfried Wilhelm Leibniz
384:In ancient works such as
36:Variable (disambiguation)
3680:Edwards, Joseph (1892).
3428:van Dalen, Dirk (2008).
3316:variable (n.), sense 1.a
2311:of the earlier function
2207:. This gives a function
1865:. One of the variables,
752:and less often from the
549:varies and tends toward
5688:Variables (mathematics)
5294:Self-verifying theories
5115:Tarski's axiomatization
4066:Tarski's undefinability
4061:incompleteness theorems
3904:. Infobase Publishing.
3802:Variable-Free Semantics
3475:Fitch, Frederic Brenton
3415:
3353:any of a set of values
3252:Landin, Joseph (1989).
3031:correlation coefficient
2808:(sometimes extended to
2773:{\displaystyle (a,b,c)}
2696:Then instead regarding
2548:{\displaystyle a,b,c,x}
1556:to the real numbers by
1414:and its application to
1396:) is the same for all.
1272:constant of integration
1166:is understood to be an
837:For example, a general
787:mathematical expression
444:Starting in the 1660s,
280:. Similarly the letter
128:{\displaystyle y=f(x).}
5703:Elementary mathematics
5668:Mathematics portal
5279:Proof of impossibility
4927:propositional variable
4237:Propositional calculus
3416:Stover & Weisstein
3164:Propositional variable
3078:normal random variable
2914:
2774:
2736:
2716:
2687:
2667:
2647:
2627:
2601:
2569:
2549:
2511:
2440:
2355:
2325:
2301:
2281:
2261:
2201:
2178:
2106:
2086:
2066:
2040:
2020:
1948:
1925:
1905:
1879:
1855:
1828:
1705:
1669:
1619:
1132:
957:
881:
716:
660:
454:infinitesimal calculus
318:
248:
228:
196:
145:; for example, in the
129:
91:domain of the function
83:argument of a function
34:. For other uses, see
5537:Kolmogorov complexity
5490:Computably enumerable
5390:Model complete theory
5182:Principia Mathematica
4242:Propositional formula
4071:Banach–Tarski paradox
3874:Stover, Christopher;
3775:10.1093/bjps/V.18.134
3655:mathworld.wolfram.com
3483:North-Holland Pub. Co
3430:"Logic and Structure"
3397:on September 15, 2019
3185:"Individual variable"
3076:, or in statistics a
3063:or the corresponding
3057:Cartesian coordinates
2915:
2870:-th coefficient of a
2775:
2737:
2717:
2688:
2668:
2648:
2628:
2602:
2600:{\displaystyle (x,y)}
2570:
2550:
2512:
2441:
2356:
2354:{\displaystyle k_{B}}
2326:
2302:
2282:
2262:
2202:
2179:
2107:
2087:
2067:
2041:
2021:
1949:
1926:
1906:
1880:
1856:
1854:{\displaystyle k_{B}}
1829:
1706:
1649:
1620:
1384:and its applications.
1133:
958:
882:
765:), another variable (
717:
661:
395:Brāhmasphuṭasiddhānta
319:
249:
229:
227:{\displaystyle a,b,c}
197:
130:
58:, "changeable") is a
5485:Church–Turing thesis
5472:Computability theory
4681:continuum hypothesis
4199:Square of opposition
4057:Gödel's completeness
3898:Tabak, John (2014).
3441:(4th ed.): 57.
3297:: 19–38 – via
2913:{\displaystyle f(x)}
2895:
2816:) for parameters or
2746:
2726:
2700:
2677:
2657:
2637:
2611:
2579:
2559:
2521:
2467:
2365:
2338:
2315:
2291:
2271:
2211:
2191:
2116:
2096:
2076:
2050:
2030:
1958:
1938:
1915:
1889:
1869:
1838:
1793:
1646:
1563:
1533:independent variable
1485:independent variable
1298:is a variable in an
1285:polynomial functions
1072:
915:
845:
669:
579:
308:
238:
234:are parameters, and
206:
152:
101:
5639:Mathematical object
5530:P versus NP problem
5495:Computable function
5289:Reverse mathematics
5215:Logical consequence
5092:primitive recursive
5087:elementary function
4860:Free/bound variable
4713:Tarski–Grothendieck
4232:Logical connectives
4162:Logical equivalence
4012:Logical consequence
3649:Weisstein, Eric W.
3576:Scientific American
3154:Observable variable
2715:{\displaystyle a,b}
2626:{\displaystyle a,b}
2309:partial application
2065:{\displaystyle V,N}
1904:{\displaystyle P,V}
1771:polynomial function
1722:(it is also called
1529:dependent variables
1329:formal power series
1317:formal power series
987:, it is called the
903:, because they are
730:mathematical object
530:continuous function
439:Scientific American
345:, their arguments,
5713:Mathematical logic
5437:Transfer principle
5400:Semantics of logic
5385:Categorical theory
5361:Non-standard model
4875:Logical connective
4002:Information theory
3951:Mathematical logic
3885:. Wolfram Research
3876:Weisstein, Eric W.
3791:Peregrin, Jaroslav
3380:"ISO 80000-2:2019"
3260:Dover Publications
3128:standard deviation
3061:Euclidean geometry
2910:
2770:
2732:
2712:
2683:
2663:
2643:
2623:
2597:
2565:
2545:
2507:
2436:
2351:
2321:
2297:
2277:
2257:
2197:
2174:
2102:
2082:
2062:
2036:
2016:
1944:
1921:
1901:
1875:
1863:Boltzmann constant
1851:
1824:
1701:
1615:
1481:dependent variable
1382:probability theory
1263:constant functions
1128:
996:naming conventions
953:
905:constant functions
877:
839:quadratic function
736:(e.g., the set of
712:
656:
366:mathematical logic
314:
302:. Even the symbol
300:degree polynomials
244:
224:
192:
147:quadratic equation
125:
5675:
5674:
5607:Abstract category
5410:Theories of truth
5220:Rule of inference
5210:Natural deduction
5191:
5190:
4736:
4735:
4441:Cartesian product
4346:
4345:
4252:Many-valued logic
4227:Boolean functions
4110:Russell's paradox
4085:diagonal argument
3982:First-order logic
3911:978-0-8160-6875-3
3883:Wolfram MathWorld
3866:978-0-19-285409-4
3820:Quine, Willard V.
3811:978-3-929979-53-4
3750:978-1-61530-219-2
3729:978-0-201-52821-3
3706:978-0-13-165711-3
3506:"Classical Logic"
3456:978-3-540-20879-2
3159:Physical constant
2946:or indices in an
2735:{\displaystyle c}
2686:{\displaystyle y}
2666:{\displaystyle x}
2646:{\displaystyle c}
2568:{\displaystyle y}
2431:
2324:{\displaystyle P}
2300:{\displaystyle V}
2280:{\displaystyle N}
2252:
2200:{\displaystyle T}
2105:{\displaystyle P}
2085:{\displaystyle T}
2039:{\displaystyle P}
2011:
1947:{\displaystyle P}
1924:{\displaystyle T}
1878:{\displaystyle N}
1733:In the theory of
1699:
1269:. For example, a
1038:for the names of
1016:physical quantity
462:variable quantity
386:Euclid's Elements
317:{\displaystyle 1}
247:{\displaystyle x}
78:of the variable.
16:(Redirected from
5720:
5666:
5665:
5617:History of logic
5612:Category of sets
5505:Decision problem
5284:Ordinal analysis
5225:Sequent calculus
5123:Boolean algebras
5063:
5062:
5037:
5008:logical/constant
4762:
4748:
4671:Zermelo–Fraenkel
4422:Set operations:
4357:
4294:
4125:
4105:Löwenheim–Skolem
3992:Formal semantics
3944:
3937:
3930:
3921:
3915:
3894:
3892:
3890:
3870:
3846:
3828:
3815:
3799:
3786:
3754:
3733:
3721:
3710:
3698:
3687:
3666:
3665:
3663:
3661:
3646:
3640:
3637:
3631:
3628:
3622:
3612:
3606:
3596:
3590:
3587:
3581:
3580:
3571:
3565:
3559:
3553:
3543:
3537:
3527:
3521:
3520:
3519:
3517:
3501:
3495:
3494:
3467:
3461:
3460:
3434:
3425:
3419:
3413:
3407:
3406:
3404:
3402:
3393:. Archived from
3384:
3376:
3370:
3365:representing an
3333:
3327:
3318:,” March 2024. "
3309:
3303:
3302:
3280:
3274:
3273:
3249:
3243:
3242:
3222:
3216:
3215:
3213:
3211:
3180:
2919:
2917:
2916:
2911:
2779:
2777:
2776:
2771:
2741:
2739:
2738:
2733:
2721:
2719:
2718:
2713:
2692:
2690:
2689:
2684:
2672:
2670:
2669:
2664:
2652:
2650:
2649:
2644:
2632:
2630:
2629:
2624:
2606:
2604:
2603:
2598:
2574:
2572:
2571:
2566:
2554:
2552:
2551:
2546:
2516:
2514:
2513:
2508:
2488:
2487:
2445:
2443:
2442:
2437:
2432:
2427:
2423:
2422:
2409:
2401:
2400:
2360:
2358:
2357:
2352:
2350:
2349:
2330:
2328:
2327:
2322:
2306:
2304:
2303:
2298:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2253:
2248:
2244:
2243:
2230:
2206:
2204:
2203:
2198:
2183:
2181:
2180:
2175:
2173:
2165:
2164:
2156:
2147:
2139:
2138:
2130:
2111:
2109:
2108:
2103:
2091:
2089:
2088:
2083:
2071:
2069:
2068:
2063:
2045:
2043:
2042:
2037:
2025:
2023:
2022:
2017:
2012:
2007:
2003:
2002:
1989:
1953:
1951:
1950:
1945:
1930:
1928:
1927:
1922:
1910:
1908:
1907:
1902:
1884:
1882:
1881:
1876:
1860:
1858:
1857:
1852:
1850:
1849:
1833:
1831:
1830:
1825:
1817:
1816:
1710:
1708:
1707:
1702:
1700:
1695:
1688:
1687:
1677:
1668:
1663:
1639:In the identity
1624:
1622:
1621:
1616:
1590:
1589:
1539:
1526:
1520:
1514:
1508:
1467:
1461:
1455:
1449:
1439:
1429:
1423:
1358:computer science
1260:
1251:
1245:
1239:
1233:
1214:
1175:
1165:
1158:
1137:
1135:
1134:
1129:
1103:
1102:
1087:
1086:
1040:random variables
1000:coordinate space
962:
960:
959:
954:
936:
935:
886:
884:
883:
878:
860:
859:
804:computer science
801:
784:
775:
764:
721:
719:
718:
713:
702:
679:
665:
663:
662:
657:
649:
635:
571:
565:
554:
548:
534:Karl Weierstrass
509:
503:
494:
488:
425:, and knowns by
330:of an arbitrary
328:identity element
325:
323:
321:
320:
315:
296:quartic function
284:
275:
268:
254:is the unknown.
253:
251:
250:
245:
233:
231:
230:
225:
201:
199:
198:
193:
167:
166:
134:
132:
131:
126:
96:
21:
5728:
5727:
5723:
5722:
5721:
5719:
5718:
5717:
5678:
5677:
5676:
5671:
5660:
5653:
5598:Category theory
5588:Algebraic logic
5571:
5542:Lambda calculus
5480:Church encoding
5466:
5442:Truth predicate
5298:
5264:Complete theory
5187:
5056:
5052:
5048:
5043:
5035:
4755: and
4751:
4746:
4732:
4708:New Foundations
4676:axiom of choice
4659:
4621:Gödel numbering
4561: and
4553:
4457:
4342:
4292:
4273:
4222:Boolean algebra
4208:
4172:Equiconsistency
4137:Classical logic
4114:
4095:Halting problem
4083: and
4059: and
4047: and
4046:
4041:Theorems (
4036:
3953:
3948:
3918:
3912:
3897:
3888:
3886:
3873:
3867:
3849:
3826:
3818:
3812:
3797:
3789:
3757:
3751:
3736:
3730:
3713:
3707:
3690:
3679:
3675:
3670:
3669:
3659:
3657:
3648:
3647:
3643:
3638:
3634:
3629:
3625:
3613:
3609:
3597:
3593:
3588:
3584:
3573:
3572:
3568:
3560:
3556:
3544:
3540:
3528:
3524:
3515:
3513:
3503:
3502:
3498:
3469:
3468:
3464:
3457:
3438:Springer-Verlag
3432:
3427:
3426:
3422:
3414:
3410:
3400:
3398:
3382:
3378:
3377:
3373:
3334:
3330:
3310:
3306:
3282:
3281:
3277:
3270:
3262:. p. 204.
3251:
3250:
3246:
3239:
3228:College algebra
3224:
3223:
3219:
3209:
3207:
3205:
3182:
3181:
3177:
3172:
3149:Lambda calculus
3145:
2893:
2892:
2856:
2849:
2841:
2834:
2827:
2790:
2744:
2743:
2724:
2723:
2698:
2697:
2693:are variables.
2675:
2674:
2655:
2654:
2635:
2634:
2609:
2608:
2577:
2576:
2557:
2556:
2519:
2518:
2479:
2465:
2464:
2457:
2451:
2414:
2410:
2392:
2363:
2362:
2341:
2336:
2335:
2313:
2312:
2289:
2288:
2269:
2268:
2235:
2231:
2209:
2208:
2189:
2188:
2151:
2125:
2114:
2113:
2094:
2093:
2074:
2073:
2048:
2047:
2028:
2027:
1994:
1990:
1956:
1955:
1936:
1935:
1913:
1912:
1887:
1886:
1867:
1866:
1841:
1836:
1835:
1808:
1791:
1790:
1787:
1679:
1678:
1644:
1643:
1581:
1561:
1560:
1546:
1535:
1522:
1516:
1510:
1491:
1463:
1457:
1451:
1445:
1435:
1425:
1419:
1408:
1402:
1377:random variable
1327:or the ring of
1325:polynomial ring
1256:
1247:
1241:
1235:
1216:
1210:
1190:, or sometimes
1171:
1160:
1142:
1094:
1078:
1070:
1069:
1060:
927:
913:
912:
851:
843:
842:
800:
790:
783:
777:
774:
766:
763:
757:
746:
667:
666:
577:
576:
567:
556:
550:
544:
505:
499:
490:
489:for a function
475:
460:variation of a
382:
306:
305:
303:
282:
271:
266:
236:
235:
204:
203:
158:
150:
149:
99:
98:
94:
39:
28:
23:
22:
18:Variable (math)
15:
12:
11:
5:
5726:
5724:
5716:
5715:
5710:
5708:Syntax (logic)
5705:
5700:
5695:
5690:
5680:
5679:
5673:
5672:
5658:
5655:
5654:
5652:
5651:
5646:
5641:
5636:
5631:
5630:
5629:
5619:
5614:
5609:
5600:
5595:
5590:
5585:
5583:Abstract logic
5579:
5577:
5573:
5572:
5570:
5569:
5564:
5562:Turing machine
5559:
5554:
5549:
5544:
5539:
5534:
5533:
5532:
5527:
5522:
5517:
5512:
5502:
5500:Computable set
5497:
5492:
5487:
5482:
5476:
5474:
5468:
5467:
5465:
5464:
5459:
5454:
5449:
5444:
5439:
5434:
5429:
5428:
5427:
5422:
5417:
5407:
5402:
5397:
5395:Satisfiability
5392:
5387:
5382:
5381:
5380:
5370:
5369:
5368:
5358:
5357:
5356:
5351:
5346:
5341:
5336:
5326:
5325:
5324:
5319:
5312:Interpretation
5308:
5306:
5300:
5299:
5297:
5296:
5291:
5286:
5281:
5276:
5266:
5261:
5260:
5259:
5258:
5257:
5247:
5242:
5232:
5227:
5222:
5217:
5212:
5207:
5201:
5199:
5193:
5192:
5189:
5188:
5186:
5185:
5177:
5176:
5175:
5174:
5169:
5168:
5167:
5162:
5157:
5137:
5136:
5135:
5133:minimal axioms
5130:
5119:
5118:
5117:
5106:
5105:
5104:
5099:
5094:
5089:
5084:
5079:
5066:
5064:
5045:
5044:
5042:
5041:
5040:
5039:
5027:
5022:
5021:
5020:
5015:
5010:
5005:
4995:
4990:
4985:
4980:
4979:
4978:
4973:
4963:
4962:
4961:
4956:
4951:
4946:
4936:
4931:
4930:
4929:
4924:
4919:
4909:
4908:
4907:
4902:
4897:
4892:
4887:
4882:
4872:
4867:
4862:
4857:
4856:
4855:
4850:
4845:
4840:
4830:
4825:
4823:Formation rule
4820:
4815:
4814:
4813:
4808:
4798:
4797:
4796:
4786:
4781:
4776:
4771:
4765:
4759:
4742:Formal systems
4738:
4737:
4734:
4733:
4731:
4730:
4725:
4720:
4715:
4710:
4705:
4700:
4695:
4690:
4685:
4684:
4683:
4678:
4667:
4665:
4661:
4660:
4658:
4657:
4656:
4655:
4645:
4640:
4639:
4638:
4631:Large cardinal
4628:
4623:
4618:
4613:
4608:
4594:
4593:
4592:
4587:
4582:
4567:
4565:
4555:
4554:
4552:
4551:
4550:
4549:
4544:
4539:
4529:
4524:
4519:
4514:
4509:
4504:
4499:
4494:
4489:
4484:
4479:
4474:
4468:
4466:
4459:
4458:
4456:
4455:
4454:
4453:
4448:
4443:
4438:
4433:
4428:
4420:
4419:
4418:
4413:
4403:
4398:
4396:Extensionality
4393:
4391:Ordinal number
4388:
4378:
4373:
4372:
4371:
4360:
4354:
4348:
4347:
4344:
4343:
4341:
4340:
4335:
4330:
4325:
4320:
4315:
4310:
4309:
4308:
4298:
4297:
4296:
4283:
4281:
4275:
4274:
4272:
4271:
4270:
4269:
4264:
4259:
4249:
4244:
4239:
4234:
4229:
4224:
4218:
4216:
4210:
4209:
4207:
4206:
4201:
4196:
4191:
4186:
4181:
4176:
4175:
4174:
4164:
4159:
4154:
4149:
4144:
4139:
4133:
4131:
4122:
4116:
4115:
4113:
4112:
4107:
4102:
4097:
4092:
4087:
4075:Cantor's
4073:
4068:
4063:
4053:
4051:
4038:
4037:
4035:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3973:
3972:
3961:
3959:
3955:
3954:
3949:
3947:
3946:
3939:
3932:
3924:
3917:
3916:
3910:
3895:
3871:
3865:
3847:
3816:
3810:
3787:
3755:
3749:
3734:
3728:
3711:
3705:
3688:
3676:
3674:
3671:
3668:
3667:
3641:
3639:Edwards Art. 6
3632:
3630:Edwards Art. 5
3623:
3607:
3591:
3589:Edwards Art. 4
3582:
3566:
3554:
3538:
3522:
3496:
3462:
3455:
3420:
3408:
3371:
3357:a symbol, esp
3328:
3304:
3275:
3268:
3244:
3237:
3217:
3203:
3174:
3173:
3171:
3168:
3167:
3166:
3161:
3156:
3151:
3144:
3141:
3140:
3139:
3130:
3124:
3115:
3105:
3103:angle measures
3080:
3074:complex number
3067:
3059:of a point in
3055:for the three
3042:
3033:
3016:
3003:
2990:
2973:
2964:
2954:
2948:indexed family
2942:) for varying
2921:
2909:
2906:
2903:
2900:
2874:
2862:-th term of a
2854:
2847:
2843:
2839:
2832:
2825:
2820:
2789:
2786:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2731:
2711:
2708:
2705:
2682:
2662:
2642:
2622:
2619:
2616:
2596:
2593:
2590:
2587:
2584:
2564:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2506:
2503:
2500:
2497:
2494:
2491:
2486:
2482:
2478:
2475:
2472:
2450:
2447:
2435:
2430:
2426:
2421:
2417:
2413:
2407:
2404:
2399:
2395:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2348:
2344:
2320:
2296:
2276:
2256:
2251:
2247:
2242:
2238:
2234:
2228:
2225:
2222:
2219:
2216:
2196:
2172:
2168:
2163:
2160:
2155:
2150:
2146:
2142:
2137:
2134:
2129:
2124:
2121:
2112:is a function
2101:
2081:
2061:
2058:
2055:
2035:
2015:
2010:
2006:
2001:
1997:
1993:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1943:
1920:
1900:
1897:
1894:
1874:
1848:
1844:
1823:
1820:
1815:
1811:
1807:
1804:
1801:
1798:
1786:
1783:
1712:
1711:
1698:
1694:
1691:
1686:
1682:
1675:
1672:
1667:
1662:
1659:
1656:
1652:
1626:
1625:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1588:
1584:
1580:
1577:
1574:
1571:
1568:
1545:
1542:
1404:Main article:
1401:
1398:
1386:
1385:
1372:
1365:
1340:
1303:
1277:antiderivative
1139:
1138:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1101:
1097:
1093:
1090:
1085:
1081:
1077:
1064:cubic equation
1059:
1056:
951:
948:
945:
942:
939:
934:
930:
926:
923:
920:
875:
872:
869:
866:
863:
858:
854:
850:
808:René Descartes
794:
781:
770:
761:
750:Latin alphabet
745:
742:
723:
722:
711:
708:
705:
701:
697:
694:
691:
688:
685:
682:
678:
674:
655:
652:
648:
644:
641:
638:
634:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
527:differentiable
522:of functions.
504:and its value
472:Leonhard Euler
411:René Descartes
403:François Viète
381:
378:
313:
288:Euler's number
286:often denotes
243:
223:
220:
217:
214:
211:
202:the variables
191:
188:
185:
182:
179:
176:
173:
170:
165:
161:
157:
124:
121:
118:
115:
112:
109:
106:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5725:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5685:
5683:
5670:
5669:
5664:
5656:
5650:
5647:
5645:
5642:
5640:
5637:
5635:
5632:
5628:
5625:
5624:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5581:
5580:
5578:
5574:
5568:
5565:
5563:
5560:
5558:
5557:Recursive set
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5531:
5528:
5526:
5523:
5521:
5518:
5516:
5513:
5511:
5508:
5507:
5506:
5503:
5501:
5498:
5496:
5493:
5491:
5488:
5486:
5483:
5481:
5478:
5477:
5475:
5473:
5469:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5426:
5423:
5421:
5418:
5416:
5413:
5412:
5411:
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5391:
5388:
5386:
5383:
5379:
5376:
5375:
5374:
5371:
5367:
5366:of arithmetic
5364:
5363:
5362:
5359:
5355:
5352:
5350:
5347:
5345:
5342:
5340:
5337:
5335:
5332:
5331:
5330:
5327:
5323:
5320:
5318:
5315:
5314:
5313:
5310:
5309:
5307:
5305:
5301:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5274:
5273:from ZFC
5270:
5267:
5265:
5262:
5256:
5253:
5252:
5251:
5248:
5246:
5243:
5241:
5238:
5237:
5236:
5233:
5231:
5228:
5226:
5223:
5221:
5218:
5216:
5213:
5211:
5208:
5206:
5203:
5202:
5200:
5198:
5194:
5184:
5183:
5179:
5178:
5173:
5172:non-Euclidean
5170:
5166:
5163:
5161:
5158:
5156:
5155:
5151:
5150:
5148:
5145:
5144:
5142:
5138:
5134:
5131:
5129:
5126:
5125:
5124:
5120:
5116:
5113:
5112:
5111:
5107:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5074:
5072:
5068:
5067:
5065:
5060:
5054:
5049:Example
5046:
5038:
5033:
5032:
5031:
5028:
5026:
5023:
5019:
5016:
5014:
5011:
5009:
5006:
5004:
5001:
5000:
4999:
4996:
4994:
4991:
4989:
4986:
4984:
4981:
4977:
4974:
4972:
4969:
4968:
4967:
4964:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4941:
4940:
4937:
4935:
4932:
4928:
4925:
4923:
4920:
4918:
4915:
4914:
4913:
4910:
4906:
4903:
4901:
4898:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4877:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4854:
4851:
4849:
4846:
4844:
4841:
4839:
4836:
4835:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4816:
4812:
4809:
4807:
4806:by definition
4804:
4803:
4802:
4799:
4795:
4792:
4791:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4766:
4763:
4760:
4758:
4754:
4749:
4743:
4739:
4729:
4726:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4693:Kripke–Platek
4691:
4689:
4686:
4682:
4679:
4677:
4674:
4673:
4672:
4669:
4668:
4666:
4662:
4654:
4651:
4650:
4649:
4646:
4644:
4641:
4637:
4634:
4633:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4606:
4602:
4598:
4595:
4591:
4588:
4586:
4583:
4581:
4578:
4577:
4576:
4572:
4569:
4568:
4566:
4564:
4560:
4556:
4548:
4545:
4543:
4540:
4538:
4537:constructible
4535:
4534:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4469:
4467:
4465:
4460:
4452:
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4423:
4421:
4417:
4414:
4412:
4409:
4408:
4407:
4404:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4383:
4379:
4377:
4374:
4370:
4367:
4366:
4365:
4362:
4361:
4358:
4355:
4353:
4349:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4307:
4304:
4303:
4302:
4299:
4295:
4290:
4289:
4288:
4285:
4284:
4282:
4280:
4276:
4268:
4265:
4263:
4260:
4258:
4255:
4254:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4219:
4217:
4215:
4214:Propositional
4211:
4205:
4202:
4200:
4197:
4195:
4192:
4190:
4187:
4185:
4182:
4180:
4177:
4173:
4170:
4169:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4148:
4145:
4143:
4142:Logical truth
4140:
4138:
4135:
4134:
4132:
4130:
4126:
4123:
4121:
4117:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4086:
4082:
4078:
4074:
4072:
4069:
4067:
4064:
4062:
4058:
4055:
4054:
4052:
4050:
4044:
4039:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3971:
3968:
3967:
3966:
3963:
3962:
3960:
3956:
3952:
3945:
3940:
3938:
3933:
3931:
3926:
3925:
3922:
3913:
3907:
3903:
3902:
3896:
3884:
3880:
3877:
3872:
3868:
3862:
3858:
3857:
3852:
3848:
3844:
3840:
3836:
3832:
3825:
3821:
3817:
3813:
3807:
3803:
3796:
3792:
3788:
3784:
3780:
3776:
3772:
3768:
3764:
3760:
3756:
3752:
3746:
3742:
3741:
3735:
3731:
3725:
3720:
3719:
3712:
3708:
3702:
3697:
3696:
3689:
3685:
3684:
3678:
3677:
3672:
3656:
3652:
3645:
3642:
3636:
3633:
3627:
3624:
3620:
3616:
3615:Foerster 2006
3611:
3608:
3604:
3600:
3595:
3592:
3586:
3583:
3578:
3577:
3570:
3567:
3564:, p. 19.
3563:
3558:
3555:
3551:
3547:
3546:Fraleigh 1989
3542:
3539:
3535:
3531:
3526:
3523:
3511:
3507:
3500:
3497:
3492:
3488:
3484:
3481:. Amsterdam:
3480:
3476:
3472:
3466:
3463:
3458:
3452:
3448:
3444:
3440:
3439:
3431:
3424:
3421:
3417:
3412:
3409:
3401:September 15,
3396:
3392:
3388:
3381:
3375:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3345:
3341:
3337:
3332:
3329:
3325:
3321:
3317:
3313:
3308:
3305:
3300:
3296:
3292:
3291:
3286:
3279:
3276:
3271:
3269:0-486-65940-2
3265:
3261:
3257:
3256:
3248:
3245:
3240:
3238:0-534-01007-5
3234:
3230:
3229:
3221:
3218:
3206:
3200:
3196:
3192:
3191:
3186:
3179:
3176:
3169:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3146:
3142:
3138:
3134:
3131:
3129:
3125:
3123:
3119:
3116:
3113:
3109:
3106:
3104:
3100:
3096:
3092:
3088:
3084:
3081:
3079:
3075:
3071:
3068:
3066:
3062:
3058:
3054:
3050:
3046:
3043:
3041:
3037:
3034:
3032:
3028:
3024:
3020:
3017:
3015:
3011:
3007:
3004:
3002:
2998:
2994:
2991:
2989:
2985:
2981:
2977:
2974:
2972:
2968:
2965:
2962:
2958:
2955:
2953:
2949:
2945:
2941:
2937:
2933:
2929:
2925:
2922:
2904:
2898:
2890:
2886:
2882:
2878:
2875:
2873:
2869:
2865:
2861:
2857:
2850:
2844:
2838:
2831:
2824:
2821:
2819:
2815:
2811:
2807:
2803:
2799:
2795:
2792:
2791:
2787:
2785:
2783:
2764:
2761:
2758:
2755:
2752:
2729:
2709:
2706:
2703:
2694:
2680:
2660:
2640:
2620:
2617:
2614:
2591:
2588:
2585:
2562:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2504:
2501:
2498:
2495:
2492:
2489:
2484:
2480:
2476:
2473:
2470:
2462:
2456:
2455:moduli spaces
2449:Moduli spaces
2448:
2446:
2433:
2428:
2424:
2419:
2415:
2411:
2405:
2397:
2393:
2389:
2386:
2383:
2380:
2377:
2374:
2368:
2346:
2342:
2332:
2318:
2310:
2294:
2274:
2254:
2249:
2245:
2240:
2236:
2232:
2226:
2220:
2214:
2194:
2185:
2161:
2158:
2148:
2140:
2135:
2132:
2122:
2119:
2099:
2079:
2059:
2056:
2053:
2033:
2013:
2008:
2004:
1999:
1995:
1991:
1985:
1979:
1976:
1973:
1970:
1967:
1961:
1941:
1932:
1918:
1898:
1895:
1892:
1872:
1864:
1846:
1842:
1821:
1818:
1813:
1809:
1805:
1802:
1799:
1796:
1784:
1782:
1780:
1776:
1772:
1768:
1764:
1760:
1756:
1752:
1748:
1744:
1740:
1736:
1731:
1729:
1725:
1721:
1717:
1714:the variable
1696:
1692:
1689:
1684:
1680:
1673:
1670:
1665:
1660:
1657:
1654:
1650:
1642:
1641:
1640:
1637:
1635:
1631:
1609:
1606:
1603:
1597:
1594:
1591:
1586:
1582:
1578:
1572:
1566:
1559:
1558:
1557:
1555:
1551:
1543:
1541:
1538:
1534:
1530:
1525:
1519:
1513:
1506:
1502:
1498:
1494:
1488:
1486:
1482:
1477:
1475:
1471:
1466:
1460:
1454:
1448:
1443:
1438:
1433:
1428:
1422:
1417:
1413:
1407:
1399:
1397:
1395:
1391:
1383:
1379:
1378:
1373:
1371:
1370:
1366:
1363:
1359:
1355:
1351:
1347:
1346:
1341:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1309:
1308:indeterminate
1304:
1301:
1297:
1293:
1292:
1291:
1288:
1286:
1282:
1278:
1274:
1273:
1268:
1264:
1259:
1253:
1250:
1244:
1238:
1231:
1227:
1223:
1219:
1213:
1208:
1204:
1199:
1197:
1193:
1189:
1188:
1183:
1179:
1174:
1169:
1163:
1157:
1153:
1149:
1145:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1099:
1095:
1091:
1088:
1083:
1079:
1075:
1068:
1067:
1066:
1065:
1057:
1055:
1053:
1049:
1045:
1041:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
992:
990:
989:constant term
986:
982:
978:
974:
970:
966:
949:
946:
943:
940:
937:
932:
928:
924:
918:
910:
906:
902:
898:
894:
890:
873:
870:
867:
864:
861:
856:
852:
848:
840:
835:
833:
829:
825:
821:
817:
813:
809:
805:
798:
793:
788:
780:
773:
769:
760:
755:
751:
743:
741:
739:
735:
731:
726:
709:
706:
703:
692:
686:
683:
680:
653:
650:
642:
639:
636:
624:
612:
609:
606:
594:
591:
588:
575:
574:
573:
570:
566:tends toward
563:
559:
553:
547:
543:
539:
535:
531:
528:
523:
521:
517:
513:
508:
502:
498:
493:
486:
482:
478:
473:
469:
468:
463:
459:
458:infinitesimal
455:
451:
447:
442:
440:
436:
432:
428:
424:
420:
416:
412:
407:
404:
399:
397:
396:
391:
387:
379:
377:
375:
371:
367:
362:
360:
356:
352:
348:
344:
340:
335:
333:
329:
311:
301:
297:
293:
289:
285:
279:
274:
269:
264:
260:
255:
241:
221:
218:
215:
212:
209:
189:
186:
183:
180:
177:
174:
171:
168:
163:
159:
155:
148:
144:
140:
135:
122:
116:
110:
107:
104:
92:
88:
84:
79:
77:
73:
69:
65:
61:
57:
56:
52:
48:
44:
37:
33:
19:
5659:
5457:Ultraproduct
5304:Model theory
5269:Independence
5205:Formal proof
5197:Proof theory
5180:
5153:
5110:real numbers
5082:second-order
5017:
4993:Substitution
4870:Metalanguage
4811:conservative
4784:Axiom schema
4728:Constructive
4698:Morse–Kelley
4664:Set theories
4643:Aleph number
4636:inaccessible
4542:Grothendieck
4426:intersection
4313:Higher-order
4301:Second-order
4247:Truth tables
4204:Venn diagram
3987:Formal proof
3900:
3889:November 22,
3887:. Retrieved
3882:
3855:
3834:
3830:
3801:
3766:
3762:
3759:Menger, Karl
3739:
3717:
3694:
3682:
3673:Bibliography
3660:February 14,
3658:. Retrieved
3654:
3644:
3635:
3626:
3610:
3594:
3585:
3575:
3569:
3557:
3541:
3525:
3516:September 1,
3514:, retrieved
3509:
3499:
3478:
3471:Feys, Robert
3465:
3436:
3423:
3411:
3399:. Retrieved
3395:the original
3386:
3374:
3362:
3358:
3354:
3346:
3343:
3331:
3323:
3319:
3307:
3294:
3288:
3278:
3258:. New York:
3254:
3247:
3227:
3220:
3210:September 5,
3208:. Retrieved
3188:
3178:
3132:
3117:
3111:
3107:
3098:
3094:
3090:
3086:
3082:
3069:
3052:
3048:
3044:
3035:
3018:
3005:
2997:prime number
2992:
2979:
2975:
2970:
2966:
2960:
2956:
2952:unit vectors
2939:
2935:
2931:
2927:
2923:
2884:
2880:
2876:
2867:
2859:
2852:
2845:
2836:
2829:
2822:
2818:coefficients
2813:
2809:
2805:
2801:
2797:
2793:
2781:
2695:
2458:
2333:
2186:
1933:
1788:
1778:
1774:
1766:
1763:coefficients
1758:
1754:
1750:
1746:
1742:
1738:
1732:
1727:
1723:
1719:
1715:
1713:
1638:
1629:
1627:
1554:real numbers
1549:
1547:
1536:
1532:
1528:
1523:
1517:
1511:
1504:
1500:
1496:
1492:
1489:
1484:
1480:
1478:
1464:
1458:
1452:
1446:
1436:
1431:
1426:
1420:
1409:
1387:
1375:
1367:
1361:
1353:
1343:
1333:power series
1306:
1295:
1289:
1270:
1266:
1257:
1254:
1248:
1242:
1236:
1229:
1225:
1221:
1217:
1211:
1202:
1200:
1191:
1187:coefficients
1185:
1181:
1177:
1172:
1167:
1161:
1155:
1151:
1147:
1143:
1140:
1061:
1051:
1047:
1043:
1035:
1031:
1027:
1011:
1007:
1003:
993:
984:
980:
976:
972:
968:
964:
908:
896:
892:
888:
836:
831:
827:
823:
819:
815:
811:
796:
791:
778:
771:
767:
758:
747:
738:real numbers
727:
724:
568:
561:
557:
551:
545:
541:
524:
511:
506:
500:
496:
491:
484:
480:
476:
465:
461:
446:Isaac Newton
443:
434:
430:
426:
422:
418:
414:
408:
400:
393:
383:
369:
363:
336:
281:
265:
263:Greek letter
256:
138:
136:
86:
80:
71:
67:
53:
46:
40:
5567:Type theory
5515:undecidable
5447:Truth value
5334:equivalence
5013:non-logical
4626:Enumeration
4616:Isomorphism
4563:cardinality
4547:Von Neumann
4512:Ultrafilter
4477:Uncountable
4411:equivalence
4328:Quantifiers
4318:Fixed-point
4287:First-order
4167:Consistency
4152:Proposition
4129:Traditional
4100:Lindström's
4090:Compactness
4032:Type theory
3977:Cardinality
3851:Sorell, Tom
3562:Sorell 2000
3367:unspecified
3344:mathematics
3320:Mathematics
3010:prime power
3001:probability
2934:(sometimes
1761:are called
1735:polynomials
1474:temperature
1281:polynomials
1020:probability
390:Brahmagupta
298:and higher
292:coefficient
43:mathematics
5682:Categories
5378:elementary
5071:arithmetic
4939:Quantifier
4917:functional
4789:Expression
4507:Transitive
4451:identities
4436:complement
4369:hereditary
4352:Set theory
3879:"Variable"
3617:, p.
3601:, p.
3599:Hosch 2010
3548:, p.
3532:, p.
3530:Tabak 2014
3342:, (noun)
3204:1402006098
3170:References
3122:eigenvalue
2453:See also:
2267:where now
1521:depend on
1354:parameters
1313:polynomial
1182:parameters
1178:an unknown
1176:is called
1042:, keeping
1026:is to use
1024:statistics
374:quantified
349:and their
278:projection
68:represents
55:variabilis
5649:Supertask
5552:Recursion
5510:decidable
5344:saturated
5322:of models
5245:deductive
5240:axiomatic
5160:Hilbert's
5147:Euclidean
5128:canonical
5051:axiomatic
4983:Signature
4912:Predicate
4801:Extension
4723:Ackermann
4648:Operation
4527:Universal
4517:Recursive
4492:Singleton
4487:Inhabited
4472:Countable
4462:Types of
4446:power set
4416:partition
4333:Predicate
4279:Predicate
4194:Syllogism
4184:Soundness
4157:Inference
4147:Tautology
4049:paradoxes
3027:remainder
2889:functions
2167:→
2149:×
2141:×
1651:∑
1598:
1552:from the
1434:variable
1432:dependent
1362:parameter
1350:mechanics
1345:parameter
1337:functions
1192:constants
922:↦
907:), while
901:constants
707:ϵ
684:−
673:⇒
654:η
640:−
622:∀
607:η
604:∃
589:ϵ
586:∀
516:arguments
441:article.
409:In 1637,
343:functions
139:parameter
64:constants
5698:Calculus
5634:Logicism
5627:timeline
5603:Concrete
5462:Validity
5432:T-schema
5425:Kripke's
5420:Tarski's
5415:semantic
5405:Strength
5354:submodel
5349:spectrum
5317:function
5165:Tarski's
5154:Elements
5141:geometry
5097:Robinson
5018:variable
5003:function
4976:spectrum
4966:Sentence
4922:variable
4865:Language
4818:Relation
4779:Automata
4769:Alphabet
4753:language
4607:-jection
4585:codomain
4571:Function
4532:Universe
4502:Infinite
4406:Relation
4189:Validity
4179:Argument
4077:theorem,
3853:(2000).
3822:(1960).
3793:(2000).
3491:67030883
3477:(1969).
3351:assigned
3340:Variable
3314:, s.v. “
3195:Springer
3143:See also
3014:quotient
2988:equation
2944:integers
2864:sequence
2858:for the
2461:parabola
1749:, where
1634:argument
1544:Examples
1470:pressure
1442:function
1412:calculus
1390:semantic
1335:and the
1321:constant
1300:equation
1267:constant
1203:variable
1196:function
979:. Since
887:, where
744:Notation
542:variable
518:and the
512:variable
497:variable
467:function
370:variable
351:elements
339:matrices
259:constant
47:variable
5693:Algebra
5576:Related
5373:Diagram
5271: (
5250:Hilbert
5235:Systems
5230:Theorem
5108:of the
5053:systems
4833:Formula
4828:Grammar
4744: (
4688:General
4401:Forcing
4386:Element
4306:Monadic
4081:paradox
4022:Theorem
3958:General
3324:Physics
3120:for an
2891:(as in
2866:or the
1416:physics
1323:in the
1296:unknown
1261:define
1168:unknown
785:) or a
555:, then
380:History
361:, etc.
355:vectors
324:
304:
143:unknown
89:in the
72:denotes
5339:finite
5102:Skolem
5055:
5030:Theory
4998:Symbol
4988:String
4971:atomic
4848:ground
4843:closed
4838:atomic
4794:ground
4757:syntax
4653:binary
4580:domain
4497:Finite
4262:finite
4120:Logics
4079:
4027:Theory
3908:
3863:
3843:985250
3841:
3808:
3783:685170
3781:
3747:
3726:
3703:
3489:
3453:
3266:
3235:
3201:
3135:for a
3110:(with
3072:for a
3023:radius
3021:for a
3008:for a
2995:for a
2986:of an
2984:degree
2978:(with
2872:series
2517:where
1861:, the
1472:, the
1394:syntax
1010:, and
520:values
495:, its
433:, and
421:, and
376:over.
359:spaces
60:symbol
49:(from
5329:Model
5077:Peano
4934:Proof
4774:Arity
4703:Naive
4590:image
4522:Fuzzy
4482:Empty
4431:union
4376:Class
4017:Model
4007:Lemma
3965:Axiom
3839:JSTOR
3827:(PDF)
3798:(PDF)
3779:JSTOR
3651:"Sum"
3433:(PDF)
3383:(PDF)
3359:x, y,
3029:or a
3012:or a
2999:or a
2950:, or
2026:Then
1773:this
1724:index
1628:then
1527:(are
1462:onto
1315:or a
782:total
754:Greek
538:limit
332:field
76:value
51:Latin
5452:Type
5255:list
5059:list
5036:list
5025:Term
4959:rank
4853:open
4747:list
4559:Maps
4464:sets
4323:Free
4293:list
4043:list
3970:list
3906:ISBN
3891:2021
3861:ISBN
3806:ISBN
3745:ISBN
3724:ISBN
3701:ISBN
3662:2022
3518:2024
3487:LCCN
3451:ISBN
3403:2019
3322:and
3264:ISBN
3233:ISBN
3212:2024
3199:ISBN
3137:mean
3101:for
3065:axes
3040:time
3038:for
3025:, a
2959:and
2887:for
2722:and
2673:and
2633:and
2555:and
2287:and
2159:>
2133:>
2072:and
1911:and
1757:and
1515:and
1283:and
1234:", "
1209:", "
1022:and
975:and
895:and
704:<
651:<
610:>
592:>
448:and
368:, a
347:sets
294:for
87:vary
45:, a
5139:of
5121:of
5069:of
4601:Sur
4575:Map
4382:Ur-
4364:Set
3835:104
3771:doi
3550:276
3443:doi
3361:or
2938:or
2851:or
2463:,
1595:sin
1444:of
1410:In
1305:An
1294:An
1252:).
1184:or
799:+ 1
740:).
734:set
364:In
97:in
70:or
41:In
5684::
5525:NP
5149::
5143::
5073::
4750:),
4605:Bi
4597:In
3833:.
3829:.
3777:.
3765:.
3653:.
3619:18
3603:71
3534:40
3508:,
3485:.
3473:;
3449:.
3435:.
3389:.
3385:.
3363:z,
3355:b.
3347:a.
3338:.
3295:24
3293:.
3287:.
3197:.
3193:.
3187:.
3097:,
3093:,
3089:,
3085:,
3051:,
3047:,
2930:,
2926:,
2883:,
2879:,
2835:,
2828:,
2812:,
2804:,
2800:,
2796:,
2784:.
2331:.
2184:.
1753:,
1745:+
1743:bx
1741:+
1739:ax
1540:.
1503:,
1499:,
1374:A
1360:,
1342:A
1224:↦
1220::
1154:,
1150:,
1146:,
1050:,
1046:,
1034:,
1030:,
1006:,
991:.
971:,
891:,
830:,
826:,
818:,
814:,
479:=
429:,
417:,
357:,
353:,
341:,
5605:/
5520:P
5275:)
5061:)
5057:(
4954:∀
4949:!
4944:∃
4905:=
4900:↔
4895:→
4890:∧
4885:∨
4880:¬
4603:/
4599:/
4573:/
4384:)
4380:(
4267:∞
4257:3
4045:)
3943:e
3936:t
3929:v
3914:.
3893:.
3869:.
3845:.
3814:.
3785:.
3773::
3767:5
3753:.
3732:.
3709:.
3664:.
3621:.
3605:.
3552:.
3536:.
3493:.
3459:.
3445::
3418:.
3405:.
3301:.
3272:.
3241:.
3214:.
3133:μ
3118:λ
3112:δ
3108:ε
3099:φ
3095:θ
3091:γ
3087:β
3083:α
3070:z
3053:z
3049:y
3045:x
3036:t
3019:r
3006:q
2993:p
2980:m
2976:n
2971:p
2967:l
2961:w
2957:l
2940:h
2936:l
2932:k
2928:j
2924:i
2920:)
2908:)
2905:x
2902:(
2899:f
2885:h
2881:g
2877:f
2868:i
2860:i
2855:i
2853:u
2848:i
2846:a
2840:2
2837:a
2833:1
2830:a
2826:0
2823:a
2814:f
2810:e
2806:d
2802:c
2798:b
2794:a
2768:)
2765:c
2762:,
2759:b
2756:,
2753:a
2750:(
2730:c
2710:b
2707:,
2704:a
2681:y
2661:x
2641:c
2621:b
2618:,
2615:a
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2592:y
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2586:x
2583:(
2563:y
2543:x
2540:,
2537:c
2534:,
2531:b
2528:,
2525:a
2505:,
2502:c
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2493:b
2490:+
2485:2
2481:x
2477:a
2474:=
2471:y
2434:.
2429:V
2425:T
2420:B
2416:k
2412:N
2406:=
2403:)
2398:B
2394:k
2390:,
2387:T
2384:,
2381:N
2378:,
2375:V
2372:(
2369:P
2347:B
2343:k
2319:P
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2255:,
2250:V
2246:T
2241:B
2237:k
2233:N
2227:=
2224:)
2221:T
2218:(
2215:P
2195:T
2171:R
2162:0
2154:R
2145:N
2136:0
2128:R
2123::
2120:P
2100:P
2080:T
2060:N
2057:,
2054:V
2034:P
2014:.
2009:V
2005:T
2000:B
1996:k
1992:N
1986:=
1983:)
1980:T
1977:,
1974:N
1971:,
1968:V
1965:(
1962:P
1942:P
1919:T
1899:V
1896:,
1893:P
1873:N
1847:B
1843:k
1822:.
1819:T
1814:B
1810:k
1806:N
1803:=
1800:V
1797:P
1779:x
1775:x
1767:x
1759:c
1755:b
1751:a
1747:c
1728:n
1720:n
1716:i
1697:2
1693:n
1690:+
1685:2
1681:n
1674:=
1671:i
1666:n
1661:1
1658:=
1655:i
1630:x
1613:)
1610:4
1607:+
1604:x
1601:(
1592:+
1587:2
1583:x
1579:=
1576:)
1573:x
1570:(
1567:f
1550:f
1537:x
1524:x
1518:z
1512:y
1507:)
1505:z
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1493:f
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1228:(
1226:f
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1126:,
1123:0
1120:=
1117:d
1114:+
1111:x
1108:c
1105:+
1100:2
1096:x
1092:b
1089:+
1084:3
1080:x
1076:a
1052:z
1048:y
1044:x
1036:Z
1032:Y
1028:X
1012:z
1008:y
1004:x
985:x
981:c
977:c
973:b
969:a
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950:c
947:+
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938:+
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929:x
925:a
919:x
909:x
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893:b
889:a
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871:+
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862:+
857:2
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816:b
812:a
797:i
795:2
792:x
789:(
779:x
772:i
768:x
762:2
759:x
710:,
700:|
696:)
693:x
690:(
687:f
681:L
677:|
647:|
643:a
637:x
633:|
628:)
625:x
619:(
616:)
613:0
601:(
598:)
595:0
583:(
569:L
564:)
562:x
560:(
558:f
552:a
546:x
507:y
501:x
492:f
487:)
485:x
483:(
481:f
477:y
435:c
431:b
427:a
423:z
419:y
415:x
312:1
283:e
273:π
267:π
242:x
222:c
219:,
216:b
213:,
210:a
190:,
187:0
184:=
181:c
178:+
175:x
172:b
169:+
164:2
160:x
156:a
123:.
120:)
117:x
114:(
111:f
108:=
105:y
95:y
38:.
20:)
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