4383:
1210:, of the properties of the sets or functions ranged over. Henkin semantics is a kind of many-sorted first-order semantics, where there are a class of models of the axioms, instead of the semantics being fixed to just the standard model as in the standard semantics. A model in Henkin semantics will provide a set of sets or set of functions as the interpretation of higher-order domains, which may be a proper subset of all sets or functions of that sort. For his axiomatisation, Henkin proved that
178:
1673:, and some countably infinite collection of sets of internal numbers, whose members we will call "internal sets", such that the domain consisting of internal numbers and internal sets satisfies exactly the same first-order sentences as are satisfied by the domain of real numbers and sets of real numbers. In particular, it satisfies a sort of least-upper-bound axiom that says, in effect:
1723:, along with the fact that all the axioms of an Archimedean complete ordered field are expressible in second-order logic. This shows that the second-order theory of the real numbers cannot be reduced to a first-order theory, in the sense that the second-order theory of the real numbers has only one model but the corresponding first-order theory has many models.
2116:
591:(MSO) is a restriction of second-order logic in which only quantification over unary relations (i.e. sets) is allowed. Quantification over functions, owing to the equivalence to relations as described above, is thus also not allowed. The second-order logic without these restrictions is sometimes called
1635:
The deductive systems considered by
Shapiro (1991) and Henkin (1950) add to the augmented first-order deductive scheme both comprehension axioms and choice axioms. These axioms are sound for standard second-order semantics. They are sound for Henkin semantics restricted to Henkin models satisfying
1731:
holds and that has no model if the continuum hypothesis does not hold (cf. Shapiro 2000, p. 105). This theory consists of a finite theory characterizing the real numbers as a complete
Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality. This
1503:
that satisfies this property is isomorphic to the real number field. On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order
1649:
real numbers. Add a new binary predicate to the language: the membership relation. Then sentences that were second-order become first-order, with the formerly second-order quantifiers ranging over the second sort instead. This reduction can be attempted in a one-sorted theory by adding unary
1583:
Certain fragments of second-order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic that allow non-linear ordering of
640:
in a particular second-order language. These are restricted, however, in that all terms that they form must be either first-order terms (which can be substituted for a first-order variable) or second-order terms (which can be substituted for a second-order variable of an appropriate sort).
1644:
One might attempt to reduce the second-order theory of the real numbers, with full second-order semantics, to the first-order theory in the following way. First expand the domain from the set of all real numbers to a two-sorted domain, with the second sort containing all
247:
As a result, second-order logic has greater expressive power than first-order logic. For example, there is no way in first-order logic to identify the set of all cubes and tetrahedrons. But the existence of this set can be asserted in second-order logic as:
1200:) Thus once the domain of the first-order variables is established, the meaning of the remaining quantifiers is fixed. It is these semantics that give second-order logic its expressive power, and they will be assumed for the remainder of this article.
1715:
Thus the first-order theory of real numbers and sets of real numbers has many models, some of which are countable. The second-order theory of the real numbers has only one model, however. This follows from the classical theorem that there is only one
1233:
For theories such as second-order arithmetic, the existence of non-standard interpretations of higher-order domains isn't just a deficiency of the particular axiomatisation derived from type theory that Henkin used, but a necessary consequence of
1616:
and logical axioms that determine which sequences of formulas constitute valid proofs. Several deductive systems can be used for second-order logic, although none can be complete for the standard semantics (see below). Each of these systems is
1184:. In each of these semantics, the interpretations of the first-order quantifiers and the logical connectives are the same as in first-order logic. Only the ranges of quantifiers over second-order variables differ in the two types of semantics
1934:
of sentences such as "Some critics admire only each other" and "Some of
Fianchetto's men went into the warehouse unaccompanied by anyone else", which he argues can only be expressed by the full force of second-order quantification. However,
1843:
made them famous. Frege used different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic. After the discovery of
1195:
sets or functions of the appropriate sort. A model with this condition is called a full model, and these are the same as models in which the range of the second-order quantifiers is the powerset of the model's first-order part.
1175:
The semantics of second-order logic establish the meaning of each sentence. Unlike first-order logic, which has only one standard semantics, there are two different semantics that are commonly used for second-order logic:
1691:
Countability of the set of all internal numbers (in conjunction with the fact that those form a densely ordered set) implies that that set does not satisfy the full least-upper-bound axiom. Countability of the set of all
1726:
There are more extreme examples showing that second-order logic with standard semantics is more expressive than first-order logic. There is a finite second-order theory whose only model is the real numbers if the
1914:
like a property). For example, it might mean " . . . is a dog." But it makes no sense to think we can quantify over something like this. (Such a position is quite consistent with Frege's own arguments on the
1206:(1950) defined an alternative kind of semantics for second-order and higher-order theories, in which the meaning of the higher-order domains is partly determined by an explicit axiomatisation, drawing on
1262:, in that the former obeys model-theoretic properties like the Lowenheim-Skolem theorem and compactness, and the latter has categoricity phenomena. For example, "we cannot meaningfully ask whether the
1105:
966:
787:
128:
374:
It is notable that while we have variables for predicates in second-order-logic, we don't have variables for properties of predicates. We cannot say, for example, that there is a property Shape(
564:
Each of the variables just defined may be universally and/or existentially quantified over, to build up formulas. Thus there are many kinds of quantifiers, two for each sort of variables. A
1544:
1879:, and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with
1943:(or branching) quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and these do not appeal to second-order quantification.
917:
1919:
distinction). So to use a predicate as a variable is to have it occupy the place of a name, which only individual variables should occupy. This reasoning has been rejected by
2762:
1157:
843:
738:
674:
1412:
1384:
1356:
1308:
1056:
1804:, and the deductive system with comprehension and choice principles is sound, complete, and effective for Henkin semantics using only models that satisfy these principles.
1018:
875:
706:
627:
1852:âeliminated this problem: sets and properties cannot be quantified over in first-order logic alone. The now-standard hierarchy of orders of logics dates from this time.
3437:
1238:: Henkin's axioms can't be supplemented further to ensure the standard interpretation is the only possible model. Henkin semantics are commonly used in the study of
1125:
986:
807:
1328:
1280:
708:) if its quantifiers (which may be universal or existential) range only over variables of first order, although it may have free variables of second order. A
3520:
2661:
1902:" is to be thought of as a variable or name denoting an object and hence can be quantified over, as in "For all things, it is the case that . . ." but the "
1848:
it was realized that something was wrong with his system. Eventually logicians found that restricting Frege's logic in various waysâto what is now called
270:
We can then assert properties of this set. For instance, the following says that the set of all cubes and tetrahedrons does not contain any dodecahedrons:
1800:
As mentioned above, Henkin proved that the standard deductive system for first-order logic is sound, complete, and effective for second-order logic with
1252:) argued that the distinction between Henkin semantics and full semantics for second-order logic is analogous to the distinction between provability in
1450:. If the domain is the set of all real numbers, the following second-order sentence (split over two lines) expresses the least upper bound property:
3834:
2524:
1577:
1831:
and whose notation is most similar to the modern form (Putnam 1982). However, today most students of logic are more familiar with the works of
2023:, and additional predicates that serve to uniquely identify which index is which (typically, one takes the graph of the successor function on
3992:
2609:
2233:
2780:
3847:
3170:
1744:
1235:
1708:
implies that the set of all subsets of a countably infinite set is an uncountably infinite set). This construction is closely related to
2266:
The proof of this corollary is that a sound, complete, and effective deduction system for standard semantics could be used to produce a
2047:
1926:
In recent years second-order logic has made something of a recovery, buoyed by Boolos' interpretation of second-order quantification as
3432:
1650:
predicates that tell whether an element is a number or a set, and taking the domain to be the union of the set of real numbers and the
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Relationships among these classes directly impact the relative expressiveness of the logics over finite structures; for example, if
571:
It's possible to forgo the introduction of function variables in the definition given above (and some authors do this) because an
4407:
4085:
3829:
2654:
2149:
3390:
3083:
2824:
2104:, then adding a transitive closure operator to second-order logic would not make it any more expressive over finite structures.
1864:
1812:
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1952:
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1849:
740:(existential second-order) formula is one additionally having some existential quantifiers over second order variables, i.e.
1061:
922:
809:
is a first-order formula. The fragment of second-order logic consisting only of existential second-order formulas is called
743:
169:(often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified.
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4124:
4041:
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79:
74:
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could be formulated as an axiomatized system within the apparatus of first-order logic (at the cost of several kinds of
395:
3417:
2038:
This identification leads to the following characterizations of variants of second-order logic over finite structures:
1963:
can be characterized by the power of the logic needed to express languages (sets of finite strings) in them. A string
3749:
3488:
2746:
2647:
2228:(hardcover). Discrete Mathematics and Its Applications (5th ed.). Boca Raton: Chapman and Hall/CRC. p. 387.
1930:
over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed
1593:
4144:
4139:
1732:
example illustrates that the question of whether a sentence in second-order logic is consistent is extremely subtle.
1624:
The weakest deductive system that can be used consists of a standard deductive system for first-order logic (such as
877:
formulas is defined dually, it is called universal second-order logic. More expressive fragments are defined for any
1511:
4073:
3663:
3057:
3025:
2716:
2154:
1628:) augmented with substitution rules for second-order terms. This deductive system is commonly used in the study of
587:
2790:
4363:
4312:
4209:
3707:
3668:
3145:
4204:
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1759:) Every provable second-order sentence is universally valid, i.e., true in all domains under standard semantics.
1223:
4134:
3673:
3525:
3508:
3231:
2711:
2447:
1940:
1774:
568:
in second-order logic, as in first-order logic, is a well-formed formula with no free variables (of any sort).
595:
to distinguish it from the monadic version. Monadic second-order logic is particularly used in the context of
4036:
4013:
3974:
3860:
3801:
3447:
3367:
3211:
3155:
2768:
2295:
1629:
1239:
1164:
630:
607:) is decidable. By contrast, full second order logic over any infinite set (or MSO logic over for example (
2630:
2199:
1887:'s adherence to first-order logic, led to a general decline in work in second (or any higher) order logic.
1426:, one can assert in first-order logic the existence of an additive inverse of each real number by writing â
1422:
Second-order logic is more expressive than first-order logic. For example, if the domain is the set of all
1245:
1218:, which hold for first-order logic, carry over to second-order logic with Henkin semantics. Since also the
4326:
4053:
4031:
3998:
3891:
3737:
3722:
3695:
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3530:
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3290:
3256:
3251:
3125:
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1956:
1936:
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150:
47:
4256:
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2325:
Boolos, George (1984). "To Be Is To Be a Value of a
Variable (or to Be Some Values of Some Variables)".
1927:
1580:
that it is not possible to characterize finiteness or countability, respectively, in first-order logic.
4382:
1951:
The expressive power of various forms of second-order logic on finite structures is intimately tied to
1793:) pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not
1446:
property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a
884:
232:
is not a sentence of first-order logic, but this is a legitimate sentence of second-order logic. Here,
1621:, which means any sentence they can be used to prove is logically valid in the appropriate semantics.
4229:
4191:
4068:
3872:
3712:
3636:
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3400:
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1860:
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1258:
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816:
711:
647:
200:) and obtain a quantified sentence by replacing the name with a variable and attaching a quantifier:
1389:
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1333:
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1023:
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2173:
Shapiro (1991) and Hinman (2005) give complete introductions to the subject, with full definitions.
2139:
1808:
1709:
1705:
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1215:
166:
131:
66:
1781:
This corollary is sometimes expressed by saying that second-order logic does not admit a complete
991:
848:
679:
610:
192:
First-order logic can quantify over individuals, but not over properties. That is, we can take an
4387:
4156:
4119:
4104:
4097:
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2565:
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2376:
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1916:
1835:, who published his work several years prior to Peirce but whose works remained less known until
1801:
1666:
637:
383:
237:
62:
51:
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3340:
2057:
2484:
1568:. To say that the domain has countable cardinality, use the sentence that says that there is a
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3939:
3929:
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3702:
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2401:
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2129:
2121:
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1585:
1505:
604:
447:
399:
241:
154:
70:
43:
17:
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2186:
1790:
1785:. In this respect second-order logic with standard semantics differs from first-order logic;
4336:
4331:
4224:
4181:
4003:
3964:
3959:
3944:
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3727:
3624:
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2431:
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2334:
2283:
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1960:
1836:
1609:
1597:
1815:
do not hold for full models of second-order logic. They do hold however for Henkin models.
1661:
sets of real numbers. That requirement cannot be reduced to a first-order sentence, as the
1110:
971:
792:
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2053:
1911:
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algorithm that can correctly decide whether a given sequence of symbols is a proof or not.
1751:) for second-order formulas that simultaneously satisfies these three desired attributes:
1613:
332:. In second-order logic we can express this by saying that every set of people containing
193:
300:
Second-order quantification is especially useful because it gives the ability to express
188:(Berlin) showing the simplest second-order sentence admitting nontrivial models, "âđđ".
2027:
or the order relation <, possibly with other arithmetic predicates). Conversely, the
1549:
In second-order logic, it is possible to write formal sentences that say "the domain is
4302:
4281:
4239:
4219:
4114:
3969:
3567:
3557:
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2689:
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2144:
1884:
1313:
1265:
443:
1766:) Every universally valid second-order formula, under standard semantics, is provable.
218:
However, we cannot do the same with the predicate. That is, the following expression:
177:
4401:
4276:
3954:
3461:
3246:
3236:
3206:
3191:
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2585:
2465:
2436:
2419:
2415:
1920:
1880:
1832:
1554:
1500:
2388:
644:
A formula in second-order logic is said to be of first-order (and sometimes denoted
4176:
4023:
3924:
3916:
3796:
3744:
3653:
3589:
3572:
3503:
3362:
3221:
2923:
2706:
2589:
2569:
2028:
1782:
600:
301:
2459:
4286:
4166:
3345:
3335:
3282:
2966:
2886:
2871:
2751:
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1891:
1868:
1786:
1735:
Additional limitations of second-order logic are described in the next section.
1720:
1557:
1423:
1207:
1203:
185:
55:
35:
3216:
3071:
3042:
2848:
2111:
1872:
1856:
1561:
1550:
1191:
In standard semantics, also called full semantics, the quantifiers range over
2318:
An
Introduction to Mathematical Logic and Type Theory: To Truth Through Proof
4368:
4271:
3324:
3241:
3201:
3165:
3101:
2913:
2903:
2876:
1876:
1756:
1651:
1618:
1569:
1565:
1560:." To say that the domain is finite, use the sentence that says that every
575:-ary function variable can be represented by a relation variable of arity
4353:
4151:
3599:
3304:
2898:
2088:
is the set of languages definable by second-order formulas with an added
2078:
is the set of languages definable by second-order formulas with an added
2056:
is the set of languages definable by existential, second-order formulas (
1589:
1447:
1443:
394:
The syntax of second-order logic tells which expressions are well formed
325:
181:
579:+1 and an appropriate formula for the uniqueness of the "result" in the
516:
elements of the domain and returning a single element of the domain. If
3949:
2741:
2561:
2380:
2346:
2085:
2046:) is the set of languages definable by monadic, second-order formulas (
2066:
is the set of languages definable by universal, second-order formulas.
1910:
for an incomplete sentence, not the name of an object (not even of an
1572:
between every two infinite subsets of the domain. It follows from the
2639:
2291:
2075:
1894:. Quine advanced the view that in predicate-language sentences like
1890:
This rejection was actively advanced by some logicians, most notably
2553:
2372:
2338:
2206:(Fall 2021 ed.), Metaphysics Research Lab, Stanford University
1863:, but nothing so bad as Russell's paradox), and this was done (see
512:
there is a sort of variables that ranges over all functions taking
3493:
2839:
2684:
2600:. Texts in Theoretical Computer Science. An EATCS Series. Berlin:
2505:
2063:
176:
31:
1823:
Predicate logic was introduced to the mathematical community by
2643:
2635:. The Stanford Encyclopedia of Philosophy (Fall 2021 Edition).
1584:
quantifier dependencies, like first-order logic extended with
1253:
636:
Just as in first-order logic, second-order logic may include
165:). Both first-order and second-order logic use the idea of a
414:
A sort of variables that range over sets of individuals. If
65:
only variables that range over individuals (elements of the
2072:
is the set of languages definable by second-order formulas.
1414:... has countable models and hence cannot be categorical."
2359:
Henkin, L. (1950). "Completeness in the theory of types".
2187:
https://faculty.washington.edu/smcohen/120/SecondOrder.pdf
1508:
satisfies the same first-order sentences in the signature
320:, then first-order logic cannot express the property that
2485:"First-Order Logic, Second-Order Logic, and Completeness"
2257:
These are the models originally studied by Henkin (1950).
1747:
that there is no deductive system (that is, no notion of
603:. The MSO theory of the complete infinite binary tree (
583:+1 argument of the relation. (Shapiro 2000, p. 63)
153:). Second-order logic also includes quantification over
27:
Form of logic that allows quantification over predicates
2248:
Such a system is used without comment by Hinman (2005).
1669:
subset of the real numbers, whose members we will call
1100:{\displaystyle \forall R_{0}\ldots \forall R_{m}\phi }
961:{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }
782:{\displaystyle \exists R_{0}\ldots \exists R_{m}\phi }
1985:
can be represented by a finite structure with domain
1514:
1392:
1364:
1336:
1316:
1288:
1268:
1133:
1113:
1064:
1026:
994:
974:
925:
887:
851:
819:
795:
746:
714:
682:
650:
613:
82:
1442:= 0) but one needs second-order logic to assert the
123:{\displaystyle \forall P\,\forall x(Px\lor \neg Px)}
69:); second-order logic, in addition, quantifies over
4295:
4190:
4022:
3915:
3767:
3460:
3383:
3277:
3181:
3070:
2997:
2932:
2847:
2838:
2760:
2677:
2525:"Second-Order Logic and Foundations of Mathematics"
1657:But notice that the domain was asserted to include
1538:
1406:
1378:
1350:
1322:
1302:
1274:
1151:
1119:
1099:
1050:
1012:
980:
960:
911:
869:
837:
801:
781:
732:
700:
668:
621:
457:there is a sort of variables that ranges over all
122:
1487:This formula is a direct formalization of "every
1665:shows. That theorem implies that there is some
1539:{\displaystyle \langle +,\cdot ,\leq \rangle }
336:and closed under the Parent relation contains
2655:
8:
1533:
1515:
446:. Sets of individuals can also be viewed as
50:. Second-order logic is in turn extended by
2320:(2nd ed.). Kluwer Academic Publishers.
2274:, which Gödel's theorem shows cannot exist.
3481:
3076:
2844:
2662:
2648:
2640:
540:are first-order terms then the expression
485:are first-order terms then the expression
422:is a first-order term then the expression
2543:
2435:
1513:
1393:
1391:
1386:, then we can note that the reformalized
1365:
1363:
1337:
1335:
1315:
1289:
1287:
1267:
1143:
1138:
1132:
1112:
1088:
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1004:
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993:
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933:
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903:
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886:
861:
856:
850:
829:
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770:
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745:
724:
719:
713:
692:
687:
681:
660:
655:
649:
615:
614:
612:
382:Cube, Tet, and Dodec. This would require
162:
89:
81:
2629:VÀÀnÀnen, Jouko. Edward N. Zalta (ed.).
2598:Finite model theory and its applications
2596:; Venema, Yde; Weinstein, Scott (2007).
2181:
2179:
2490:. In V. Hendricks; et al. (eds.).
2204:The Stanford Encyclopedia of Philosophy
2166:
2031:of any finite structure (over a finite
1696:sets implies that it is not the set of
402:, second-order logic includes many new
2442:. Reprinted in Putnam, Hilary (1990),
1564:function from the domain to itself is
461:-ary relations on the individuals. If
2584:GrÀdel, Erich; Kolaitis, Phokion G.;
2200:"Second-order and Higher-order Logic"
2035:) can be encoded by a finite string.
1640:Non-reducibility to first-order logic
1636:the comprehension and choice axioms.
7:
1947:Relation to computational complexity
1226:imports that Henkin models are just
2632:Second-order and Higher-order Logic
2185:Professor Marc Cohen lecture notes
304:properties. For example, if Parent(
161:, and other variables (see section
2398:Fundamentals of Mathematical Logic
2290:, trans. Ruy J. G. B. de Queiroz (
2226:Introduction to Mathematical Logic
1400:
1397:
1394:
1372:
1369:
1366:
1344:
1341:
1338:
1296:
1293:
1290:
1163:for the analogous construction of
1135:
1081:
1065:
1028:
996:
942:
926:
889:
853:
845:, or even as âSO. The fragment of
821:
763:
747:
716:
684:
652:
378:) that is true for the predicates
108:
90:
83:
46:, which itself is an extension of
25:
1249:
1197:
1185:
912:{\displaystyle \Sigma _{k+1}^{1}}
599:, an algorithmic meta-theorem in
4381:
2150:Second-order propositional logic
2114:
2048:BĂŒchi-Elgot-Trakhtenbrot theorem
73:. For example, the second-order
1953:computational complexity theory
1578:upward LöwenheimâSkolem theorem
1152:{\displaystyle \Sigma _{k}^{1}}
838:{\displaystyle \Sigma _{1}^{1}}
733:{\displaystyle \Sigma _{1}^{1}}
669:{\displaystyle \Sigma _{0}^{1}}
418:is a variable of this sort and
1745:Gödel's incompleteness theorem
1407:{\displaystyle \mathrm {ZFC} }
1379:{\displaystyle \mathrm {ZFC} }
1351:{\displaystyle \mathrm {ZFC} }
1303:{\displaystyle \mathrm {ZFC} }
1236:Gödel's incompleteness theorem
1051:{\displaystyle \Pi _{k+1}^{1}}
811:existential second-order logic
117:
96:
18:Existential second-order logic
1:
4342:History of mathematical logic
2202:, in Zalta, Edward N. (ed.),
2008:, satisfied by those indices
4267:Primitive recursive function
2437:10.1016/0315-0860(82)90123-9
1959:studies which computational
1906:" is to be thought of as an
1499:." It can be shown that any
1228:disguised first-order models
1212:Gödel's completeness theorem
1013:{\displaystyle \Pi _{k}^{1}}
881:> 0 by mutual recursion:
870:{\displaystyle \Pi _{1}^{1}}
701:{\displaystyle \Pi _{0}^{1}}
629:,+)) can interpret the true
622:{\displaystyle \mathbb {N} }
2508:. Oxford: Clarendon Press.
2492:First-order logic revisited
1865:ZermeloâFraenkel set theory
1594:independence-friendly logic
1222:hold for Henkin semantics,
813:and abbreviated as ESO, as
524:-ary function variable and
469:-ary relation variable and
442:to save parentheses) is an
410:) of variables. These are:
400:syntax of first-order logic
4429:
3331:SchröderâBernstein theorem
3058:Monadic predicate calculus
2717:Foundations of mathematics
2532:Bulletin of Symbolic Logic
2224:Mendelson, Elliot (2009).
2155:Monadic second-order logic
1937:generalized quantification
1819:History and disputed value
1700:subsets of the set of all
588:Monadic second-order logic
4377:
4364:Philosophy of mathematics
4313:Automated theorem proving
3484:
3438:Von NeumannâBernaysâGödel
3079:
2444:Realism with a Human Face
2361:Journal of Symbolic Logic
1867:), as sets are vital for
1220:SkolemâLöwenheim theorems
2448:Harvard University Press
2198:VÀÀnÀnen, Jouko (2021),
1813:LöwenheimâSkolem theorem
1684:upper bound has a least
1663:LöwenheimâSkolem theorem
1612:for a logic is a set of
1504:logic. (In fact, every
1330:. But if we reformalize
560:) is a first-order term.
508:For each natural number
453:For each natural number
4408:Systems of formal logic
4014:Self-verifying theories
3835:Tarski's axiomatization
2786:Tarski's undefinability
2781:incompleteness theorems
2494:. Berlin: Logos-Verlag.
2351:. Reprinted in Boolos,
2316:Andrews, Peter (2002).
1630:second-order arithmetic
1553:" or "the domain is of
1497:has a least upper bound
1240:second-order arithmetic
1165:second-order arithmetic
631:second-order arithmetic
593:full second-order logic
505:) is an atomic formula.
149:) is true (this is the
137:, and every individual
4413:Charles Sanders Peirce
4388:Mathematics portal
3999:Proof of impossibility
3647:propositional variable
2957:Propositional calculus
2452:pp. 252–260
2353:Logic, Logic and Logic
2268:recursively enumerable
1957:descriptive complexity
1932:nonfirstorderizability
1841:Alfred North Whitehead
1827:, who coined the term
1721:complete ordered field
1546:as the real numbers.)
1540:
1408:
1380:
1352:
1324:
1304:
1276:
1153:
1121:
1101:
1052:
1020:formula, and similar,
1014:
982:
962:
913:
871:
839:
803:
783:
734:
702:
670:
623:
240:and is semantically a
189:
151:law of excluded middle
124:
4257:Kolmogorov complexity
4210:Computably enumerable
4110:Model complete theory
3902:Principia Mathematica
2962:Propositional formula
2791:BanachâTarski paradox
2523:VÀÀnÀnen, J. (2001).
2483:Rossberg, M. (2004).
2420:"Peirce the Logician"
2327:Journal of Philosophy
1981:in a finite alphabet
1928:plural quantification
1797:, properly speaking.
1743:It is a corollary of
1654:of the real numbers.
1541:
1409:
1381:
1353:
1325:
1305:
1277:
1154:
1122:
1120:{\displaystyle \phi }
1102:
1053:
1015:
983:
981:{\displaystyle \phi }
963:
914:
872:
840:
804:
802:{\displaystyle \phi }
784:
735:
703:
671:
624:
398:. In addition to the
180:
125:
4205:ChurchâTuring thesis
4192:Computability theory
3401:continuum hypothesis
2919:Square of opposition
2777:Gödel's completeness
2458:W. V. Quine (1970).
2424:Historia Mathematica
1993:}, unary predicates
1989: = {1,...,
1729:continuum hypothesis
1512:
1470:)( ∧ →
1390:
1362:
1334:
1314:
1286:
1266:
1161:analytical hierarchy
1131:
1111:
1062:
1024:
992:
972:
923:
885:
849:
817:
793:
744:
712:
680:
648:
611:
390:Syntax and fragments
130:says that for every
80:
4359:Mathematical object
4250:P versus NP problem
4215:Computable function
4009:Reverse mathematics
3935:Logical consequence
3812:primitive recursive
3807:elementary function
3580:Free/bound variable
3433:TarskiâGrothendieck
2952:Logical connectives
2882:Logical equivalence
2732:Logical consequence
2461:Philosophy of Logic
2396:Hinman, P. (2005).
1809:compactness theorem
1739:Metalogical results
1574:compactness theorem
1224:Lindström's theorem
1216:compactness theorem
1148:
1047:
1009:
908:
866:
834:
729:
697:
665:
638:non-logical symbols
597:Courcelle's theorem
361:⧠Parent(a, b)) â P
167:domain of discourse
67:domain of discourse
48:propositional logic
42:is an extension of
4157:Transfer principle
4120:Semantics of logic
4105:Categorical theory
4081:Non-standard model
3595:Logical connective
2722:Information theory
2671:Mathematical logic
2135:Higher-order logic
2080:transitive closure
1961:complexity classes
1855:It was found that
1829:second-order logic
1667:countably infinite
1586:Henkin quantifiers
1536:
1404:
1376:
1348:
1320:
1300:
1272:
1178:standard semantics
1149:
1134:
1117:
1097:
1048:
1027:
1010:
995:
978:
958:
909:
888:
867:
852:
835:
820:
799:
779:
730:
715:
698:
683:
666:
651:
619:
406:(sometimes called
238:predicate variable
190:
120:
61:First-order logic
52:higher-order logic
40:second-order logic
4395:
4394:
4327:Abstract category
4130:Theories of truth
3940:Rule of inference
3930:Natural deduction
3911:
3910:
3456:
3455:
3161:Cartesian product
3066:
3065:
2972:Many-valued logic
2947:Boolean functions
2830:Russell's paradox
2805:diagonal argument
2702:First-order logic
2611:978-3-540-00428-8
2588:; Maarten, Marx;
2235:978-1-58488-876-5
2130:First-order logic
2122:Philosophy portal
2090:least fixed point
2044:regular languages
1941:partially ordered
1850:first-order logic
1846:Russell's paradox
1626:natural deduction
1604:Deductive systems
1596:, and VÀÀnÀnen's
1506:real-closed field
1444:least-upper-bound
1323:{\displaystyle V}
1275:{\displaystyle V}
384:third-order logic
44:first-order logic
16:(Redirected from
4420:
4386:
4385:
4337:History of logic
4332:Category of sets
4225:Decision problem
4004:Ordinal analysis
3945:Sequent calculus
3843:Boolean algebras
3783:
3782:
3757:
3728:logical/constant
3482:
3468:
3391:ZermeloâFraenkel
3142:Set operations:
3077:
3014:
2845:
2825:LöwenheimâSkolem
2712:Formal semantics
2664:
2657:
2650:
2641:
2636:
2623:
2573:
2547:
2529:
2519:
2495:
2489:
2479:
2441:
2439:
2411:
2392:
2350:
2321:
2303:
2281:
2275:
2272:Peano arithmetic
2264:
2258:
2255:
2249:
2246:
2240:
2239:
2220:
2214:
2213:
2212:
2211:
2195:
2189:
2183:
2174:
2171:
2140:Löwenheim number
2124:
2119:
2118:
2117:
1837:Bertrand Russell
1802:Henkin semantics
1710:Skolem's paradox
1706:Cantor's theorem
1680:set that has an
1671:internal numbers
1610:deductive system
1598:dependence logic
1545:
1543:
1542:
1537:
1498:
1494:
1490:
1481:
1479:
1456:
1418:Expressive power
1413:
1411:
1410:
1405:
1403:
1385:
1383:
1382:
1377:
1375:
1357:
1355:
1354:
1349:
1347:
1329:
1327:
1326:
1321:
1309:
1307:
1306:
1301:
1299:
1281:
1279:
1278:
1273:
1182:Henkin semantics
1158:
1156:
1155:
1150:
1147:
1142:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1093:
1092:
1077:
1076:
1057:
1055:
1054:
1049:
1046:
1041:
1019:
1017:
1016:
1011:
1008:
1003:
987:
985:
984:
979:
967:
965:
964:
959:
954:
953:
938:
937:
918:
916:
915:
910:
907:
902:
876:
874:
873:
868:
865:
860:
844:
842:
841:
836:
833:
828:
808:
806:
805:
800:
788:
786:
785:
780:
775:
774:
759:
758:
739:
737:
736:
731:
728:
723:
707:
705:
704:
699:
696:
691:
675:
673:
672:
667:
664:
659:
628:
626:
625:
620:
618:
370:
296:
266:
244:of individuals.
228:
214:
129:
127:
126:
121:
21:
4428:
4427:
4423:
4422:
4421:
4419:
4418:
4417:
4398:
4397:
4396:
4391:
4380:
4373:
4318:Category theory
4308:Algebraic logic
4291:
4262:Lambda calculus
4200:Church encoding
4186:
4162:Truth predicate
4018:
3984:Complete theory
3907:
3776:
3772:
3768:
3763:
3755:
3475: and
3471:
3466:
3452:
3428:New Foundations
3396:axiom of choice
3379:
3341:Gödel numbering
3281: and
3273:
3177:
3062:
3012:
2993:
2942:Boolean algebra
2928:
2892:Equiconsistency
2857:Classical logic
2834:
2815:Halting problem
2803: and
2779: and
2767: and
2766:
2761:Theorems (
2756:
2673:
2668:
2628:
2612:
2602:Springer-Verlag
2594:Vardi, Moshe Y.
2583:
2580:
2578:Further reading
2554:10.2307/2687796
2527:
2522:
2516:
2498:
2487:
2482:
2476:
2457:
2414:
2408:
2395:
2373:10.2307/2266967
2358:
2339:10.2307/2026308
2324:
2315:
2312:
2307:
2306:
2296:Clarendon Press
2282:
2278:
2265:
2261:
2256:
2252:
2247:
2243:
2236:
2223:
2221:
2217:
2209:
2207:
2197:
2196:
2192:
2184:
2177:
2172:
2168:
2163:
2120:
2115:
2113:
2110:
2058:Fagin's theorem
2017:
1998:
1979:
1973:
1955:. The field of
1949:
1912:abstract object
1821:
1741:
1704:numbers (since
1689:
1676:Every nonempty
1642:
1614:inference rules
1606:
1510:
1509:
1496:
1492:
1488:
1461:
1459:
1454:
1420:
1388:
1387:
1360:
1359:
1332:
1331:
1312:
1311:
1284:
1283:
1264:
1263:
1186:(VÀÀnÀnen 2001)
1173:
1129:
1128:
1109:
1108:
1084:
1068:
1060:
1059:
1022:
1021:
990:
989:
970:
969:
945:
929:
921:
920:
883:
882:
847:
846:
815:
814:
791:
790:
766:
750:
742:
741:
710:
709:
678:
677:
646:
645:
609:
608:
559:
550:
539:
530:
504:
495:
484:
475:
448:unary relations
392:
344:
316:is a parent of
312:) denotes that
274:
252:
222:
204:
194:atomic sentence
175:
145:is true or not(
78:
77:
28:
23:
22:
15:
12:
11:
5:
4426:
4424:
4416:
4415:
4410:
4400:
4399:
4393:
4392:
4378:
4375:
4374:
4372:
4371:
4366:
4361:
4356:
4351:
4350:
4349:
4339:
4334:
4329:
4320:
4315:
4310:
4305:
4303:Abstract logic
4299:
4297:
4293:
4292:
4290:
4289:
4284:
4282:Turing machine
4279:
4274:
4269:
4264:
4259:
4254:
4253:
4252:
4247:
4242:
4237:
4232:
4222:
4220:Computable set
4217:
4212:
4207:
4202:
4196:
4194:
4188:
4187:
4185:
4184:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4148:
4147:
4142:
4137:
4127:
4122:
4117:
4115:Satisfiability
4112:
4107:
4102:
4101:
4100:
4090:
4089:
4088:
4078:
4077:
4076:
4071:
4066:
4061:
4056:
4046:
4045:
4044:
4039:
4032:Interpretation
4028:
4026:
4020:
4019:
4017:
4016:
4011:
4006:
4001:
3996:
3986:
3981:
3980:
3979:
3978:
3977:
3967:
3962:
3952:
3947:
3942:
3937:
3932:
3927:
3921:
3919:
3913:
3912:
3909:
3908:
3906:
3905:
3897:
3896:
3895:
3894:
3889:
3888:
3887:
3882:
3877:
3857:
3856:
3855:
3853:minimal axioms
3850:
3839:
3838:
3837:
3826:
3825:
3824:
3819:
3814:
3809:
3804:
3799:
3786:
3784:
3765:
3764:
3762:
3761:
3760:
3759:
3747:
3742:
3741:
3740:
3735:
3730:
3725:
3715:
3710:
3705:
3700:
3699:
3698:
3693:
3683:
3682:
3681:
3676:
3671:
3666:
3656:
3651:
3650:
3649:
3644:
3639:
3629:
3628:
3627:
3622:
3617:
3612:
3607:
3602:
3592:
3587:
3582:
3577:
3576:
3575:
3570:
3565:
3560:
3550:
3545:
3543:Formation rule
3540:
3535:
3534:
3533:
3528:
3518:
3517:
3516:
3506:
3501:
3496:
3491:
3485:
3479:
3462:Formal systems
3458:
3457:
3454:
3453:
3451:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3415:
3410:
3405:
3404:
3403:
3398:
3387:
3385:
3381:
3380:
3378:
3377:
3376:
3375:
3365:
3360:
3359:
3358:
3351:Large cardinal
3348:
3343:
3338:
3333:
3328:
3314:
3313:
3312:
3307:
3302:
3287:
3285:
3275:
3274:
3272:
3271:
3270:
3269:
3264:
3259:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3188:
3186:
3179:
3178:
3176:
3175:
3174:
3173:
3168:
3163:
3158:
3153:
3148:
3140:
3139:
3138:
3133:
3123:
3118:
3116:Extensionality
3113:
3111:Ordinal number
3108:
3098:
3093:
3092:
3091:
3080:
3074:
3068:
3067:
3064:
3063:
3061:
3060:
3055:
3050:
3045:
3040:
3035:
3030:
3029:
3028:
3018:
3017:
3016:
3003:
3001:
2995:
2994:
2992:
2991:
2990:
2989:
2984:
2979:
2969:
2964:
2959:
2954:
2949:
2944:
2938:
2936:
2930:
2929:
2927:
2926:
2921:
2916:
2911:
2906:
2901:
2896:
2895:
2894:
2884:
2879:
2874:
2869:
2864:
2859:
2853:
2851:
2842:
2836:
2835:
2833:
2832:
2827:
2822:
2817:
2812:
2807:
2795:Cantor's
2793:
2788:
2783:
2773:
2771:
2758:
2757:
2755:
2754:
2749:
2744:
2739:
2734:
2729:
2724:
2719:
2714:
2709:
2704:
2699:
2694:
2693:
2692:
2681:
2679:
2675:
2674:
2669:
2667:
2666:
2659:
2652:
2644:
2638:
2637:
2625:
2624:
2610:
2586:Libkin, Leonid
2579:
2576:
2575:
2574:
2545:10.1.1.25.5579
2538:(4): 504â520.
2520:
2514:
2496:
2480:
2474:
2455:
2430:(3): 290â301.
2416:Putnam, Hilary
2412:
2406:
2400:. A K Peters.
2393:
2356:
2322:
2311:
2308:
2305:
2304:
2276:
2270:completion of
2259:
2250:
2241:
2234:
2215:
2190:
2175:
2165:
2164:
2162:
2159:
2158:
2157:
2152:
2147:
2145:Omega language
2142:
2137:
2132:
2126:
2125:
2109:
2106:
2094:
2093:
2083:
2073:
2067:
2061:
2051:
2015:
1996:
1977:
1971:
1948:
1945:
1917:concept-object
1820:
1817:
1779:
1778:
1775:proof-checking
1767:
1760:
1740:
1737:
1675:
1641:
1638:
1605:
1602:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1485:
1484:
1483:
1482:
1419:
1416:
1402:
1399:
1396:
1374:
1371:
1368:
1346:
1343:
1340:
1319:
1298:
1295:
1292:
1282:as defined in
1271:
1246:Jouko VÀÀnÀnen
1172:
1169:
1159:formula. (See
1146:
1141:
1137:
1116:
1096:
1091:
1087:
1083:
1080:
1075:
1071:
1067:
1045:
1040:
1037:
1034:
1030:
1007:
1002:
998:
977:
957:
952:
948:
944:
941:
936:
932:
928:
906:
901:
898:
895:
891:
864:
859:
855:
832:
827:
823:
798:
778:
773:
769:
765:
762:
757:
753:
749:
727:
722:
718:
695:
690:
686:
663:
658:
654:
617:
562:
561:
555:
548:
535:
528:
506:
500:
493:
480:
473:
451:
450:on the domain.
444:atomic formula
430:(also written
391:
388:
372:
371:
298:
297:
268:
267:
230:
229:
216:
215:
174:
171:
119:
116:
113:
110:
107:
104:
101:
98:
95:
92:
88:
85:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4425:
4414:
4411:
4409:
4406:
4405:
4403:
4390:
4389:
4384:
4376:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4348:
4345:
4344:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4300:
4298:
4294:
4288:
4285:
4283:
4280:
4278:
4277:Recursive set
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4251:
4248:
4246:
4243:
4241:
4238:
4236:
4233:
4231:
4228:
4227:
4226:
4223:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4203:
4201:
4198:
4197:
4195:
4193:
4189:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4146:
4143:
4141:
4138:
4136:
4133:
4132:
4131:
4128:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4106:
4103:
4099:
4096:
4095:
4094:
4091:
4087:
4086:of arithmetic
4084:
4083:
4082:
4079:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4051:
4050:
4047:
4043:
4040:
4038:
4035:
4034:
4033:
4030:
4029:
4027:
4025:
4021:
4015:
4012:
4010:
4007:
4005:
4002:
4000:
3997:
3994:
3993:from ZFC
3990:
3987:
3985:
3982:
3976:
3973:
3972:
3971:
3968:
3966:
3963:
3961:
3958:
3957:
3956:
3953:
3951:
3948:
3946:
3943:
3941:
3938:
3936:
3933:
3931:
3928:
3926:
3923:
3922:
3920:
3918:
3914:
3904:
3903:
3899:
3898:
3893:
3892:non-Euclidean
3890:
3886:
3883:
3881:
3878:
3876:
3875:
3871:
3870:
3868:
3865:
3864:
3862:
3858:
3854:
3851:
3849:
3846:
3845:
3844:
3840:
3836:
3833:
3832:
3831:
3827:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3794:
3792:
3788:
3787:
3785:
3780:
3774:
3769:Example
3766:
3758:
3753:
3752:
3751:
3748:
3746:
3743:
3739:
3736:
3734:
3731:
3729:
3726:
3724:
3721:
3720:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3697:
3694:
3692:
3689:
3688:
3687:
3684:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3661:
3660:
3657:
3655:
3652:
3648:
3645:
3643:
3640:
3638:
3635:
3634:
3633:
3630:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3597:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3555:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3532:
3529:
3527:
3526:by definition
3524:
3523:
3522:
3519:
3515:
3512:
3511:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3486:
3483:
3480:
3478:
3474:
3469:
3463:
3459:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3413:KripkeâPlatek
3411:
3409:
3406:
3402:
3399:
3397:
3394:
3393:
3392:
3389:
3388:
3386:
3382:
3374:
3371:
3370:
3369:
3366:
3364:
3361:
3357:
3354:
3353:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3329:
3326:
3322:
3318:
3315:
3311:
3308:
3306:
3303:
3301:
3298:
3297:
3296:
3292:
3289:
3288:
3286:
3284:
3280:
3276:
3268:
3265:
3263:
3260:
3258:
3257:constructible
3255:
3254:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3189:
3187:
3185:
3180:
3172:
3169:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3143:
3141:
3137:
3134:
3132:
3129:
3128:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3103:
3099:
3097:
3094:
3090:
3087:
3086:
3085:
3082:
3081:
3078:
3075:
3073:
3069:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3027:
3024:
3023:
3022:
3019:
3015:
3010:
3009:
3008:
3005:
3004:
3002:
3000:
2996:
2988:
2985:
2983:
2980:
2978:
2975:
2974:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2939:
2937:
2935:
2934:Propositional
2931:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2893:
2890:
2889:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2862:Logical truth
2860:
2858:
2855:
2854:
2852:
2850:
2846:
2843:
2841:
2837:
2831:
2828:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2802:
2798:
2794:
2792:
2789:
2787:
2784:
2782:
2778:
2775:
2774:
2772:
2770:
2764:
2759:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2691:
2688:
2687:
2686:
2683:
2682:
2680:
2676:
2672:
2665:
2660:
2658:
2653:
2651:
2646:
2645:
2642:
2634:
2633:
2627:
2626:
2621:
2617:
2613:
2607:
2603:
2599:
2595:
2591:
2590:Spencer, Joel
2587:
2582:
2581:
2577:
2571:
2567:
2563:
2559:
2555:
2551:
2546:
2541:
2537:
2533:
2526:
2521:
2517:
2515:0-19-825029-0
2511:
2507:
2504:
2501:
2497:
2493:
2486:
2481:
2477:
2475:9780674665637
2471:
2467:
2466:Prentice Hall
2463:
2462:
2456:
2453:
2449:
2445:
2438:
2433:
2429:
2425:
2421:
2417:
2413:
2409:
2407:1-56881-262-0
2403:
2399:
2394:
2390:
2386:
2382:
2378:
2374:
2370:
2366:
2362:
2357:
2354:
2348:
2344:
2340:
2336:
2333:(8): 430â50.
2332:
2328:
2323:
2319:
2314:
2313:
2309:
2301:
2297:
2293:
2289:
2285:
2280:
2277:
2273:
2269:
2263:
2260:
2254:
2251:
2245:
2242:
2237:
2231:
2227:
2219:
2216:
2205:
2201:
2194:
2191:
2188:
2182:
2180:
2176:
2170:
2167:
2160:
2156:
2153:
2151:
2148:
2146:
2143:
2141:
2138:
2136:
2133:
2131:
2128:
2127:
2123:
2112:
2107:
2105:
2103:
2100: =
2099:
2091:
2087:
2084:
2081:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2055:
2052:
2049:
2045:
2041:
2040:
2039:
2036:
2034:
2030:
2029:Cayley tables
2026:
2022:
2019: =
2018:
2011:
2007:
2004: â
2003:
1999:
1992:
1988:
1984:
1980:
1970:
1967: =
1966:
1962:
1958:
1954:
1946:
1944:
1942:
1938:
1933:
1929:
1924:
1922:
1921:George Boolos
1918:
1913:
1909:
1905:
1901:
1897:
1893:
1888:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1858:
1853:
1851:
1847:
1842:
1838:
1834:
1830:
1826:
1818:
1816:
1814:
1810:
1805:
1803:
1798:
1796:
1792:
1788:
1784:
1776:
1773:) There is a
1772:
1771:Effectiveness
1768:
1765:
1761:
1758:
1754:
1753:
1752:
1750:
1746:
1738:
1736:
1733:
1730:
1724:
1722:
1719:
1713:
1711:
1707:
1703:
1699:
1695:
1687:
1683:
1679:
1674:
1672:
1668:
1664:
1660:
1655:
1653:
1648:
1639:
1637:
1633:
1631:
1627:
1622:
1620:
1615:
1611:
1603:
1601:
1599:
1595:
1591:
1587:
1581:
1579:
1575:
1571:
1567:
1563:
1559:
1556:
1552:
1547:
1530:
1527:
1524:
1521:
1518:
1507:
1502:
1501:ordered field
1477:
1473:
1469:
1465:
1458:
1457:
1455:(∀ A) (
1453:
1452:
1451:
1449:
1445:
1441:
1437:
1433:
1429:
1425:
1417:
1415:
1317:
1269:
1261:
1260:
1256:and truth in
1255:
1251:
1247:
1243:
1241:
1237:
1231:
1229:
1225:
1221:
1217:
1213:
1209:
1205:
1201:
1199:
1194:
1189:
1187:
1183:
1179:
1170:
1168:
1166:
1162:
1144:
1139:
1114:
1094:
1089:
1085:
1078:
1073:
1069:
1058:has the form
1043:
1038:
1035:
1032:
1005:
1000:
975:
955:
950:
946:
939:
934:
930:
919:has the form
904:
899:
896:
893:
880:
862:
857:
830:
825:
812:
796:
776:
771:
767:
760:
755:
751:
725:
720:
693:
688:
661:
656:
642:
639:
634:
632:
606:
602:
598:
594:
590:
589:
584:
582:
578:
574:
569:
567:
558:
554:
547:
543:
538:
534:
527:
523:
519:
515:
511:
507:
503:
499:
492:
488:
483:
479:
472:
468:
464:
460:
456:
452:
449:
445:
441:
437:
433:
429:
425:
421:
417:
413:
412:
411:
409:
405:
401:
397:
389:
387:
385:
381:
377:
368:
364:
360:
356:
352:
348:
343:
342:
341:
339:
335:
331:
327:
323:
319:
315:
311:
307:
303:
294:
290:
286:
282:
278:
273:
272:
271:
264:
260:
256:
251:
250:
249:
245:
243:
239:
235:
226:
221:
220:
219:
212:
208:
203:
202:
201:
199:
195:
187:
183:
179:
172:
170:
168:
164:
160:
156:
152:
148:
144:
140:
136:
133:
114:
111:
105:
102:
99:
93:
86:
76:
72:
68:
64:
59:
57:
53:
49:
45:
41:
37:
33:
19:
4379:
4177:Ultraproduct
4024:Model theory
3989:Independence
3925:Formal proof
3917:Proof theory
3900:
3873:
3830:real numbers
3802:second-order
3713:Substitution
3590:Metalanguage
3531:conservative
3504:Axiom schema
3448:Constructive
3418:MorseâKelley
3384:Set theories
3363:Aleph number
3356:inaccessible
3262:Grothendieck
3146:intersection
3033:Higher-order
3021:Second-order
3020:
2967:Truth tables
2924:Venn diagram
2707:Formal proof
2631:
2597:
2535:
2531:
2506:
2503:
2491:
2460:
2443:
2427:
2423:
2397:
2367:(2): 81â91.
2364:
2360:
2352:
2330:
2326:
2317:
2288:Model Theory
2287:
2279:
2262:
2253:
2244:
2225:
2218:
2208:, retrieved
2203:
2193:
2169:
2101:
2097:
2095:
2037:
2024:
2020:
2013:
2009:
2005:
2001:
1994:
1990:
1986:
1982:
1975:
1968:
1964:
1950:
1925:
1908:abbreviation
1907:
1903:
1899:
1895:
1889:
1861:completeness
1854:
1828:
1825:C. S. Peirce
1822:
1806:
1799:
1794:
1783:proof theory
1780:
1764:Completeness
1748:
1742:
1734:
1725:
1714:
1701:
1697:
1693:
1690:
1688:upper bound.
1685:
1681:
1677:
1670:
1658:
1656:
1646:
1643:
1634:
1623:
1607:
1592:and Sandu's
1582:
1548:
1486:
1475:
1471:
1467:
1463:
1439:
1435:
1431:
1427:
1424:real numbers
1421:
1310:is the real
1257:
1244:
1232:
1227:
1202:
1192:
1190:
1181:
1177:
1174:
878:
810:
643:
635:
601:graph theory
592:
586:
585:
580:
576:
572:
570:
565:
563:
556:
552:
545:
541:
536:
532:
525:
521:
517:
513:
509:
501:
497:
490:
486:
481:
477:
470:
466:
462:
458:
454:
439:
435:
431:
427:
423:
419:
415:
407:
403:
393:
379:
375:
373:
366:
362:
358:
354:
350:
346:
337:
333:
329:
321:
317:
313:
309:
305:
302:reachability
299:
292:
291:(Px ⧠Dodec(
288:
284:
280:
279:(Px â (Cube(
276:
269:
262:
258:
257:(Px â (Cube(
254:
246:
233:
231:
224:
217:
210:
206:
197:
191:
146:
142:
138:
134:
60:
39:
29:
4287:Type theory
4235:undecidable
4167:Truth value
4054:equivalence
3733:non-logical
3346:Enumeration
3336:Isomorphism
3283:cardinality
3267:Von Neumann
3232:Ultrafilter
3197:Uncountable
3131:equivalence
3048:Quantifiers
3038:Fixed-point
3007:First-order
2887:Consistency
2872:Proposition
2849:Traditional
2820:Lindström's
2810:Compactness
2752:Type theory
2697:Cardinality
2500:Shapiro, S.
2284:Manzano, M.
1892:W. V. Quine
1869:mathematics
1749:provability
1718:Archimedean
1558:cardinality
1208:type theory
1204:Leon Henkin
56:type theory
36:mathematics
4402:Categories
4098:elementary
3791:arithmetic
3659:Quantifier
3637:functional
3509:Expression
3227:Transitive
3171:identities
3156:complement
3089:hereditary
3072:Set theory
2620:1133.03001
2310:References
2210:2022-05-03
2012:such that
1873:Arithmetic
1857:set theory
1562:surjective
1466:)(∀
520:is such a
465:is such a
196:like Cube(
63:quantifies
4369:Supertask
4272:Recursion
4230:decidable
4064:saturated
4042:of models
3965:deductive
3960:axiomatic
3880:Hilbert's
3867:Euclidean
3848:canonical
3771:axiomatic
3703:Signature
3632:Predicate
3521:Extension
3443:Ackermann
3368:Operation
3247:Universal
3237:Recursive
3212:Singleton
3207:Inhabited
3192:Countable
3182:Types of
3166:power set
3136:partition
3053:Predicate
2999:Predicate
2914:Syllogism
2904:Soundness
2877:Inference
2867:Tautology
2769:paradoxes
2540:CiteSeerX
2502:(2000) .
2298:, 1999),
2092:operator.
2082:operator.
2042:REG (the
2033:signature
2000:for each
1877:mereology
1791:pp. 90â91
1757:Soundness
1652:power set
1570:bijection
1566:injective
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1171:Semantics
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