652:
7684:
them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction
Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834).
1489:
6866:
1025:
6494:, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory.
1484:{\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}
3750:
203:
5087:
3360:
6852:
7384:, pp. 145, 147, "It is rather well-known, through Peano's own acknowledgement, that Peano made extensive use of Grassmann's work in his development of the axioms. It is not so well-known that Grassmann had essentially the characterization of the set of all integers, now customary in texts of modern algebra, that it forms an ordered
7587:
An illustration of 'interpretation' is
Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of
5149:
The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as the successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternative uses an order relation symbol instead of the successor
4316:
Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or
7683:
So
Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of
4429:
of induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom. The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of
4880:
6607:. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.
6482:
shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism. This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.
3745:{\displaystyle {\begin{aligned}0&=\emptyset \\1&=s(0)=s(\emptyset )=\emptyset \cup \{\emptyset \}=\{\emptyset \}=\{0\}\\2&=s(1)=s(\{0\})=\{0\}\cup \{\{0\}\}=\{0,\{0\}\}=\{0,1\}\\3&=s(2)=s(\{0,1\})=\{0,1\}\cup \{\{0,1\}\}=\{0,1,\{0,1\}\}=\{0,1,2\}\end{aligned}}}
4375:
that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true
4344:. Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε
1014:
4356:, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.
8048:
7540:
6957:
The non-contiguous set satisfies axiom 1 as it has a 0 element, 2–5 as it doesn't affect equality relations, 6 & 8 as all pieces have a successor, bar the zero element and axiom 7 as no two dominos topple, or are toppled by, the same
3242:
1822:
4239:
6086:
5082:{\displaystyle \forall {\bar {y}}{\Bigg (}{\bigg (}\varphi (0,{\bar {y}})\land \forall x{\Big (}\varphi (x,{\bar {y}})\Rightarrow \varphi (S(x),{\bar {y}}){\Big )}{\bigg )}\Rightarrow \forall x\varphi (x,{\bar {y}}){\Bigg )}}
2352:
4367:, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be
5779:
5404:
6168:
5568:
7104:
6235:
691:(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0.
5995:
307:(∈, which comes from Peano's ε). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the
5912:
7664:
4577:
4826:
4141:
3365:
3124:
1732:
1030:
909:
5639:
2466:
161:
The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about
5465:
5244:
7242:
2531:
1938:
904:
7154:
6837:
4703:
6757:
6513:. This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible
5835:
7292:
4433:
First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from the
7012:
5696:
4751:
4499:
5309:
2643:
6282:
6704:
4317:
stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using
4625:
3337:
7388:
in wihich each set of positive elements has a least member. was probably the first serious and rather successful attempt to put numbers on a more or less axiomatic basis.".
7186:
2136:
2043:
1580:
364:
6352:
4359:
The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as
4289:; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything. In 1900,
7324:
6656:
5119:
3803:), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.
303:
was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for
7935:
4401:
1638:
3119:
7356:
6330:
1727:
1524:
4437:, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in the
2385:
2194:
2165:
2075:
2002:
1970:
1854:
144:
proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book
8896:
4415:. The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above is
6310:
4403:
theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a
Hilbert-style proof of "0=1").
2095:
1658:
1600:
154:
4136:
3264:
7882:
5137:. The first-order induction schema includes every instance of the first-order induction axiom; that is, it includes the induction axiom for every formula
8655:]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. Fratres Bocca. pp. 83–97.
7864:
6002:
165:; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic". The next three axioms are
2812:, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are
3026:, i.e., "What are the numbers and what are they good for?") that any two models of the Peano axioms (including the second-order induction axiom) are
7507:
403:. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.
651:
4430:
natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).
9021:
2199:
8880:
8856:
8830:
8806:
8617:
8593:
8434:
8407:
8381:
8357:
8233:
8202:
8170:
7958:
544:
The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "
8394:
6378:
5707:
5320:
6093:
5472:
8917:
7017:
6175:
2711:
The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤":
8393:
Meseguer, José; Goguen, Joseph A. (Dec 1986). "Initiality, induction, and computability". In
Maurice Nivat and John C. Reynolds (ed.).
6509:, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is
9026:
6393:
but are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such as
4423:
over predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-order
6401:
833:
induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section
8557:
8525:
8329:
8267:
4313:, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself, if Peano arithmetic is consistent.
286:
5923:
322:
177:. A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing the
8185:(1983). "Mathematics, the Empirical Facts, and Logical Necessity". In Hempel, Carl G.; Putnam, Hilary; Essler, Wilhelm K. (eds.).
6899:
6502:
6457:
3252:
6491:
6479:
3278:
169:
statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a
6408:
for deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence,
5846:
2387:
is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication is commutative and
317:, published in 1879. Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of
8700:
7591:
4505:
224:
4360:
4326:
4757:
267:
8948:
8585:
8349:
6543:, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
6444:(more quantifier alternations) than existential formulas are more expressive, and define sets in the higher levels of the
4338:
4310:
220:
6904:
239:
9016:
7856:
5579:
2397:
6497:
It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as
9031:
9011:
8943:
8033:
6929:
5411:
5160:
1009:{\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}
7191:
4298:
4268:
867:
and addition and multiplication are often added as axioms. The respective functions and relations are constructed in
173:
statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to
2475:
1866:
246:
8162:
6879:
6421:
4832:
7109:
6789:
4831:
In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a
4631:
4444:
The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of
387:
Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in
213:
8288:
6712:
5790:
3354:
that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:
1972:
is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined:
5915:
7247:
253:
6970:
5654:
4709:
4454:
8321:
8114:
7950:
7491:
6924:
6506:
5255:
4372:
3792:
2716:
769:
178:
174:
2608:
8462:
8245:
6889:
6445:
6246:
3820:
1661:
664:
549:
400:
397:
389:
162:
133:
129:
7504:
6665:
235:
124:, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the
8797:
Buss, Samuel R. (1998). "Chapter II: First-Order Proof Theory of
Arithmetic". In Buss, Samuel R. (ed.).
7994:
6909:
6425:
5571:
4583:
4330:
4054:
3347:
1721:
is a function mapping two natural numbers to another one. Given addition, it is defined recursively as:
1687:
573:
536:
304:
8995:
6865:
6478:
implies that the existence of nonstandard elements cannot be excluded in first-order logic. The upward
4302:
671:
to the chain of light pieces ("no junk") as only light dominoes will fall when the nearest is toppled.
8976:
5154:(axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness):
3295:
6413:
5312:
5247:
3834:
676:
95:
8842:
8688:
8667:
8605:
7868:
7159:
6919:
6510:
6475:
6394:
5782:
4445:
2100:
2007:
1698:
1529:
503:
344:
8938:
6335:
8890:
8777:
8761:
8720:
8712:
8549:
8486:
8208:
8060:
8015:
7929:
7921:
7516:
6871:
6471:
5838:
4434:
4420:
4416:
3800:
3788:
3760:
3248:
872:
826:
700:
625:
545:
458:
423:
367:
300:
170:
125:
20:
7679:
7588:
natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form
7297:
6884:
6632:
5095:
7583:
6529:
be the order type of the rationals, the order type of any countable nonstandard model of PA is
3237:{\displaystyle {\begin{aligned}f(0_{A})&=0_{B}\\f(S_{A}(n))&=S_{B}(f(n))\end{aligned}}}
8957:
8876:
8852:
8826:
8818:
8802:
8613:
8589:
8553:
8521:
8430:
8403:
8377:
8369:
8353:
8348:. Derives the basic number systems from the Peano axioms. English/German vocabulary included.
8325:
8273:
8263:
8229:
8198:
8166:
8154:
8134:
7954:
6914:
6857:
6386:
6238:
5646:
4438:
4412:
4379:
4353:
4282:
3784:
1817:{\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}
887:
830:
663:
satisfied by the set of all dominoes — whether light or dark — taken together. The 9th axiom (
335:
166:
121:
91:
8509:
4441:
of Peano arithmetic, and axioms are included that relate the three operations to each other.
1605:
8753:
8739:"Self-verifying axiom systems, the incompleteness theorem and related reflection principles"
8704:
8630:
8478:
8420:
8300:
8223:
8190:
8182:
8118:
8110:
8052:
8007:
7878:
6237:, i.e. zero and one are distinct and there is no element between them. In other words, 0 is
5699:
5642:
4368:
4352:
describing a suitable order on the integers, or more abstractly as consisting of the finite
4278:
3282:
3011:
2388:
1857:
698:
sufficiently many times to zero requires an additional axiom, which is sometimes called the
141:
137:
58:
30:
8773:
8498:
6948:
the nearest light piece corresponding to 0, and a neighbor piece corresponding to successor
6315:
4281:
and others agreed that these axioms implicitly defined what we mean by a "natural number".
1497:
655:
The chain of light dominoes on the right, starting with the nearest, can represent the set
8769:
8494:
8249:
7989:
7511:
7385:
6441:
6429:
4322:
3824:
3812:
3019:
2361:
2170:
2141:
2051:
1978:
1946:
1830:
1690:
858:
330:
309:
260:
83:
2708:; because there is no natural number between 0 and 1, it is a discrete ordered semiring.
2549:
relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:
8646:
8450:
3255:. (This is not the case with any first-order reformulation of the Peano axioms, below.)
8642:
8578:
8542:
8537:
8446:
8123:
8092:
8034:"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"
7976:
7803:
7485:
6463:
6295:
6170:, i.e. given any two distinct elements, the larger is the smaller plus another element.
4411:
All of the Peano axioms except the ninth axiom (the induction axiom) are statements in
4349:
4334:
3780:
3274:
2080:
1718:
1643:
1585:
850:
149:
102:
79:
75:
9005:
8868:
8573:
8514:
8341:
8212:
8088:
8064:
8029:
8019:
7912:
4364:
4306:
4290:
4234:{\displaystyle {\begin{aligned}u(0)&=0_{X},\\u(Sx)&=S_{X}(ux).\end{aligned}}}
2947:
2828:
883:
314:
87:
8724:
8305:
8781:
6894:
6432:, and thus definable by existentially quantified formulas (with free variables) of
6088:, i.e. the ordering is preserved under multiplication by the same positive element.
6081:{\displaystyle \forall x,y,z\ (0<z\land x<y\Rightarrow x\cdot z<y\cdot z)}
4425:
3087:
2970:
2705:
318:
182:
8960:
7014:" can be proven from the other axioms (in first-order logic) as follows. Firstly,
6292:. It is also incomplete and one of its important properties is that any structure
8846:
8424:
8138:
8049:
On
Formally Undecidable Propositions of Principia Mathematica and Related Systems
7944:
8734:
8668:"Introduction to Peano Arithmetic (Gödel Incompleteness and Nonstandard Models)"
8194:
8150:
4363:. A small number of philosophers and mathematicians, some of whom also advocate
4318:
4272:
3027:
2546:
1672:
854:
202:
7541:
File:Inductive proofs of properties of add, mult from recursive definitions.pdf
8991:
8823:
Introduction to
Mathematical Logic (Discrete Mathematics and Its Applications)
8637:. On p. 100, he restates and defends his axioms of 1888. pp. 98–103.
8374:
Introduction to
Mathematical Logic (Discrete Mathematics and Its Applications)
8097:
6847:
6514:
6487:
3270:
2813:
868:
117:
106:
8626:
Contains translations of the following two papers, with valuable commentary:
8277:
8965:
8928:
8738:
8466:
6405:
3796:
2817:
1694:
895:
8093:"Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes"
8068:
679:
of the intuitive notion of natural numbers: the number 1 can be defined as
694:
The intuitive notion that each natural number can be obtained by applying
370:
for the axioms consist of a constant symbol 0 and a unary function symbol
7351:
5151:
4294:
4285:
was more cautious, saying they only defined natural numbers if they were
3783:
with several weak systems of set theory. One such system is ZFC with the
2534:
2347:{\displaystyle S(0)\cdot S(a)=S(0)+S(0)\cdot a=S(0)+a=a+S(0)=S(a+0)=S(a)}
846:
86:
investigations, including research into fundamental questions of whether
7294:
by substitution, contradicting irreflexivity. Therefore it must be that
829:. It is now common to replace this second-order principle with a weaker
8765:
8716:
8490:
8056:
8011:
1706:
7946:
From Kant to
Hilbert: A Source Book in the Foundations of Mathematics
6498:
5774:{\displaystyle \forall x,y,z\ (x<y\land y<z\Rightarrow x<z)}
4836:
3285:, starts from a definition of 0 as the empty set, ∅, and an operator
2820:
1675:
845:
If we use the second-order induction axiom, it is possible to define
8757:
8708:
8482:
6385:(if consistent) is incomplete. Consequently, there are sentences of
5997:, i.e. the ordering is preserved under addition of the same element.
5399:{\displaystyle \forall x,y,z\ ((x\cdot y)\cdot z=x\cdot (y\cdot z))}
8923:
Includes a discussion of Poincaré's critique of the Peano's axioms.
8610:
From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931
6163:{\displaystyle \forall x,y\ (x<y\Rightarrow \exists z\ (x+z=y))}
5563:{\displaystyle \forall x,y,z\ (x\cdot (y+z)=(x\cdot y)+(x\cdot z))}
646:
is false. That is, there is no natural number whose successor is 0.
8912:
7099:{\displaystyle x\cdot 0+x\cdot 0=x\cdot (0+0)=x\cdot 0=x\cdot 0+0}
6230:{\displaystyle 0<1\land \forall x\ (x>0\Rightarrow x\geq 1)}
189:
is sometimes used for specifically naming this restricted system.
71:
6416:. Undecidability arises already for the existential sentences of
377:
The first axiom states that the constant 0 is a natural number:
2605:
This relation is stable under addition and multiplication: for
759:
The induction axiom is sometimes stated in the following form:
8565:
8512:. In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.).
7444:
7442:
196:
82:. These axioms have been used nearly unchanged in a number of
7992:(1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie".
7244:
by addition of the same element and commutativity, and hence
5990:{\displaystyle \forall x,y,z\ (x<y\Rightarrow x+z<y+z)}
4348:
can be encoded in terms of finite objects (for example, as a
42:
8987:
8927:
Podnieks, Karlis (2015-01-25). "3. First Order Arithmetic".
8455:(V ed.). Turin, Bocca frères, Ch. Clausen. p. 27.
8402:. Cambridge: Cambridge University Press. pp. 459–541.
7666:
of which the series of the natural numbers is one instance.
4053:) has an initial object; this initial object is known as a
834:
535:
is also a natural number. That is, the natural numbers are
48:
36:
6312:
satisfying this theory has an initial segment (ordered by
4293:
posed the problem of proving their consistency using only
3787:
replaced by its negation. Another such system consists of
825:
In Peano's original formulation, the induction axiom is a
8105:(3–4). Reprinted in English translation in 1990. Gödel's
8977:"What are numbers, and what is their meaning?: Dedekind"
6400:
Closely related to the above incompleteness result (via
6369:
is obtained by adding the first-order induction schema.
5907:{\displaystyle \forall x,y\ (x<y\lor x=y\lor y<x)}
875:, and can be shown to be unique using the Peano axioms.
572:) is a natural number. That is, the natural numbers are
8848:
The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs
8653:
The principles of arithmetic, presented by a new method
7659:{\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n},\ldots }
4572:{\displaystyle \forall x,y\ (S(x)=S(y)\Rightarrow x=y)}
328:
The Peano axioms define the arithmetical properties of
1428:
1379:
1342:
1268:
1228:
1191:
1141:
1107:
1070:
146:
The principles of arithmetic presented by a new method
7978:
Bertrand Russell's construction of the external world
7594:
7300:
7250:
7194:
7162:
7112:
7020:
6973:
6792:
6715:
6668:
6635:
6338:
6318:
6298:
6249:
6241:
by 1, which suggests that these numbers are discrete.
6178:
6096:
6005:
5926:
5849:
5793:
5710:
5657:
5582:
5475:
5414:
5323:
5258:
5163:
5098:
4883:
4821:{\displaystyle \forall x,y\ (x\cdot S(y)=x\cdot y+x)}
4760:
4712:
4634:
4586:
4508:
4457:
4382:
4139:
3363:
3298:
3122:
2611:
2478:
2400:
2364:
2202:
2173:
2144:
2103:
2083:
2054:
2010:
1981:
1949:
1869:
1833:
1730:
1646:
1608:
1588:
1532:
1500:
1028:
907:
347:
45:
8187:
Methodology, Epistemology, and Philosophy of Science
7751:
6501:
in 1933 provided an explicit construction of such a
3281:. The standard construction of the naturals, due to
3251:. This means that the second-order Peano axioms are
78:
presented by the 19th-century Italian mathematician
33:
7106:by distributivity and additive identity. Secondly,
6706:is a formula in the language of arithmetic so that
2915:Thus, by the strong induction principle, for every
227:. Unsourced material may be challenged and removed.
39:
8577:
8541:
8513:
7975:
7658:
7318:
7286:
7236:
7180:
7148:
7098:
7006:
6831:
6751:
6698:
6650:
6346:
6324:
6304:
6276:
6229:
6162:
6080:
5989:
5906:
5829:
5773:
5690:
5634:{\displaystyle \forall x\ (x+0=x\land x\cdot 0=0)}
5633:
5562:
5459:
5398:
5303:
5238:
5113:
5081:
4820:
4745:
4697:
4619:
4571:
4493:
4395:
4233:
4045:is said to satisfy the Dedekind–Peano axioms if US
3744:
3331:
3236:
2637:
2525:
2461:{\displaystyle a\cdot (b+c)=(a\cdot b)+(a\cdot c)}
2460:
2379:
2346:
2188:
2159:
2130:
2089:
2069:
2037:
1996:
1964:
1932:
1848:
1816:
1652:
1632:
1594:
1574:
1518:
1483:
1008:
358:
299:When Peano formulated his axioms, the language of
8691:(June 1957). "The Axiomatization of Arithmetic".
6486:When interpreted as a proof within a first-order
5074:
5031:
5024:
4951:
4908:
4901:
2907:, for otherwise it would be the least element of
8996:Creative Commons Attribution/Share-Alike License
8140:Lehrbuch der Arithmetik für höhere Lehranstalten
5460:{\displaystyle \forall x,y\ (x\cdot y=y\cdot x)}
5239:{\displaystyle \forall x,y,z\ ((x+y)+z=x+(y+z))}
1494:To prove commutativity of addition, first prove
865:with first-order induction, this is not possible
8930:What is Mathematics: Gödel's Theorem and Around
8580:A Formalization of Set Theory without Variables
8051:for details on English translations.: 173–198.
8002:. Reprinted in English translation in his 1969
7237:{\displaystyle x\cdot 0+x\cdot 0>x\cdot 0+0}
6467:
6288:The theory defined by these axioms is known as
4244:This is precisely the recursive definition of 0
8516:Studies in the Logic of Charles Sanders Peirce
7527:
7448:
7433:
7421:
3811:The Peano axioms can also be understood using
2526:{\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))}
1933:{\displaystyle a\cdot S(0)=a+(a\cdot 0)=a+0=a}
8648:Arithmetices principia, nova methodo exposita
8584:. AMS Colloquium Publications. Vol. 41.
8293:Bulletin of the American Mathematical Society
6389:(FOL) that are true in the standard model of
835:§ Peano arithmetic as first-order theory
155:Arithmetices principia, nova methodo exposita
109:provided by Peano axioms is commonly called
8:
8520:. Indiana University Press. pp. 43–52.
7934:: CS1 maint: multiple names: authors list (
7567:
7357:Random House Webster's Unabridged Dictionary
7149:{\displaystyle x\cdot 0=0\lor x\cdot 0>0}
6832:{\displaystyle M\vDash \phi (c,{\bar {a}}).}
4698:{\displaystyle \forall x,y\ (x+S(y)=S(x+y))}
3735:
3717:
3711:
3708:
3696:
3681:
3675:
3672:
3660:
3657:
3651:
3639:
3630:
3618:
3580:
3568:
3562:
3559:
3553:
3544:
3538:
3535:
3529:
3526:
3520:
3514:
3505:
3499:
3461:
3455:
3449:
3443:
3437:
3431:
3326:
3320:
841:Defining arithmetic operations and relations
659:of natural numbers. However, axioms 1–8 are
8299:(10). Translated by Winton, Maby: 437–479.
6752:{\displaystyle M\vDash \phi (b,{\bar {a}})}
6611:
4277:When the Peano axioms were first proposed,
3346:is defined as the intersection of all sets
3265:Set-theoretic definition of natural numbers
2831:—one can reason as follows. Let a nonempty
120:was not well appreciated until the work of
8895:: CS1 maint: location missing publisher (
8510:"3. Peirce's Axiomatization of Arithmetic"
7865:Courant Institute of Mathematical Sciences
6521:be the order type of the natural numbers,
6517:of a countable nonstandard model. Letting
6470:, there are other models as well (called "
6358:elements, while other elements are called
5830:{\displaystyle \forall x\ (\neg (x<x))}
5649:for multiplication (actually superfluous).
8304:
8122:
7775:
7644:
7625:
7612:
7599:
7593:
7530:, sections 2.3 (p. 464) and 4.1 (p. 471).
7369:
7299:
7249:
7193:
7161:
7111:
7019:
6972:
6812:
6811:
6791:
6735:
6734:
6714:
6682:
6681:
6667:
6637:
6636:
6634:
6340:
6339:
6337:
6317:
6297:
6248:
6177:
6095:
6004:
5925:
5848:
5792:
5709:
5656:
5581:
5474:
5413:
5322:
5257:
5162:
5100:
5099:
5097:
5073:
5072:
5058:
5057:
5030:
5029:
5023:
5022:
5008:
5007:
4969:
4968:
4950:
4949:
4926:
4925:
4907:
4906:
4900:
4899:
4888:
4887:
4882:
4866:in the language of Peano arithmetic, the
4759:
4711:
4633:
4585:
4507:
4456:
4387:
4381:
4206:
4167:
4140:
4138:
4096:is any other object, then the unique map
3364:
3362:
3297:
3206:
3177:
3157:
3137:
3123:
3121:
2808:This form of the induction axiom, called
2631:
2630:
2610:
2483:
2482:
2477:
2399:
2363:
2201:
2172:
2143:
2102:
2082:
2053:
2009:
1980:
1948:
1868:
1832:
1731:
1729:
1645:
1607:
1587:
1531:
1499:
1471:
1427:
1378:
1341:
1267:
1227:
1190:
1140:
1106:
1069:
1029:
1027:
996:
995:
937:
936:
908:
906:
349:
348:
346:
287:Learn how and when to remove this message
8986:This article incorporates material from
7889:What are and what should the numbers be?
7763:
7287:{\displaystyle x\cdot 0+0>x\cdot 0+0}
650:
7739:
7728:
7693:
7556:
7409:
7344:
7007:{\displaystyle \forall x\ (x\cdot 0=0)}
6941:
6525:be the order type of the integers, and
5691:{\displaystyle \forall x\ (x\cdot 1=x)}
4746:{\displaystyle \forall x\ (x\cdot 0=0)}
4494:{\displaystyle \forall x\ (0\neq S(x))}
3086:of the Peano axioms, there is a unique
140:of natural-number arithmetic. In 1888,
8888:
8875:(Second ed.). Mineola, New York.
8825:(6th ed.). Chapman and Hall/CRC.
8225:The Logical Foundations of Mathematics
8159:Henri Poincaré: A scientific biography
7927:
7814:
7799:
7552:
7397:
6779:that is greater than every element of
6354:. Elements in that segment are called
5467:, i.e., multiplication is commutative.
5406:, i.e., multiplication is associative.
5304:{\displaystyle \forall x,y\ (x+y=y+x)}
4407:Peano arithmetic as first-order theory
7717:
7705:
7579:
7472:
7460:
6621:be a nonstandard model of PA and let
6404:for FOL) it follows that there is no
3269:The Peano axioms can be derived from
2638:{\displaystyle a,b,c\in \mathbb {N} }
890:two natural numbers (two elements of
57:
7:
8143:. Verlag von Theod. Chr. Fr. Enslin.
7861:Computability. Notes by Barry Jacobs
7838:
7826:
7787:
7675:
7381:
6603:is a cut that is a proper subset of
6277:{\displaystyle \forall x\ (x\geq 0)}
863:directly using the axioms. However,
225:adding citations to reliable sources
8918:Internet Encyclopedia of Philosophy
8426:Mathematical Methods in Linguistics
7884:Was sind und was sollen die Zahlen?
6699:{\displaystyle \phi (x,{\bar {a}})}
6284:, i.e. zero is the minimum element.
3024:Was sind und was sollen die Zahlen?
816:) is true for every natural number
193:Historical second-order formulation
8260:Introduction to Mathematical Logic
8124:10.1111/j.1746-8361.1958.tb01464.x
7802:, VI.4.3, presenting a theorem of
7752:Partee, Ter Meulen & Wall 2012
6974:
6250:
6191:
6127:
6097:
6006:
5927:
5850:
5806:
5794:
5711:
5658:
5583:
5476:
5415:
5324:
5259:
5164:
5039:
4943:
4884:
4761:
4713:
4635:
4620:{\displaystyle \forall x\ (x+0=x)}
4587:
4509:
4458:
4448:, is sufficient for this purpose:
4384:
3446:
3434:
3425:
3416:
3378:
3030:. In particular, given two models
2849:Because 0 is the least element of
2358:Therefore, by the induction axiom
2354:, using commutativity of addition.
396:The next four axioms describe the
14:
8291:[Mathematical Problems].
8240:Derives the Peano axioms (called
7911:Beman, Wooster, Woodruff (1901).
6373:Undecidability and incompleteness
3807:Interpretation in category theory
3277:and axioms of set theory such as
3016:The Nature and Meaning of Numbers
2989:is a (necessarily infinite) set,
2577:if and only if there exists some
1602:. Using both results, then prove
6900:Non-standard model of arithmetic
6864:
6850:
6458:Non-standard model of arithmetic
6420:, due to the negative answer to
3332:{\displaystyle s(a)=a\cup \{a\}}
2973:of the Peano axioms is a triple
894:) to another one. It is defined
201:
29:
8471:American Journal of Mathematics
8306:10.1090/s0002-9904-1902-00923-3
7914:Essays on the Theory of Numbers
6424:, whose proof implies that all
6379:Gödel's incompleteness theorems
5914:, i.e., the ordering satisfies
5837:, i.e., the '<' operator is
5781:, i.e., the '<' operator is
1701:. The smallest group embedding
212:needs additional citations for
8994:, which is licensed under the
8981:Commentary on Dedekind's work.
8801:. New York: Elsevier Science.
8701:Association for Symbolic Logic
8666:Van Oosten, Jaap (June 1999).
8564:Derives the Peano axioms from
8396:Algebraic Methods in Semantics
7974:Fritz, Charles A. Jr. (1952).
7063:
7051:
7001:
6983:
6823:
6817:
6802:
6746:
6740:
6725:
6693:
6687:
6672:
6642:
6271:
6259:
6224:
6212:
6200:
6157:
6154:
6136:
6124:
6112:
6075:
6051:
6027:
5984:
5960:
5948:
5901:
5865:
5824:
5821:
5809:
5803:
5768:
5756:
5732:
5685:
5667:
5628:
5592:
5557:
5554:
5542:
5536:
5524:
5518:
5506:
5497:
5454:
5430:
5393:
5390:
5378:
5360:
5348:
5345:
5298:
5274:
5233:
5230:
5218:
5200:
5188:
5185:
5150:operation and the language of
5105:
5069:
5063:
5048:
5036:
5019:
5013:
5001:
4995:
4989:
4983:
4980:
4974:
4959:
4937:
4931:
4916:
4893:
4815:
4794:
4788:
4776:
4740:
4722:
4692:
4689:
4677:
4668:
4662:
4650:
4614:
4596:
4566:
4554:
4551:
4545:
4536:
4530:
4524:
4488:
4485:
4479:
4467:
4221:
4212:
4192:
4183:
4153:
4147:
3633:
3615:
3606:
3600:
3508:
3496:
3487:
3481:
3419:
3413:
3404:
3398:
3308:
3302:
3227:
3224:
3218:
3212:
3192:
3189:
3183:
3170:
3143:
3130:
2520:
2517:
2511:
2479:
2455:
2443:
2437:
2425:
2419:
2407:
2374:
2368:
2341:
2335:
2326:
2314:
2305:
2299:
2278:
2272:
2257:
2251:
2242:
2236:
2227:
2221:
2212:
2206:
2183:
2177:
2154:
2148:
2113:
2107:
2064:
2058:
2020:
2014:
1991:
1985:
1959:
1953:
1909:
1897:
1885:
1879:
1843:
1837:
1804:
1792:
1776:
1770:
1569:
1557:
1542:
1536:
1464:
1461:
1455:
1449:
1422:
1419:
1416:
1410:
1404:
1398:
1373:
1361:
1336:
1330:
1295:
1289:
1262:
1259:
1253:
1247:
1222:
1210:
1185:
1179:
1132:
1126:
1101:
1089:
1064:
1058:
987:
975:
962:
956:
754:contains every natural number.
116:The importance of formalizing
16:Axioms for the natural numbers
1:
9022:Formal theories of arithmetic
8746:The Journal of Symbolic Logic
8693:The Journal of Symbolic Logic
8586:American Mathematical Society
8222:Hatcher, William S. (2014) .
8006:, M. E. Szabo, ed.: 132–213.
7982:. New York, Humanities Press.
7181:{\displaystyle x\cdot 0>0}
6599:is closed under successor. A
4311:second incompleteness theorem
3833:, and define the category of
2167:is also the left identity of
2131:{\displaystyle S(0)\cdot a=a}
2038:{\displaystyle S(0)\cdot 0=0}
1575:{\displaystyle S(a)+b=S(a+b)}
359:{\displaystyle \mathbb {N} .}
8612:. Harvard University Press.
7539:For formal proofs, see e.g.
6402:Gödel's completeness theorem
6347:{\displaystyle \mathbb {N} }
5152:discretely ordered semirings
4073:is this initial object, and
3776:satisfies the Peano axioms.
3010:satisfies the axioms above.
8975:Burris, Stanley N. (2001).
8944:Encyclopedia of Mathematics
8262:. Hochschultext. Springer.
8195:10.1007/978-94-015-7676-5_8
6930:Typographical Number Theory
4868:first-order induction axiom
3342:The set of natural numbers
2953:being a nonempty subset of
2004:is the left identity of 0:
675:Axioms 1, 6, 7, 8 define a
334:, usually represented as a
9048:
8376:(4th ed.). Springer.
8318:Models of Peano arithmetic
8163:Princeton University Press
8135:Grassmann, Hermann Günther
8041:Monatshefte für Mathematik
7943:Ewald, William B. (1996).
7907:Two English translations:
7528:Meseguer & Goguen 1986
7319:{\displaystyle x\cdot 0=0}
6880:Foundations of mathematics
6658:is a tuple of elements of
6651:{\displaystyle {\bar {a}}}
6455:
6365:Finally, Peano arithmetic
5145:Equivalent axiomatizations
5114:{\displaystyle {\bar {y}}}
4835:and even decidable set of
4327:a proof of the consistency
4266:
3262:
793:) being true implies that
9027:Logic in computer science
8576:; Givant, Steven (1987).
4329:of Peano's axioms, using
3014:proved in his 1888 book,
1678:with identity element 0.
781:for every natural number
723:for every natural number
631:For every natural number
560:For every natural number
408:For every natural number
181:axiom with a first-order
8799:Handbook of Proof Theory
8467:"On the Logic of Number"
8289:"Mathematische Probleme"
7568:Tarski & Givant 1987
6905:Paris–Harrington theorem
6480:Löwenheim–Skolem theorem
4396:{\displaystyle \Pi _{1}}
4269:Hilbert's second problem
3759:together with 0 and the
2077:is the left identity of
583:For all natural numbers
521:is a natural number and
464:For all natural numbers
429:For all natural numbers
8423:; Wall, Robert (2012).
8346:Grundlagen Der Analysis
8322:Oxford University Press
8287:Hilbert, David (1902).
8115:Oxford University Press
7951:Oxford University Press
7492:Simon Fraser University
6925:Second-order arithmetic
6555:in a nonstandard model
6422:Hilbert's tenth problem
5570:, i.e., multiplication
5121:is an abbreviation for
4373:self-verifying theories
4371:. Curiously, there are
1827:It is easy to see that
1633:{\displaystyle a+b=b+a}
1582:, each by induction on
855:total (linear) ordering
762:
707:
558:
502:. That is, equality is
457:. That is, equality is
422:. That is, equality is
406:
380:
175:second-order arithmetic
8851:. Dover Publications.
8508:Shields, Paul (1997).
8452:Formulario Mathematico
8350:AMS Chelsea Publishing
8316:Kaye, Richard (1991).
8246:axiomatic set theories
7660:
7505:Mathematical Induction
7487:Mathematical Induction
7320:
7288:
7238:
7182:
7150:
7100:
7008:
6833:
6753:
6700:
6652:
6505:. On the other hand,
6466:satisfy the axioms of
6446:arithmetical hierarchy
6348:
6326:
6306:
6278:
6231:
6164:
6082:
5991:
5908:
5831:
5775:
5692:
5635:
5564:
5461:
5400:
5305:
5240:
5115:
5083:
4833:recursively enumerable
4822:
4747:
4699:
4621:
4573:
4495:
4397:
4235:
3746:
3333:
3238:
3023:
2845:has no least element.
2639:
2527:
2462:
2381:
2348:
2190:
2161:
2132:
2091:
2071:
2039:
1998:
1966:
1934:
1856:is the multiplicative
1850:
1818:
1654:
1634:
1596:
1576:
1520:
1485:
1010:
672:
390:Formulario mathematico
382:0 is a natural number.
360:
179:second-order induction
153:
134:Charles Sanders Peirce
8258:Hermes, Hans (1973).
7995:Mathematische Annalen
7661:
7321:
7289:
7239:
7183:
7151:
7101:
7009:
6910:Presburger arithmetic
6834:
6754:
6701:
6653:
6559:is a nonempty subset
6426:computably enumerable
6349:
6327:
6325:{\displaystyle \leq }
6307:
6279:
6232:
6165:
6083:
5992:
5909:
5832:
5776:
5693:
5645:for addition, and an
5636:
5565:
5462:
5401:
5306:
5241:
5116:
5084:
4823:
4748:
4700:
4622:
4574:
4496:
4398:
4331:transfinite induction
4303:twenty-three problems
4267:Further information:
4236:
4055:natural number object
3835:pointed unary systems
3747:
3334:
3273:constructions of the
3239:
2961:has a least element.
2640:
2528:
2463:
2382:
2349:
2191:
2162:
2133:
2092:
2072:
2040:
1999:
1967:
1935:
1851:
1819:
1655:
1635:
1597:
1577:
1521:
1519:{\displaystyle 0+b=b}
1486:
1011:
654:
361:
64:Dedekind–Peano axioms
62:), also known as the
8843:Smullyan, Raymond M.
8673:. Utrecht University
8635:Letter to Keferstein
8606:Van Heijenoort, Jean
8544:Axiomatic Set Theory
8189:. pp. 167–192.
7953:. pp. 787–832.
7592:
7298:
7248:
7192:
7160:
7110:
7018:
6971:
6790:
6713:
6666:
6633:
6571:is downward closed (
6507:Tennenbaum's theorem
6412:is an example of an
6336:
6316:
6296:
6247:
6176:
6094:
6003:
5924:
5847:
5791:
5708:
5655:
5580:
5473:
5412:
5321:
5311:, i.e., addition is
5256:
5246:, i.e., addition is
5161:
5096:
4881:
4758:
4710:
4632:
4584:
4506:
4455:
4380:
4137:
3779:Peano arithmetic is
3361:
3296:
3289:on sets defined as:
3259:Set-theoretic models
3120:
2873:, suppose for every
2841:be given and assume
2749:) is true for every
2696:Thus, the structure
2609:
2476:
2398:
2380:{\displaystyle S(0)}
2362:
2200:
2189:{\displaystyle S(a)}
2171:
2160:{\displaystyle S(0)}
2142:
2101:
2081:
2070:{\displaystyle S(0)}
2052:
2008:
1997:{\displaystyle S(0)}
1979:
1965:{\displaystyle S(0)}
1947:
1867:
1849:{\displaystyle S(0)}
1831:
1728:
1644:
1606:
1586:
1530:
1498:
1026:
905:
713:is a set such that:
677:unary representation
345:
221:improve this article
9017:Mathematical axioms
7869:New York University
7449:Van Heijenoort 1967
7434:Van Heijenoort 1967
7422:Van Heijenoort 1967
6920:Robinson arithmetic
6890:Goodstein's theorem
6625:be a proper cut of
6615: —
6476:compactness theorem
6472:non-standard models
6462:Although the usual
6395:Robinson arithmetic
5702:for multiplication.
5641:, i.e., zero is an
4839:. For each formula
4446:Robinson arithmetic
4435:successor operation
3801:axiom of adjunction
3795:, existence of the
3755:and so on. The set
886:is a function that
368:non-logical symbols
126:successor operation
9032:Mathematical logic
9012:1889 introductions
8958:Weisstein, Eric W.
8933:. pp. 93–121.
8819:Mendelson, Elliott
8550:Dover Publications
8463:Peirce, C. S.
8372:(December 1997) .
8370:Mendelson, Elliott
8057:10.1007/bf01700692
8012:10.1007/bf01565428
7922:Dover Publications
7656:
7517:Harvard University
7510:2 May 2013 at the
7502:Gerardo con Diaz,
7316:
7284:
7234:
7178:
7146:
7096:
7004:
6872:Mathematics portal
6829:
6749:
6696:
6648:
6613:
6452:Nonstandard models
6414:undecidable theory
6344:
6322:
6302:
6274:
6227:
6160:
6078:
5987:
5904:
5827:
5771:
5698:, i.e., one is an
5688:
5631:
5560:
5457:
5396:
5301:
5236:
5111:
5079:
4818:
4743:
4695:
4617:
4569:
4491:
4393:
4231:
4229:
3789:general set theory
3761:successor function
3742:
3740:
3329:
3234:
3232:
2853:, it must be that
2635:
2523:
2458:
2377:
2344:
2186:
2157:
2128:
2087:
2067:
2035:
1994:
1962:
1930:
1846:
1814:
1812:
1650:
1630:
1592:
1572:
1516:
1481:
1479:
1432:
1383:
1346:
1272:
1232:
1195:
1145:
1111:
1074:
1006:
1004:
873:second-order logic
827:second-order axiom
701:axiom of induction
673:
356:
301:mathematical logic
21:mathematical logic
8882:978-0-486-49073-1
8858:978-0-486-49705-1
8845:(December 2013).
8832:978-1-4822-3772-6
8808:978-0-444-89840-1
8631:Dedekind, Richard
8619:978-0-674-32449-7
8595:978-0-8218-1041-5
8436:978-94-009-2213-6
8421:Ter Meulen, Alice
8419:Partee, Barbara;
8409:978-0-521-26793-9
8383:978-0-412-80830-2
8359:978-0-8284-0141-8
8235:978-1-4831-8963-5
8204:978-90-481-8389-0
8183:Harsanyi, John C.
8172:978-0-691-15271-4
7960:978-0-19-853271-2
7879:Dedekind, Richard
7841:, pp. 70ff..
7790:, pp. 16–18.
6982:
6915:Skolem arithmetic
6858:Philosophy portal
6820:
6743:
6690:
6645:
6503:nonstandard model
6387:first-order logic
6305:{\displaystyle M}
6258:
6199:
6135:
6111:
6026:
5947:
5864:
5802:
5731:
5666:
5647:absorbing element
5591:
5496:
5429:
5344:
5273:
5184:
5108:
5066:
5016:
4977:
4934:
4896:
4775:
4721:
4649:
4595:
4523:
4466:
4413:first-order logic
3849:The objects of US
3785:axiom of infinity
2533:is a commutative
2090:{\displaystyle a}
1653:{\displaystyle b}
1595:{\displaystyle b}
1474:
1431:
1382:
1345:
1271:
1231:
1194:
1144:
1110:
1073:
999:
940:
297:
296:
289:
271:
122:Hermann Grassmann
59:[peˈaːno]
9039:
8980:
8971:
8970:
8961:"Peano's Axioms"
8952:
8934:
8922:
8913:"Henri Poincaré"
8900:
8894:
8886:
8862:
8836:
8812:
8785:
8743:
8728:
8682:
8680:
8678:
8672:
8656:
8638:
8623:
8599:
8583:
8563:
8547:
8531:
8519:
8502:
8456:
8440:
8413:
8401:
8387:
8363:
8335:
8310:
8308:
8281:
8239:
8216:
8176:
8144:
8128:
8126:
8111:Solomon Feferman
8082:
8080:
8079:
8073:
8067:. Archived from
8038:
8023:
7990:Gentzen, Gerhard
7983:
7981:
7964:
7939:
7933:
7925:
7919:
7904:
7902:
7900:
7894:
7872:
7842:
7836:
7830:
7824:
7818:
7812:
7806:
7797:
7791:
7785:
7779:
7773:
7767:
7761:
7755:
7749:
7743:
7737:
7731:
7726:
7720:
7715:
7709:
7703:
7697:
7691:
7685:
7673:
7667:
7665:
7663:
7662:
7657:
7649:
7648:
7630:
7629:
7617:
7616:
7604:
7603:
7577:
7571:
7565:
7559:
7550:
7544:
7537:
7531:
7525:
7519:
7500:
7494:
7482:
7476:
7470:
7464:
7458:
7452:
7446:
7437:
7431:
7425:
7419:
7413:
7407:
7401:
7395:
7389:
7379:
7373:
7367:
7361:
7349:
7327:
7325:
7323:
7322:
7317:
7293:
7291:
7290:
7285:
7243:
7241:
7240:
7235:
7187:
7185:
7184:
7179:
7156:by Axiom 15. If
7155:
7153:
7152:
7147:
7105:
7103:
7102:
7097:
7013:
7011:
7010:
7005:
6980:
6965:
6959:
6955:
6949:
6946:
6874:
6869:
6868:
6860:
6855:
6854:
6853:
6838:
6836:
6835:
6830:
6822:
6821:
6813:
6771:Then there is a
6758:
6756:
6755:
6750:
6745:
6744:
6736:
6705:
6703:
6702:
6697:
6692:
6691:
6683:
6657:
6655:
6654:
6649:
6647:
6646:
6638:
6616:
6542:
6430:diophantine sets
6381:, the theory of
6353:
6351:
6350:
6345:
6343:
6332:) isomorphic to
6331:
6329:
6328:
6323:
6311:
6309:
6308:
6303:
6283:
6281:
6280:
6275:
6256:
6236:
6234:
6233:
6228:
6197:
6169:
6167:
6166:
6161:
6133:
6109:
6087:
6085:
6084:
6079:
6024:
5996:
5994:
5993:
5988:
5945:
5913:
5911:
5910:
5905:
5862:
5836:
5834:
5833:
5828:
5800:
5780:
5778:
5777:
5772:
5729:
5697:
5695:
5694:
5689:
5664:
5640:
5638:
5637:
5632:
5589:
5569:
5567:
5566:
5561:
5494:
5466:
5464:
5463:
5458:
5427:
5405:
5403:
5402:
5397:
5342:
5310:
5308:
5307:
5302:
5271:
5245:
5243:
5242:
5237:
5182:
5120:
5118:
5117:
5112:
5110:
5109:
5101:
5088:
5086:
5085:
5080:
5078:
5077:
5068:
5067:
5059:
5035:
5034:
5028:
5027:
5018:
5017:
5009:
4979:
4978:
4970:
4955:
4954:
4936:
4935:
4927:
4912:
4911:
4905:
4904:
4898:
4897:
4889:
4874:is the sentence
4865:
4827:
4825:
4824:
4819:
4773:
4752:
4750:
4749:
4744:
4719:
4704:
4702:
4701:
4696:
4647:
4626:
4624:
4623:
4618:
4593:
4578:
4576:
4575:
4570:
4521:
4500:
4498:
4497:
4492:
4464:
4402:
4400:
4399:
4394:
4392:
4391:
4279:Bertrand Russell
4240:
4238:
4237:
4232:
4230:
4211:
4210:
4172:
4171:
4129:
4095:
4072:
4036:
4010:
3992:
3925:
3906:
3884:is an object of
3879:
3775:
3751:
3749:
3748:
3743:
3741:
3338:
3336:
3335:
3330:
3283:John von Neumann
3243:
3241:
3240:
3235:
3233:
3211:
3210:
3182:
3181:
3162:
3161:
3142:
3141:
3112:
3085:
3057:
3009:
2995:
2984:
2945:
2934:
2924:
2906:
2892:
2882:
2872:
2859:
2840:
2810:strong induction
2793:
2768:
2758:
2740:
2728:(0) is true, and
2706:ordered semiring
2703:
2702:, +, ·, 1, 0, ≤)
2654:
2644:
2642:
2641:
2636:
2634:
2600:
2586:
2576:
2566:
2532:
2530:
2529:
2524:
2486:
2467:
2465:
2464:
2459:
2389:distributes over
2386:
2384:
2383:
2378:
2353:
2351:
2350:
2345:
2195:
2193:
2192:
2187:
2166:
2164:
2163:
2158:
2137:
2135:
2134:
2129:
2096:
2094:
2093:
2088:
2076:
2074:
2073:
2068:
2044:
2042:
2041:
2036:
2003:
2001:
2000:
1995:
1971:
1969:
1968:
1963:
1939:
1937:
1936:
1931:
1855:
1853:
1852:
1847:
1823:
1821:
1820:
1815:
1813:
1685:
1670:
1659:
1657:
1656:
1651:
1640:by induction on
1639:
1637:
1636:
1631:
1601:
1599:
1598:
1593:
1581:
1579:
1578:
1573:
1525:
1523:
1522:
1517:
1490:
1488:
1487:
1482:
1480:
1477:
1475:
1472:
1433:
1429:
1388:
1384:
1380:
1351:
1347:
1343:
1301:
1273:
1269:
1237:
1233:
1229:
1200:
1196:
1192:
1150:
1146:
1142:
1116:
1112:
1108:
1079:
1075:
1071:
1015:
1013:
1012:
1007:
1005:
1001:
1000:
997:
942:
941:
938:
866:
778:(0) is true, and
645:
619:
609:
530:
501:
456:
446:
421:
365:
363:
362:
357:
352:
292:
285:
281:
278:
272:
270:
229:
205:
197:
187:Peano arithmetic
142:Richard Dedekind
111:Peano arithmetic
84:metamathematical
68:Peano postulates
61:
55:
54:
51:
50:
47:
44:
41:
38:
35:
9047:
9046:
9042:
9041:
9040:
9038:
9037:
9036:
9002:
9001:
8974:
8956:
8955:
8937:
8926:
8910:
8907:
8887:
8883:
8867:
8859:
8841:
8833:
8817:
8809:
8796:
8793:
8791:Further reading
8788:
8758:10.2307/2695030
8741:
8735:Willard, Dan E.
8733:
8709:10.2307/2964176
8687:
8676:
8674:
8670:
8665:
8643:Peano, Giuseppe
8641:
8629:
8620:
8604:
8596:
8572:
8560:
8538:Suppes, Patrick
8536:
8528:
8507:
8483:10.2307/2369151
8461:
8447:Peano, Giuseppe
8445:
8437:
8418:
8410:
8399:
8392:
8384:
8368:
8360:
8340:
8332:
8315:
8286:
8270:
8257:
8250:category theory
8244:) from several
8236:
8221:
8205:
8181:
8173:
8165:. p. 133.
8149:
8133:
8107:Collected Works
8087:
8077:
8075:
8071:
8036:
8028:
8004:Collected works
7988:
7973:
7961:
7942:
7926:
7917:
7910:
7898:
7896:
7892:
7877:
7855:
7851:
7846:
7845:
7837:
7833:
7829:, Section 11.3.
7825:
7821:
7813:
7809:
7798:
7794:
7786:
7782:
7774:
7770:
7764:Harsanyi (1983)
7762:
7758:
7750:
7746:
7738:
7734:
7727:
7723:
7716:
7712:
7704:
7700:
7692:
7688:
7682:
7674:
7670:
7640:
7621:
7608:
7595:
7590:
7589:
7586:
7578:
7574:
7566:
7562:
7551:
7547:
7538:
7534:
7526:
7522:
7512:Wayback Machine
7501:
7497:
7483:
7479:
7471:
7467:
7459:
7455:
7447:
7440:
7432:
7428:
7420:
7416:
7408:
7404:
7396:
7392:
7386:integral domain
7380:
7376:
7368:
7364:
7350:
7346:
7341:
7336:
7331:
7330:
7296:
7295:
7246:
7245:
7190:
7189:
7158:
7157:
7108:
7107:
7016:
7015:
6969:
6968:
6966:
6962:
6956:
6952:
6947:
6943:
6938:
6885:Frege's theorem
6870:
6863:
6856:
6851:
6849:
6846:
6841:
6788:
6787:
6711:
6710:
6664:
6663:
6631:
6630:
6629:. Suppose that
6614:
6612:Overspill lemma
6549:
6530:
6464:natural numbers
6460:
6454:
6442:quantifier rank
6375:
6334:
6333:
6314:
6313:
6294:
6293:
6245:
6244:
6174:
6173:
6092:
6091:
6001:
6000:
5922:
5921:
5845:
5844:
5789:
5788:
5706:
5705:
5653:
5652:
5578:
5577:
5471:
5470:
5410:
5409:
5319:
5318:
5254:
5253:
5159:
5158:
5147:
5136:
5127:
5094:
5093:
4879:
4878:
4863:
4854:
4840:
4756:
4755:
4708:
4707:
4630:
4629:
4582:
4581:
4504:
4503:
4453:
4452:
4409:
4383:
4378:
4377:
4361:Gentzen's proof
4347:
4342:
4323:Gerhard Gentzen
4297:methods as the
4275:
4265:
4258:
4249:
4228:
4227:
4202:
4195:
4177:
4176:
4163:
4156:
4135:
4134:
4127:
4118:
4097:
4093:
4084:
4074:
4062:
4048:
4032:
4023:
4012:
4009:
4003:
3994:
3980:
3974:
3965:
3955:
3946:
3916:
3908:
3901:
3895:
3889:
3877:
3868:
3858:
3852:
3840:
3832:
3825:terminal object
3813:category theory
3809:
3763:
3739:
3738:
3590:
3584:
3583:
3471:
3465:
3464:
3388:
3382:
3381:
3371:
3359:
3358:
3294:
3293:
3275:natural numbers
3267:
3261:
3231:
3230:
3202:
3195:
3173:
3164:
3163:
3153:
3146:
3133:
3118:
3117:
3111:
3102:
3090:
3083:
3074:
3068:
3059:
3055:
3046:
3040:
3031:
2997:
2990:
2974:
2967:
2936:
2926:
2916:
2894:
2884:
2874:
2864:
2854:
2832:
2785:
2784:then for every
2760:
2750:
2732:
2697:
2646:
2607:
2606:
2588:
2578:
2568:
2554:
2543:
2474:
2473:
2396:
2395:
2360:
2359:
2198:
2197:
2169:
2168:
2140:
2139:
2099:
2098:
2079:
2078:
2050:
2049:
2006:
2005:
1977:
1976:
1945:
1944:
1865:
1864:
1829:
1828:
1811:
1810:
1779:
1758:
1757:
1744:
1726:
1725:
1715:
1679:
1664:
1642:
1641:
1604:
1603:
1584:
1583:
1528:
1527:
1496:
1495:
1478:
1476:
1468:
1467:
1425:
1386:
1385:
1376:
1349:
1348:
1339:
1314:
1302:
1299:
1298:
1265:
1235:
1234:
1225:
1198:
1197:
1188:
1163:
1151:
1148:
1147:
1138:
1114:
1113:
1104:
1077:
1076:
1067:
1042:
1024:
1023:
1003:
1002:
993:
965:
944:
943:
934:
921:
903:
902:
881:
864:
843:
823:
757:
649:
636:
611:
592:
542:
539:under equality.
522:
493:
448:
438:
413:
385:
343:
342:
331:natural numbers
310:Begriffsschrift
293:
282:
276:
273:
230:
228:
218:
206:
195:
76:natural numbers
32:
28:
17:
12:
11:
5:
9045:
9043:
9035:
9034:
9029:
9024:
9019:
9014:
9004:
9003:
8983:
8982:
8972:
8953:
8939:"Peano axioms"
8935:
8924:
8911:Murzi, Mauro.
8906:
8905:External links
8903:
8902:
8901:
8881:
8869:Takeuti, Gaisi
8864:
8863:
8857:
8838:
8837:
8831:
8821:(June 2015) .
8814:
8813:
8807:
8792:
8789:
8787:
8786:
8752:(2): 536–596.
8730:
8729:
8684:
8683:
8662:
8661:
8660:
8659:
8658:
8657:
8639:
8618:
8601:
8600:
8594:
8574:Tarski, Alfred
8569:
8568:
8558:
8533:
8532:
8526:
8504:
8503:
8458:
8457:
8442:
8441:
8435:
8415:
8414:
8408:
8389:
8388:
8382:
8365:
8364:
8358:
8342:Landau, Edmund
8337:
8336:
8330:
8312:
8311:
8283:
8282:
8268:
8254:
8253:
8234:
8218:
8217:
8203:
8178:
8177:
8171:
8155:"The Essayist"
8146:
8145:
8130:
8129:
8084:
8083:
8025:
8024:
7985:
7984:
7970:
7969:
7968:
7967:
7966:
7965:
7959:
7940:
7874:
7873:
7852:
7850:
7847:
7844:
7843:
7831:
7819:
7807:
7804:Thoralf Skolem
7792:
7780:
7778:, p. 155.
7776:Mendelson 1997
7768:
7756:
7754:, p. 215.
7744:
7732:
7721:
7710:
7698:
7686:
7668:
7655:
7652:
7647:
7643:
7639:
7636:
7633:
7628:
7624:
7620:
7615:
7611:
7607:
7602:
7598:
7572:
7570:, Section 7.6.
7560:
7545:
7532:
7520:
7495:
7477:
7465:
7453:
7438:
7426:
7414:
7402:
7390:
7374:
7370:Grassmann 1861
7362:
7343:
7342:
7340:
7337:
7335:
7332:
7329:
7328:
7315:
7312:
7309:
7306:
7303:
7283:
7280:
7277:
7274:
7271:
7268:
7265:
7262:
7259:
7256:
7253:
7233:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7177:
7174:
7171:
7168:
7165:
7145:
7142:
7139:
7136:
7133:
7130:
7127:
7124:
7121:
7118:
7115:
7095:
7092:
7089:
7086:
7083:
7080:
7077:
7074:
7071:
7068:
7065:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7038:
7035:
7032:
7029:
7026:
7023:
7003:
7000:
6997:
6994:
6991:
6988:
6985:
6979:
6976:
6960:
6950:
6940:
6939:
6937:
6934:
6933:
6932:
6927:
6922:
6917:
6912:
6907:
6902:
6897:
6892:
6887:
6882:
6876:
6875:
6861:
6845:
6842:
6840:
6839:
6828:
6825:
6819:
6816:
6810:
6807:
6804:
6801:
6798:
6795:
6769:
6768:
6748:
6742:
6739:
6733:
6730:
6727:
6724:
6721:
6718:
6695:
6689:
6686:
6680:
6677:
6674:
6671:
6644:
6641:
6609:
6548:
6545:
6456:Main article:
6453:
6450:
6436:. Formulas of
6374:
6371:
6342:
6321:
6301:
6286:
6285:
6273:
6270:
6267:
6264:
6261:
6255:
6252:
6242:
6226:
6223:
6220:
6217:
6214:
6211:
6208:
6205:
6202:
6196:
6193:
6190:
6187:
6184:
6181:
6171:
6159:
6156:
6153:
6150:
6147:
6144:
6141:
6138:
6132:
6129:
6126:
6123:
6120:
6117:
6114:
6108:
6105:
6102:
6099:
6089:
6077:
6074:
6071:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6023:
6020:
6017:
6014:
6011:
6008:
5998:
5986:
5983:
5980:
5977:
5974:
5971:
5968:
5965:
5962:
5959:
5956:
5953:
5950:
5944:
5941:
5938:
5935:
5932:
5929:
5919:
5903:
5900:
5897:
5894:
5891:
5888:
5885:
5882:
5879:
5876:
5873:
5870:
5867:
5861:
5858:
5855:
5852:
5842:
5826:
5823:
5820:
5817:
5814:
5811:
5808:
5805:
5799:
5796:
5786:
5770:
5767:
5764:
5761:
5758:
5755:
5752:
5749:
5746:
5743:
5740:
5737:
5734:
5728:
5725:
5722:
5719:
5716:
5713:
5703:
5687:
5684:
5681:
5678:
5675:
5672:
5669:
5663:
5660:
5650:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5609:
5606:
5603:
5600:
5597:
5594:
5588:
5585:
5575:
5574:over addition.
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5538:
5535:
5532:
5529:
5526:
5523:
5520:
5517:
5514:
5511:
5508:
5505:
5502:
5499:
5493:
5490:
5487:
5484:
5481:
5478:
5468:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5426:
5423:
5420:
5417:
5407:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5359:
5356:
5353:
5350:
5347:
5341:
5338:
5335:
5332:
5329:
5326:
5316:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5279:
5276:
5270:
5267:
5264:
5261:
5251:
5235:
5232:
5229:
5226:
5223:
5220:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5181:
5178:
5175:
5172:
5169:
5166:
5146:
5143:
5132:
5125:
5107:
5104:
5090:
5089:
5076:
5071:
5065:
5062:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5033:
5026:
5021:
5015:
5012:
5006:
5003:
5000:
4997:
4994:
4991:
4988:
4985:
4982:
4976:
4973:
4967:
4964:
4961:
4958:
4953:
4948:
4945:
4942:
4939:
4933:
4930:
4924:
4921:
4918:
4915:
4910:
4903:
4895:
4892:
4886:
4859:
4852:
4829:
4828:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4772:
4769:
4766:
4763:
4753:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4718:
4715:
4705:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4646:
4643:
4640:
4637:
4627:
4616:
4613:
4610:
4607:
4604:
4601:
4598:
4592:
4589:
4579:
4568:
4565:
4562:
4559:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4520:
4517:
4514:
4511:
4501:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4463:
4460:
4408:
4405:
4390:
4386:
4350:Turing machine
4345:
4340:
4283:Henri Poincaré
4264:
4261:
4254:
4245:
4242:
4241:
4226:
4223:
4220:
4217:
4214:
4209:
4205:
4201:
4198:
4196:
4194:
4191:
4188:
4185:
4182:
4179:
4178:
4175:
4170:
4166:
4162:
4159:
4157:
4155:
4152:
4149:
4146:
4143:
4142:
4123:
4114:
4089:
4080:
4046:
4039:
4038:
4028:
4019:
4005:
3999:
3970:
3961:
3951:
3942:
3931:
3912:
3897:
3891:
3873:
3864:
3857:) are triples
3850:
3845:) as follows:
3838:
3828:
3808:
3805:
3793:extensionality
3781:equiconsistent
3753:
3752:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3591:
3589:
3586:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3472:
3470:
3467:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3389:
3387:
3384:
3383:
3380:
3377:
3374:
3372:
3370:
3367:
3366:
3340:
3339:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3263:Main article:
3260:
3257:
3245:
3244:
3229:
3226:
3223:
3220:
3217:
3214:
3209:
3205:
3201:
3198:
3196:
3194:
3191:
3188:
3185:
3180:
3176:
3172:
3169:
3166:
3165:
3160:
3156:
3152:
3149:
3147:
3145:
3140:
3136:
3132:
3129:
3126:
3125:
3107:
3098:
3079:
3070:
3064:
3051:
3042:
3036:
2966:
2963:
2913:
2912:
2861:
2806:
2805:
2804:
2803:
2782:
2729:
2694:
2693:
2675:
2633:
2629:
2626:
2623:
2620:
2617:
2614:
2603:
2602:
2542:
2539:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2485:
2481:
2470:
2469:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2376:
2373:
2370:
2367:
2356:
2355:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2185:
2182:
2179:
2176:
2156:
2153:
2150:
2147:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2086:
2066:
2063:
2060:
2057:
2046:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
1993:
1990:
1987:
1984:
1961:
1958:
1955:
1952:
1941:
1940:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1858:right identity
1845:
1842:
1839:
1836:
1825:
1824:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1780:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1759:
1756:
1753:
1750:
1747:
1745:
1743:
1740:
1737:
1734:
1733:
1719:multiplication
1714:
1713:Multiplication
1711:
1649:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1591:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1515:
1512:
1509:
1506:
1503:
1492:
1491:
1470:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1426:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1389:
1387:
1377:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1352:
1350:
1340:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1315:
1313:
1310:
1307:
1304:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1266:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1238:
1236:
1226:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1201:
1199:
1189:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1164:
1162:
1159:
1156:
1153:
1152:
1149:
1139:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1117:
1115:
1105:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1080:
1078:
1068:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1043:
1041:
1038:
1035:
1032:
1031:
1017:
1016:
994:
992:
989:
986:
983:
980:
977:
974:
971:
968:
966:
964:
961:
958:
955:
952:
949:
946:
945:
935:
933:
930:
927:
924:
922:
920:
917:
914:
911:
910:
880:
877:
851:multiplication
842:
839:
822:
821:
807:
806:
779:
761:
756:
755:
749:
748:
721:
706:
648:
647:
629:
581:
557:
541:
540:
507:
462:
427:
405:
393:include zero.
384:
383:
379:
355:
351:
305:set membership
295:
294:
236:"Peano axioms"
209:
207:
200:
194:
191:
138:axiomatization
103:axiomatization
80:Giuseppe Peano
15:
13:
10:
9:
6:
4:
3:
2:
9044:
9033:
9030:
9028:
9025:
9023:
9020:
9018:
9015:
9013:
9010:
9009:
9007:
9000:
8999:
8997:
8993:
8989:
8978:
8973:
8968:
8967:
8962:
8959:
8954:
8950:
8946:
8945:
8940:
8936:
8932:
8931:
8925:
8920:
8919:
8914:
8909:
8908:
8904:
8898:
8892:
8884:
8878:
8874:
8870:
8866:
8865:
8860:
8854:
8850:
8849:
8844:
8840:
8839:
8834:
8828:
8824:
8820:
8816:
8815:
8810:
8804:
8800:
8795:
8794:
8790:
8783:
8779:
8775:
8771:
8767:
8763:
8759:
8755:
8751:
8747:
8740:
8736:
8732:
8731:
8726:
8722:
8718:
8714:
8710:
8706:
8702:
8698:
8694:
8690:
8686:
8685:
8669:
8664:
8663:
8654:
8650:
8649:
8644:
8640:
8636:
8632:
8628:
8627:
8625:
8624:
8621:
8615:
8611:
8607:
8603:
8602:
8597:
8591:
8587:
8582:
8581:
8575:
8571:
8570:
8567:
8561:
8559:0-486-61630-4
8555:
8551:
8546:
8545:
8539:
8535:
8534:
8529:
8527:0-253-33020-3
8523:
8518:
8517:
8511:
8506:
8505:
8500:
8496:
8492:
8488:
8484:
8480:
8476:
8472:
8468:
8464:
8460:
8459:
8454:
8453:
8448:
8444:
8443:
8438:
8432:
8428:
8427:
8422:
8417:
8416:
8411:
8405:
8398:
8397:
8391:
8390:
8385:
8379:
8375:
8371:
8367:
8366:
8361:
8355:
8351:
8347:
8343:
8339:
8338:
8333:
8331:0-19-853213-X
8327:
8323:
8319:
8314:
8313:
8307:
8302:
8298:
8294:
8290:
8285:
8284:
8279:
8275:
8271:
8269:3-540-05819-2
8265:
8261:
8256:
8255:
8251:
8247:
8243:
8237:
8231:
8227:
8226:
8220:
8219:
8214:
8210:
8206:
8200:
8196:
8192:
8188:
8184:
8180:
8179:
8174:
8168:
8164:
8160:
8156:
8152:
8148:
8147:
8142:
8141:
8136:
8132:
8131:
8125:
8120:
8116:
8113:et al., eds.
8112:
8108:
8104:
8100:
8099:
8094:
8090:
8086:
8085:
8074:on 2018-04-11
8070:
8066:
8062:
8058:
8054:
8050:
8046:
8042:
8035:
8031:
8027:
8026:
8021:
8017:
8013:
8009:
8005:
8001:
7997:
7996:
7991:
7987:
7986:
7980:
7979:
7972:
7971:
7962:
7956:
7952:
7948:
7947:
7941:
7937:
7931:
7923:
7916:
7915:
7909:
7908:
7906:
7905:
7890:
7886:
7885:
7880:
7876:
7875:
7870:
7866:
7862:
7858:
7857:Davis, Martin
7854:
7853:
7848:
7840:
7835:
7832:
7828:
7823:
7820:
7816:
7811:
7808:
7805:
7801:
7796:
7793:
7789:
7784:
7781:
7777:
7772:
7769:
7765:
7760:
7757:
7753:
7748:
7745:
7741:
7736:
7733:
7730:
7725:
7722:
7719:
7714:
7711:
7707:
7702:
7699:
7695:
7690:
7687:
7681:
7677:
7672:
7669:
7653:
7650:
7645:
7641:
7637:
7634:
7631:
7626:
7622:
7618:
7613:
7609:
7605:
7600:
7596:
7585:
7581:
7576:
7573:
7569:
7564:
7561:
7558:
7554:
7549:
7546:
7542:
7536:
7533:
7529:
7524:
7521:
7518:
7514:
7513:
7509:
7506:
7499:
7496:
7493:
7489:
7488:
7481:
7478:
7475:, p. 27.
7474:
7469:
7466:
7462:
7457:
7454:
7451:, p. 83.
7450:
7445:
7443:
7439:
7435:
7430:
7427:
7424:, p. 94.
7423:
7418:
7415:
7411:
7406:
7403:
7399:
7394:
7391:
7387:
7383:
7378:
7375:
7371:
7366:
7363:
7359:
7358:
7353:
7348:
7345:
7338:
7333:
7313:
7310:
7307:
7304:
7301:
7281:
7278:
7275:
7272:
7269:
7266:
7263:
7260:
7257:
7254:
7251:
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7210:
7207:
7204:
7201:
7198:
7195:
7175:
7172:
7169:
7166:
7163:
7143:
7140:
7137:
7134:
7131:
7128:
7125:
7122:
7119:
7116:
7113:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7060:
7057:
7054:
7048:
7045:
7042:
7039:
7036:
7033:
7030:
7027:
7024:
7021:
6998:
6995:
6992:
6989:
6986:
6977:
6964:
6961:
6954:
6951:
6945:
6942:
6935:
6931:
6928:
6926:
6923:
6921:
6918:
6916:
6913:
6911:
6908:
6906:
6903:
6901:
6898:
6896:
6893:
6891:
6888:
6886:
6883:
6881:
6878:
6877:
6873:
6867:
6862:
6859:
6848:
6843:
6826:
6814:
6808:
6805:
6799:
6796:
6793:
6786:
6785:
6784:
6782:
6778:
6774:
6766:
6762:
6737:
6731:
6728:
6722:
6719:
6716:
6709:
6708:
6707:
6684:
6678:
6675:
6669:
6661:
6639:
6628:
6624:
6620:
6608:
6606:
6602:
6598:
6594:
6590:
6586:
6582:
6578:
6574:
6570:
6566:
6562:
6558:
6554:
6546:
6544:
6541:
6537:
6533:
6528:
6524:
6520:
6516:
6512:
6508:
6504:
6500:
6495:
6493:
6489:
6484:
6481:
6477:
6473:
6469:
6465:
6459:
6451:
6449:
6447:
6443:
6439:
6435:
6431:
6427:
6423:
6419:
6415:
6411:
6407:
6403:
6398:
6396:
6392:
6388:
6384:
6380:
6377:According to
6372:
6370:
6368:
6363:
6361:
6357:
6319:
6299:
6291:
6268:
6265:
6262:
6253:
6243:
6240:
6221:
6218:
6215:
6209:
6206:
6203:
6194:
6188:
6185:
6182:
6179:
6172:
6151:
6148:
6145:
6142:
6139:
6130:
6121:
6118:
6115:
6106:
6103:
6100:
6090:
6072:
6069:
6066:
6063:
6060:
6057:
6054:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6021:
6018:
6015:
6012:
6009:
5999:
5981:
5978:
5975:
5972:
5969:
5966:
5963:
5957:
5954:
5951:
5942:
5939:
5936:
5933:
5930:
5920:
5917:
5898:
5895:
5892:
5889:
5886:
5883:
5880:
5877:
5874:
5871:
5868:
5859:
5856:
5853:
5843:
5840:
5818:
5815:
5812:
5797:
5787:
5784:
5765:
5762:
5759:
5753:
5750:
5747:
5744:
5741:
5738:
5735:
5726:
5723:
5720:
5717:
5714:
5704:
5701:
5682:
5679:
5676:
5673:
5670:
5661:
5651:
5648:
5644:
5625:
5622:
5619:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5586:
5576:
5573:
5551:
5548:
5545:
5539:
5533:
5530:
5527:
5521:
5515:
5512:
5509:
5503:
5500:
5491:
5488:
5485:
5482:
5479:
5469:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5424:
5421:
5418:
5408:
5387:
5384:
5381:
5375:
5372:
5369:
5366:
5363:
5357:
5354:
5351:
5339:
5336:
5333:
5330:
5327:
5317:
5314:
5295:
5292:
5289:
5286:
5283:
5280:
5277:
5268:
5265:
5262:
5252:
5249:
5227:
5224:
5221:
5215:
5212:
5209:
5206:
5203:
5197:
5194:
5191:
5179:
5176:
5173:
5170:
5167:
5157:
5156:
5155:
5153:
5144:
5142:
5140:
5135:
5131:
5124:
5102:
5060:
5054:
5051:
5045:
5042:
5010:
5004:
4998:
4992:
4986:
4971:
4965:
4962:
4956:
4946:
4940:
4928:
4922:
4919:
4913:
4890:
4877:
4876:
4875:
4873:
4869:
4862:
4858:
4851:
4847:
4843:
4838:
4834:
4812:
4809:
4806:
4803:
4800:
4797:
4791:
4785:
4782:
4779:
4770:
4767:
4764:
4754:
4737:
4734:
4731:
4728:
4725:
4716:
4706:
4686:
4683:
4680:
4674:
4671:
4665:
4659:
4656:
4653:
4644:
4641:
4638:
4628:
4611:
4608:
4605:
4602:
4599:
4590:
4580:
4563:
4560:
4557:
4548:
4542:
4539:
4533:
4527:
4518:
4515:
4512:
4502:
4482:
4476:
4473:
4470:
4461:
4451:
4450:
4449:
4447:
4442:
4440:
4436:
4431:
4428:
4427:
4422:
4418:
4414:
4406:
4404:
4388:
4374:
4370:
4366:
4365:ultrafinitism
4362:
4357:
4355:
4351:
4343:
4336:
4332:
4328:
4324:
4320:
4314:
4312:
4308:
4304:
4300:
4296:
4292:
4291:David Hilbert
4288:
4284:
4280:
4274:
4270:
4262:
4260:
4257:
4253:
4248:
4224:
4218:
4215:
4207:
4203:
4199:
4197:
4189:
4186:
4180:
4173:
4168:
4164:
4160:
4158:
4150:
4144:
4133:
4132:
4131:
4130:is such that
4126:
4122:
4117:
4112:
4108:
4104:
4100:
4092:
4088:
4083:
4078:
4070:
4066:
4060:
4056:
4052:
4044:
4035:
4031:
4027:
4022:
4018:
4015:
4008:
4002:
3997:
3991:
3987:
3983:
3978:
3973:
3969:
3964:
3959:
3954:
3950:
3945:
3940:
3936:
3932:
3929:
3924:
3920:
3915:
3911:
3905:
3900:
3894:
3887:
3883:
3876:
3872:
3867:
3862:
3856:
3848:
3847:
3846:
3844:
3836:
3831:
3826:
3822:
3818:
3814:
3806:
3804:
3802:
3798:
3794:
3790:
3786:
3782:
3777:
3774:
3770:
3766:
3762:
3758:
3732:
3729:
3726:
3723:
3720:
3714:
3705:
3702:
3699:
3693:
3690:
3687:
3684:
3678:
3669:
3666:
3663:
3654:
3648:
3645:
3642:
3636:
3627:
3624:
3621:
3612:
3609:
3603:
3597:
3594:
3592:
3587:
3577:
3574:
3571:
3565:
3556:
3550:
3547:
3541:
3532:
3523:
3517:
3511:
3502:
3493:
3490:
3484:
3478:
3475:
3473:
3468:
3458:
3452:
3440:
3428:
3422:
3410:
3407:
3401:
3395:
3392:
3390:
3385:
3375:
3373:
3368:
3357:
3356:
3355:
3353:
3349:
3345:
3323:
3317:
3314:
3311:
3305:
3299:
3292:
3291:
3290:
3288:
3284:
3280:
3276:
3272:
3271:set theoretic
3266:
3258:
3256:
3254:
3250:
3221:
3215:
3207:
3203:
3199:
3197:
3186:
3178:
3174:
3167:
3158:
3154:
3150:
3148:
3138:
3134:
3127:
3116:
3115:
3114:
3110:
3106:
3101:
3097:
3093:
3089:
3082:
3078:
3073:
3067:
3063:
3054:
3050:
3045:
3039:
3035:
3029:
3025:
3021:
3017:
3013:
3008:
3004:
3000:
2994:
2988:
2982:
2978:
2972:
2964:
2962:
2960:
2956:
2952:
2949:
2943:
2939:
2933:
2929:
2923:
2919:
2910:
2905:
2901:
2897:
2891:
2887:
2881:
2877:
2871:
2867:
2862:
2858:
2852:
2848:
2847:
2846:
2844:
2839:
2835:
2830:
2829:least element
2826:
2822:
2819:
2815:
2811:
2801:
2797:
2792:
2788:
2783:
2780:
2776:
2772:
2767:
2763:
2757:
2753:
2748:
2744:
2739:
2735:
2730:
2727:
2724:
2723:
2721:
2718:
2714:
2713:
2712:
2709:
2707:
2701:
2691:
2687:
2683:
2679:
2676:
2673:
2669:
2665:
2661:
2658:
2657:
2656:
2653:
2649:
2627:
2624:
2621:
2618:
2615:
2612:
2599:
2595:
2591:
2585:
2581:
2575:
2571:
2565:
2561:
2557:
2552:
2551:
2550:
2548:
2540:
2538:
2536:
2514:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2452:
2449:
2446:
2440:
2434:
2431:
2428:
2422:
2416:
2413:
2410:
2404:
2401:
2394:
2393:
2392:
2390:
2371:
2365:
2338:
2332:
2329:
2323:
2320:
2317:
2311:
2308:
2302:
2296:
2293:
2290:
2287:
2284:
2281:
2275:
2269:
2266:
2263:
2260:
2254:
2248:
2245:
2239:
2233:
2230:
2224:
2218:
2215:
2209:
2203:
2180:
2174:
2151:
2145:
2125:
2122:
2119:
2116:
2110:
2104:
2084:
2061:
2055:
2047:
2032:
2029:
2026:
2023:
2017:
2011:
1988:
1982:
1975:
1974:
1973:
1956:
1950:
1943:To show that
1927:
1924:
1921:
1918:
1915:
1912:
1906:
1903:
1900:
1894:
1891:
1888:
1882:
1876:
1873:
1870:
1863:
1862:
1861:
1859:
1840:
1834:
1807:
1801:
1798:
1795:
1789:
1786:
1783:
1781:
1773:
1767:
1764:
1761:
1754:
1751:
1748:
1746:
1741:
1738:
1735:
1724:
1723:
1722:
1720:
1712:
1710:
1708:
1704:
1700:
1696:
1692:
1689:
1683:
1677:
1674:
1668:
1663:
1647:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1589:
1566:
1563:
1560:
1554:
1551:
1548:
1545:
1539:
1533:
1513:
1510:
1507:
1504:
1501:
1458:
1452:
1446:
1443:
1440:
1437:
1434:
1413:
1407:
1401:
1395:
1392:
1390:
1370:
1367:
1364:
1358:
1355:
1353:
1344:by definition
1333:
1327:
1324:
1321:
1318:
1316:
1311:
1308:
1305:
1292:
1286:
1283:
1280:
1277:
1274:
1256:
1250:
1244:
1241:
1239:
1219:
1216:
1213:
1207:
1204:
1202:
1193:by definition
1182:
1176:
1173:
1170:
1167:
1165:
1160:
1157:
1154:
1135:
1129:
1123:
1120:
1118:
1098:
1095:
1092:
1086:
1083:
1081:
1072:by definition
1061:
1055:
1052:
1049:
1046:
1044:
1039:
1036:
1033:
1022:
1021:
1020:
1019:For example:
990:
984:
981:
978:
972:
969:
967:
959:
953:
950:
947:
931:
928:
925:
923:
918:
915:
912:
901:
900:
899:
897:
893:
889:
885:
878:
876:
874:
870:
862:
861:
856:
852:
848:
840:
838:
836:
832:
828:
819:
815:
811:
804:
800:
796:
792:
788:
784:
780:
777:
774:
773:
771:
767:
763:
760:
753:
746:
742:
738:
735:implies that
734:
730:
726:
722:
719:
715:
714:
712:
708:
705:
703:
702:
697:
692:
690:
686:
682:
678:
670:
666:
662:
658:
653:
643:
639:
634:
630:
627:
623:
618:
614:
607:
603:
599:
595:
590:
586:
582:
579:
575:
571:
567:
563:
559:
556:
554:
551:
547:
538:
534:
529:
525:
520:
516:
512:
508:
505:
500:
496:
491:
487:
483:
479:
475:
471:
467:
463:
460:
455:
451:
445:
441:
436:
432:
428:
425:
420:
416:
411:
407:
404:
402:
399:
394:
392:
391:
381:
378:
375:
373:
369:
353:
340:
337:
333:
332:
326:
324:
320:
316:
315:Gottlob Frege
312:
311:
306:
302:
291:
288:
280:
269:
266:
262:
259:
255:
252:
248:
245:
241:
238: –
237:
233:
232:Find sources:
226:
222:
216:
215:
210:This article
208:
204:
199:
198:
192:
190:
188:
184:
180:
176:
172:
168:
164:
159:
157:
156:
151:
147:
143:
139:
135:
131:
127:
123:
119:
114:
112:
108:
104:
99:
97:
93:
89:
88:number theory
85:
81:
77:
73:
69:
65:
60:
53:
26:
22:
8985:
8984:
8964:
8942:
8929:
8916:
8873:Proof theory
8872:
8847:
8822:
8798:
8749:
8745:
8696:
8692:
8675:. Retrieved
8652:
8647:
8634:
8609:
8579:
8543:
8515:
8477:(1): 85–95.
8474:
8470:
8451:
8429:. Springer.
8425:
8395:
8373:
8345:
8317:
8296:
8292:
8259:
8241:
8228:. Elsevier.
8224:
8186:
8158:
8151:Gray, Jeremy
8139:
8106:
8102:
8096:
8076:. Retrieved
8069:the original
8044:
8040:
8003:
7999:
7993:
7977:
7945:
7913:
7897:. Retrieved
7888:
7883:
7860:
7834:
7822:
7810:
7795:
7783:
7771:
7759:
7747:
7740:Willard 2001
7735:
7729:Gentzen 1936
7724:
7713:
7701:
7694:Hilbert 1902
7689:
7671:
7575:
7563:
7557:Hatcher 2014
7548:
7535:
7523:
7503:
7498:
7486:
7484:Matt DeVos,
7480:
7468:
7463:, p. 1.
7456:
7436:, p. 2.
7429:
7417:
7410:Shields 1997
7405:
7393:
7377:
7365:
7355:
7347:
6963:
6953:
6944:
6895:Neo-logicism
6780:
6776:
6772:
6770:
6764:
6760:
6659:
6626:
6622:
6618:
6610:
6604:
6600:
6596:
6592:
6588:
6584:
6580:
6576:
6572:
6568:
6564:
6560:
6556:
6552:
6550:
6539:
6535:
6531:
6526:
6522:
6518:
6496:
6485:
6461:
6440:with higher
6437:
6433:
6417:
6409:
6399:
6390:
6382:
6376:
6366:
6364:
6359:
6355:
6289:
6287:
5148:
5138:
5133:
5129:
5122:
5091:
4871:
4867:
4860:
4856:
4849:
4845:
4841:
4830:
4443:
4432:
4426:axiom schema
4424:
4417:second-order
4410:
4358:
4315:
4286:
4276:
4255:
4251:
4246:
4243:
4124:
4120:
4115:
4110:
4106:
4102:
4098:
4090:
4086:
4081:
4076:
4068:
4064:
4058:
4050:
4042:
4040:
4033:
4029:
4025:
4020:
4016:
4013:
4006:
4000:
3995:
3989:
3985:
3981:
3976:
3971:
3967:
3962:
3957:
3952:
3948:
3943:
3938:
3934:
3927:
3922:
3918:
3913:
3909:
3903:
3898:
3892:
3885:
3881:
3874:
3870:
3865:
3860:
3854:
3842:
3829:
3816:
3810:
3778:
3772:
3768:
3764:
3756:
3754:
3351:
3343:
3341:
3286:
3268:
3247:and it is a
3246:
3108:
3104:
3099:
3095:
3091:
3088:homomorphism
3080:
3076:
3071:
3065:
3061:
3052:
3048:
3043:
3037:
3033:
3015:
3006:
3002:
2998:
2992:
2986:
2980:
2976:
2968:
2958:
2954:
2950:
2941:
2937:
2931:
2927:
2921:
2917:
2914:
2908:
2903:
2899:
2895:
2889:
2885:
2879:
2875:
2869:
2865:
2856:
2850:
2842:
2837:
2833:
2824:
2814:well-ordered
2809:
2807:
2799:
2795:
2790:
2786:
2778:
2774:
2770:
2765:
2761:
2755:
2751:
2746:
2742:
2737:
2733:
2725:
2719:
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2699:
2695:
2689:
2685:
2681:
2677:
2671:
2667:
2663:
2659:
2651:
2647:
2604:
2597:
2593:
2589:
2583:
2579:
2573:
2569:
2563:
2559:
2555:
2544:
2541:Inequalities
2471:
2357:
1942:
1826:
1716:
1702:
1688:cancellative
1681:
1666:
1493:
1018:
891:
882:
859:
844:
824:
817:
813:
809:
802:
798:
794:
790:
786:
782:
775:
765:
758:
751:
744:
740:
736:
732:
728:
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717:
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699:
695:
693:
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684:
680:
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660:
656:
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637:
632:
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616:
612:
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601:
597:
593:
588:
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569:
565:
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514:
510:
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485:
481:
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465:
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439:
434:
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418:
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395:
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386:
376:
371:
338:
329:
327:
308:
298:
283:
274:
264:
257:
250:
243:
231:
219:Please help
214:verification
211:
186:
183:axiom schema
171:second-order
160:
145:
136:provided an
115:
110:
100:
67:
63:
25:Peano axioms
24:
18:
8703:: 145–158.
8677:2 September
8117:: 280–287.
8089:Gödel, Kurt
8030:Gödel, Kurt
7815:Hermes 1973
7800:Hermes 1973
7553:Suppes 1960
7398:Peirce 1881
6362:elements.
6360:nonstandard
5839:irreflexive
5572:distributes
5313:commutative
5248:associative
4419:, since it
4321:. In 1936,
4319:type theory
4309:proved his
4305:. In 1931,
4273:Consistency
4263:Consistency
3933:A morphism
3930:-morphisms.
3253:categorical
3113:satisfying
2948:contradicts
2781:)) is true,
2547:total order
1717:Similarly,
1693:, and thus
1673:commutative
1430:using
1270:using
896:recursively
831:first-order
805:)) is true,
772:such that:
768:is a unary
620:. That is,
185:. The term
167:first-order
132:. In 1881,
9006:Categories
8992:PlanetMath
8109:, Vol II.
8098:Dialectica
8078:2013-10-31
7718:Gödel 1958
7706:Gödel 1931
7580:Fritz 1952
7473:Peano 1908
7461:Peano 1889
7334:References
6783:such that
6601:proper cut
6515:order type
6511:computable
6490:, such as
6488:set theory
5916:trichotomy
5783:transitive
4421:quantifies
4307:Kurt Gödel
4295:finitistic
4287:consistent
3979:-morphism
3799:, and the
3028:isomorphic
2802:) is true.
2759:such that
2731:for every
2587:such that
2545:The usual
2391:addition:
1695:embeddable
1686:is also a
869:set theory
683:(0), 2 as
504:transitive
247:newspapers
118:arithmetic
107:arithmetic
92:consistent
8966:MathWorld
8949:EMS Press
8891:cite book
8689:Wang, Hao
8278:1431-4657
8248:and from
8213:121297669
8065:197663120
8020:122719892
7930:cite book
7839:Kaye 1991
7827:Kaye 1991
7817:, VI.3.1.
7788:Kaye 1991
7676:Gray 2013
7654:…
7635:…
7382:Wang 1957
7339:Citations
7305:⋅
7273:⋅
7255:⋅
7223:⋅
7211:⋅
7199:⋅
7167:⋅
7135:⋅
7129:∨
7117:⋅
7085:⋅
7073:⋅
7049:⋅
7037:⋅
7025:⋅
6990:⋅
6975:∀
6818:¯
6800:ϕ
6797:⊨
6741:¯
6723:ϕ
6720:⊨
6688:¯
6670:ϕ
6643:¯
6547:Overspill
6428:sets are
6406:algorithm
6320:≤
6266:≥
6251:∀
6219:≥
6213:⇒
6192:∀
6189:∧
6128:∃
6125:⇒
6098:∀
6070:⋅
6058:⋅
6052:⇒
6040:∧
6007:∀
5961:⇒
5928:∀
5890:∨
5878:∨
5851:∀
5807:¬
5795:∀
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5745:∧
5712:∀
5674:⋅
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5617:⋅
5611:∧
5584:∀
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5531:⋅
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5376:⋅
5364:⋅
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5325:∀
5260:∀
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5040:∀
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4984:⇒
4975:¯
4957:φ
4944:∀
4941:∧
4932:¯
4914:φ
4894:¯
4885:∀
4804:⋅
4783:⋅
4762:∀
4729:⋅
4714:∀
4636:∀
4588:∀
4555:⇒
4510:∀
4474:≠
4459:∀
4439:signature
4385:Π
4333:up to an
4101: : (
3937: : (
3896: : 1
3797:empty set
3655:∪
3524:∪
3447:∅
3435:∅
3429:∪
3426:∅
3417:∅
3379:∅
3318:∪
3249:bijection
2717:predicate
2628:∈
2503:⋅
2450:⋅
2432:⋅
2405:⋅
2261:⋅
2216:⋅
2117:⋅
2097:(that is
2024:⋅
1904:⋅
1874:⋅
1799:⋅
1765:⋅
1739:⋅
1662:structure
1381:using (2)
1230:using (2)
1143:using (1)
1109:using (2)
770:predicate
731:being in
696:successor
667:) limits
665:induction
626:injection
546:successor
459:symmetric
424:reflexive
130:induction
8871:(2013).
8737:(2001).
8725:26896458
8645:(1889).
8633:(1890).
8608:(1967).
8540:(1960).
8465:(1881).
8449:(1908).
8344:(1965).
8153:(2013).
8137:(1861).
8091:(1958).
8032:(1931).
7895:. Vieweg
7881:(1888).
7859:(1974).
7508:Archived
6844:See also
6763:∈
6759:for all
6591:∈
6583:∈
6567:so that
6474:"); the
6356:standard
5700:identity
5643:identity
3984: :
3917: :
3821:category
3767: :
3094: :
3012:Dedekind
2985:, where
2946:, which
2935:. Thus,
2863:For any
2818:nonempty
2715:For any
2655:, then:
2553:For all
2535:semiring
2138:), then
1707:integers
884:Addition
879:Addition
847:addition
743:) is in
716:0 is in
550:function
509:For all
401:relation
398:equality
323:Schröder
277:May 2024
163:equality
96:complete
74:for the
8951:, 2001
8782:2822314
8774:1833464
8766:2695030
8717:2964176
8499:1507856
8491:2369151
7849:Sources
7352:"Peano"
6239:covered
4855:, ...,
4337:called
4335:ordinal
4301:of his
3975:) is a
2957:. Thus
2893:. Then
2816:—every
2769:, then
1705:is the
837:below.
610:, then
531:, then
492:, then
447:, then
261:scholar
66:or the
8879:
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8047:. See
8018:
7957:
7899:4 July
7891:]
7680:p. 133
7584:p. 137
6981:
6958:piece.
6595:) and
6499:Skolem
6257:
6198:
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6025:
5946:
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5183:
5092:where
4837:axioms
4774:
4720:
4648:
4594:
4522:
4465:
4299:second
3888:, and
3880:where
3815:. Let
3350:under
3348:closed
3020:German
2965:Models
2827:has a
2821:subset
2704:is an
2472:Thus,
1676:monoid
1660:. The
853:, and
624:is an
576:under
574:closed
537:closed
263:
256:
249:
242:
234:
72:axioms
70:, are
23:, the
8778:S2CID
8762:JSTOR
8742:(PDF)
8721:S2CID
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8671:(PDF)
8651:[
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7893:(PDF)
7887:[
7188:then
6936:Notes
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5128:,...,
4369:total
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4109:) → (
4105:, 0,
4067:, 0,
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3956:) → (
3823:with
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2979:, 0,
2971:model
2741:, if
2722:, if
2674:, and
2645:, if
1699:group
1697:in a
1691:magma
1671:is a
808:then
750:then
720:, and
644:) = 0
591:, if
517:, if
476:, if
437:, if
319:Boole
268:JSTOR
254:books
150:Latin
8897:link
8877:ISBN
8853:ISBN
8827:ISBN
8803:ISBN
8679:2023
8614:ISBN
8590:ISBN
8554:ISBN
8522:ISBN
8431:ISBN
8404:ISBN
8378:ISBN
8354:ISBN
8326:ISBN
8274:ISSN
8264:ISBN
8230:ISBN
8199:ISBN
8167:ISBN
7955:ISBN
7936:link
7901:2016
7267:>
7217:>
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6579:and
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4271:and
4250:and
4011:and
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3907:and
3837:, US
3058:and
2996:and
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1526:and
1473:etc.
898:as:
888:maps
661:also
600:) =
587:and
513:and
484:and
472:and
433:and
366:The
321:and
240:news
128:and
101:The
94:and
8990:on
8754:doi
8705:doi
8566:ZFC
8479:doi
8301:doi
8191:doi
8119:doi
8053:doi
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998:(2)
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871:or
857:on
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341:or
336:set
313:by
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90:is
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5754:z
5748:y
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5680:=
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5668:(
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5629:)
5626:0
5623:=
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5614:x
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5599:+
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5587:x
5558:)
5555:)
5552:z
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5543:(
5540:+
5537:)
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5528:x
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5522:=
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5516:z
5513:+
5510:y
5507:(
5501:x
5498:(
5492:z
5489:,
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5434:x
5431:(
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5422:,
5419:x
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5388:z
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3402:0
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3306:a
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3300:s
3287:s
3228:)
3225:)
3222:n
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3213:(
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3200:=
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3187:n
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2402:a
2375:)
2372:0
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2366:S
2342:)
2339:a
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2324:0
2321:+
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2309:=
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2303:0
2300:(
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2294:+
2291:a
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2282:+
2279:)
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2267:=
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2258:)
2255:0
2252:(
2249:S
2246:+
2243:)
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2237:(
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2231:=
2228:)
2225:a
2222:(
2219:S
2213:)
2210:0
2207:(
2204:S
2184:)
2181:a
2178:(
2175:S
2155:)
2152:0
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2126:a
2123:=
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2114:)
2111:0
2108:(
2105:S
2085:a
2065:)
2062:0
2059:(
2056:S
2045:.
2033:0
2030:=
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2021:)
2018:0
2015:(
2012:S
1992:)
1989:0
1986:(
1983:S
1960:)
1957:0
1954:(
1951:S
1928:a
1925:=
1922:0
1919:+
1916:a
1913:=
1910:)
1907:0
1901:a
1898:(
1895:+
1892:a
1889:=
1886:)
1883:0
1880:(
1877:S
1871:a
1844:)
1841:0
1838:(
1835:S
1808:.
1805:)
1802:b
1796:a
1793:(
1790:+
1787:a
1784:=
1777:)
1774:b
1771:(
1768:S
1762:a
1755:,
1752:0
1749:=
1742:0
1736:a
1703:N
1682:N
1680:(
1667:N
1665:(
1648:b
1628:a
1625:+
1622:b
1619:=
1616:b
1613:+
1610:a
1590:b
1570:)
1567:b
1564:+
1561:a
1558:(
1555:S
1552:=
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1546:+
1543:)
1540:a
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1534:S
1514:b
1511:=
1508:b
1505:+
1502:0
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1462:)
1459:a
1456:(
1453:S
1450:(
1447:S
1444:=
1441:2
1438:+
1435:a
1423:)
1420:)
1417:)
1414:a
1411:(
1408:S
1405:(
1402:S
1399:(
1396:S
1393:=
1374:)
1371:2
1368:+
1365:a
1362:(
1359:S
1356:=
1337:)
1334:2
1331:(
1328:S
1325:+
1322:a
1319:=
1312:3
1309:+
1306:a
1296:)
1293:a
1290:(
1287:S
1284:=
1281:1
1278:+
1275:a
1263:)
1260:)
1257:a
1254:(
1251:S
1248:(
1245:S
1242:=
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1205:=
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1183:1
1180:(
1177:S
1174:+
1171:a
1168:=
1161:2
1158:+
1155:a
1136:,
1133:)
1130:a
1127:(
1124:S
1121:=
1102:)
1099:0
1096:+
1093:a
1090:(
1087:S
1084:=
1065:)
1062:0
1059:(
1056:S
1053:+
1050:a
1047:=
1040:1
1037:+
1034:a
991:.
988:)
985:b
982:+
979:a
976:(
973:S
970:=
963:)
960:b
957:(
954:S
951:+
948:a
932:,
929:a
926:=
919:0
916:+
913:a
892:N
860:N
820:.
818:n
814:n
812:(
810:φ
803:n
801:(
799:S
797:(
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791:n
789:(
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783:n
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752:K
747:,
745:K
741:n
739:(
737:S
733:K
729:n
725:n
718:K
711:K
689:S
687:(
685:S
681:S
669:N
657:N
642:n
640:(
638:S
633:n
628:.
622:S
617:n
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604:(
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598:m
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589:n
585:m
580:.
578:S
570:n
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562:n
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533:a
528:b
524:a
519:b
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511:a
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499:z
495:x
490:z
486:y
482:y
478:x
474:z
470:y
466:x
461:.
454:x
450:y
444:y
440:x
435:y
431:x
426:.
419:x
415:x
410:x
372:S
354:.
350:N
339:N
290:)
284:(
279:)
275:(
265:·
258:·
251:·
244:·
217:.
148:(
52:/
46:n
40:ˈ
37:i
34:p
31:/
27:(
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