3538:, …) and this identification preserves the corresponding algebraic operations of the reals. The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat arbitrary, must be self-consistent and well defined. For example, we may have two sequences that differ in their first
5902:
4873:(because you can add no more sets to it without breaking it). The only explicitly known example of an ultrafilter is the family of sets containing a given element (in our case, say, the number 10). Such ultrafilters are called trivial, and if we use it in our construction, we come back to the ordinary real numbers. Any ultrafilter containing a finite set is trivial. It is known that any filter can be extended to an ultrafilter, but the proof uses the
140:
43:
7095:
4351:. It will contain the infinitesimals in addition to the ordinary real numbers, as well as infinitely large numbers (the reciprocals of infinitesimals, including those represented by sequences diverging to infinity). Also every hyperreal that is not infinitely large will be infinitely close to an ordinary real, in other words, it will be the sum of an ordinary real and an infinitesimal.
952:, is true for the real numbers, and it is in the form required by the transfer principle, so it is also true for the hyperreal numbers. This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals.
7199:
766:, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, function, and relation has its natural hyperreal extension, satisfying the same first-order properties. The kinds of logical sentences that obey this restriction on quantification are referred to as statements in
2384:
Using hyperreal numbers for differentiation allows for a more algebraically manipulable approach to derivatives. In standard differentiation, partial differentials and higher-order differentials are not independently manipulable through algebraic techniques. However, using the hyperreals, a system
3183:
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed
3816:
717:
that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number x ..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number
955:
Similarly, the casual use of 1/0 = ∞ is invalid, since the transfer principle applies to the statement that zero has no multiplicative inverse. The rigorous counterpart of such a calculation would be that if ε is a non-zero infinitesimal, then 1/ε is infinite.
3275:, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic. Hyper-real fields were in fact originally introduced by
535:
proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that
Robinson delineated.
3498:
2740:
4347: = 0, at least one of them should be declared zero. Surprisingly enough, there is a consistent way to do it. As a result, the equivalence classes of sequences that differ by some sequence declared zero will form a field, which is called a hyperreal
900:
1658:
662:
4282:. Recall that the sequences converging to zero are sometimes called infinitely small. These are almost the infinitesimals in a sense; the true infinitesimals include certain classes of sequences that contain a sequence converging to zero.
2261:
2065:
3821:
but here we run into trouble, since some entries of the first sequence may be bigger than the corresponding entries of the second sequence, and some others may be smaller. It follows that the relation defined in this way is only a
1296:
5557:
5624:
4185:
2155:
3826:. To get around this, we have to specify which positions matter. Since there are infinitely many indices, we don't want finite sets of indices to matter. A consistent choice of index sets that matter is given by any free
3604:
3080:
This shows that using hyperreal numbers, Leibniz's notation for the definite integral can actually be interpreted as a meaningful algebraic expression (just as the derivative can be interpreted as a meaningful quotient).
7397:
4289:, that is, one can multiply, add and subtract them, but not necessarily divide by a non-zero element. The real numbers are considered as the constant sequences, the sequence is zero if it is identically zero, that is,
1949:
5696:
5395:
5471:
4362:
of rationals and declared all the sequences that converge to zero to be zero. The result is the reals. To continue the construction of hyperreals, consider the zero sets of our sequences, that is, the
2317:
2832:
3297:
2625:
2361:
are a number system based on this idea. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the
5018:
5739:
2986:
1864:
2468:
2436:
4263:, and in ZFC with the negation of continuum hypothesis we can prove that there are non-order-isomorphic pairs of fields that are both countably indexed ultrapowers of the reals.
4420:
5072:
All the arithmetical expressions and formulas make sense for hyperreals and hold true if they are true for the ordinary reals. It turns out that any finite (that is, such that
795:
6337:
Keisler, H. Jerome (1994) The hyperreal line. Real numbers, generalizations of the reals, and theories of continua, 207—237, Synthese Lib., 242, Kluwer Acad. Publ., Dordrecht.
2357:
The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity.
4222:
483:
433:
6592:
5488:
5321:. This operation is an order-preserving homomorphism and hence is well-behaved both algebraically and order theoretically. It is order-preserving though not isotonic; i.e.
6907:
6830:
6791:
6753:
6725:
6697:
6669:
6557:
6524:
6496:
6468:
5955: – Generalization of the real numbers – Surreal numbers are a much larger class of numbers, that contains the hyperreals as well as other classes of non-real numbers.
5561:
3573:
5047:
1779:
305:
3593:
3152:
hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the
5106:
2509:
1541:
1034:
1007:
685:
221:
5345:
4502:
2924:
2165:
1403:
5421:
5214:
is an infinitesimal. It can be proven by bisection method used in proving the
Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial.
5067:
2590:
1438:
1130:
979:
only infinitesimally. The standard part function can also be defined for infinite hyperreal numbers as follows: If x is a positive infinite hyperreal number, set st(
3018:
2769:
2617:
2561:
2535:
1959:
4806:
4589:
4560:
4531:
4449:
2892:
1467:
7402:
5172:
4777:
4732:
4341:
3076:
2348:
1687:
566:
4643:
1802:
1490:
1345:
1322:
1173:
1095:
1072:
5212:
5192:
5146:
5126:
4945:
4925:
4905:
4706:
4686:
4663:
4620:
4469:
3038:
2854:
1727:
1707:
1533:
1513:
1365:
1150:
506:
325:
261:
241:
185:
1181:
4088:
4884:
Now if we take a nontrivial ultrafilter (which is an extension of the Fréchet filter) and do our construction, we get the hyperreal numbers as a result.
3811:{\displaystyle (a_{0},a_{1},a_{2},\ldots )\leq (b_{0},b_{1},b_{2},\ldots )\iff (a_{0}\leq b_{0})\wedge (a_{1}\leq b_{1})\wedge (a_{2}\leq b_{2})\ldots }
3216:. Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where
2075:
5635:
7131:
6418:
3542:
members, but are equal after that; such sequences should clearly be considered as representing the same hyperreal number. Similarly, most sequences
2397:
Another key use of the hyperreal number system is to give a precise meaning to the integral sign ∫ used by
Leibniz to define the definite integral.
1870:
6309:
5069:
relative to our ultrafilter", two sequences being in the same class if and only if the zero set of their difference belongs to our ultrafilter.
7453:
2373:
is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity
1036:(the idea is that an infinite hyperreal number should be smaller than the "true" absolute infinity but closer to it than any real number is).
6354:
6329:
6256:
6209:
6178:
6049:
6015:
3598:
Comparing sequences is thus a delicate matter. We could, for example, try to define a relation between sequences in a componentwise fashion:
4278:
The following is an intuitive way of understanding the hyperreal numbers. The approach taken here is very close to the one in the book by
762:..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over
5350:
691:, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable
5426:
3184:
outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an
7345:
7238:
7182:
7169:
6293:
5926:
5915:
3124:
is somewhat misleading in that there is not a unique ordered field that is referred to in most treatments. However, a 2003 paper by
126:
60:
3229:
7228:
6980:
107:
5706:
7428:
5745:
79:
64:
2271:
3204:
introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as
929:
Informal notations for non-real quantities have historically appeared in calculus in two contexts: as infinitesimals, like
7433:
7233:
6941:
6302:
3221:
7438:
7310:
7124:
6411:
4075:
86:
2778:
7243:
6567:
3493:{\displaystyle (a_{0},a_{1},a_{2},\ldots )+(b_{0},b_{1},b_{2},\ldots )=(a_{0}+b_{0},a_{1}+b_{1},a_{2}+b_{2},\ldots )}
2735:{\displaystyle \int _{a}^{b}(\varepsilon ,dx):=\operatorname {st} \left(\sum _{n=0}^{N}\varepsilon (a+n\ dx)\right),}
7443:
7259:
6201:
6105:
93:
53:
7361:
6975:
6931:
5749:
3543:
3251:
showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of
555:
in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of
339:
7098:
6970:
5887:
4953:
4870:
3827:
3185:
7188:
6393:. Includes an axiomatic treatment of the hyperreals, and is freely available under a Creative Commons license
2932:
1044:
One of the key uses of the hyperreal number system is to give a precise meaning to the differential operator
75:
7117:
6404:
4286:
3145:
1814:
3271:, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers
2441:
7290:
7058:
6562:
5307:
2403:
1737:
895:{\displaystyle 1<\omega ,\quad 1+1<\omega ,\quad 1+1+1<\omega ,\quad 1+1+1+1<\omega ,\ldots .}
731:
688:
3837:; these can be characterized as ultrafilters that do not contain any finite sets. (The good news is that
7376:
7043:
6879:
6607:
6602:
4365:
3241:
3209:
540:
7159:
5901:
513:
7218:
7213:
6796:
6529:
5931:
4252:
4193:
3508:
3252:
3153:
984:
914:
544:
528:
445:
395:
6573:
4285:
Let us see where these classes come from. Consider first the sequences of real numbers. They form a
7223:
7007:
6917:
6874:
6856:
6634:
4862:
4348:
3961:
3220:
is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see
31:
6890:
6813:
6774:
6736:
6708:
6680:
6652:
6540:
6507:
6479:
6451:
7295:
6912:
6624:
6346:
6138:
6120:
6075:
5907:
3552:
3256:
1653:{\displaystyle {\frac {df(x,dx)}{dx}}=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}
934:
751:
704:
335:
7407:
6074:
Fite, Isabelle (2022). "Total and
Partial Differentials as Algebraically Manipulable Entities".
5026:
2256:{\displaystyle =\operatorname {st} \left({\frac {2x\cdot dx}{dx}}+{\frac {(dx)^{2}}{dx}}\right)}
1742:
266:
6285:
6279:
3578:
7448:
7330:
7320:
7305:
7070:
7033:
6997:
6936:
6922:
6617:
6597:
6350:
6325:
6289:
6252:
6205:
6174:
6045:
6011:
5937:
5251:
5075:
4878:
4866:
4260:
3201:
3160:
2473:
2386:
2060:{\displaystyle =\operatorname {st} \left({\frac {x^{2}+2x\cdot dx+(dx)^{2}-x^{2}}{dx}}\right)}
1016:
989:
767:
763:
667:
385:
347:
343:
190:
100:
5324:
4474:
2897:
1370:
539:
The application of hyperreal numbers and in particular the transfer principle to problems of
7458:
7371:
7366:
7017:
6926:
6835:
6801:
6642:
6612:
6534:
6437:
6380:
6275:
6193:
6130:
6044:. Princeton legacy library. Princeton, New Jersey: Princeton University Press. p. 474.
5400:
5267:
5052:
4279:
3504:
3248:
3237:
3125:
2566:
1408:
1100:
657:{\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+\Delta x)-f(x)}{\Delta x}}\right)}
547:. One immediate application is the definition of the basic concepts of analysis such as the
532:
6042:
Abraham
Robinson: the creation of nonstandard analysis: a personal and mathematical odyssey
4354:
This construction is parallel to the construction of the reals from the rationals given by
2994:
2748:
2595:
2540:
2514:
1324:" used to denote any infinitesimal is consistent with the above definition of the operator
7340:
7325:
7164:
6965:
6869:
6501:
6321:
6248:
5940: – Non algebraically closed field whose extension by sqrt(–1) is algebraically closed
5863:
5765:
4874:
4782:
4565:
4536:
4507:
4425:
4359:
3549:
forever, and we must find some way of taking such a sequence and interpreting it as, say,
3546:
3233:
3213:
3168:
3133:
3102:
2859:
1443:
359:
156:
139:
5151:
4756:
4711:
4320:
3838:
3043:
2327:
1666:
1291:{\displaystyle df(x,dx):=\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)\ dx.}
4625:
1784:
1472:
1327:
1304:
1155:
1077:
1054:
750:." This ability to carry over statements from the reals to the hyperreals is called the
7381:
7274:
7012:
7002:
6987:
6806:
6674:
6445:
5952:
5855:
5235:
5197:
5177:
5131:
5111:
4930:
4910:
4890:
4691:
4671:
4648:
4605:
4454:
3921:
3834:
3205:
3172:
3137:
3129:
3110:
3023:
2839:
1712:
1692:
1518:
1498:
1350:
1135:
491:
310:
246:
226:
170:
17:
5552:{\displaystyle \operatorname {st} (x+y)=\operatorname {st} (x)+\operatorname {st} (y)}
5286:
nonstandard real number is "very close" to a unique real number, in the sense that if
7422:
7140:
7075:
7048:
6957:
6225:
5785:
3988:
3823:
3141:
3094:
964:
164:
6142:
5619:{\displaystyle \operatorname {st} (xy)=\operatorname {st} (x)\operatorname {st} (y)}
4231:
One question we might ask is whether, if we had chosen a different free ultrafilter
4180:{\displaystyle (2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}^{2}}=2^{\aleph _{0}},}
7315:
7300:
7038:
6840:
5920:
5879:
5828:
4599:
One of the sequences that vanish on two complementary sets should be declared zero.
4355:
4267:
4059:
3276:
3260:
3197:
3106:
2772:
940:
As an example of the transfer principle, the statement that for any nonzero number
692:
523:
of arguments involving infinitesimals date back to ancient Greek mathematics, with
486:
328:
27:
Element of a nonstandard model of the reals, which can be infinite or infinitesimal
4947:
naturally extends to a hyperreal function of a hyperreal variable by composition:
2381:; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.
2150:{\displaystyle =\operatorname {st} \left({\frac {2x\cdot dx+(dx)^{2}}{dx}}\right)}
3212:. Nonetheless these concepts were from the beginning seen as suspect, notably by
7269:
7175:
6864:
6646:
5793:
5290:
is a finite nonstandard real, then there exists one and only one real number st(
4071:
3925:
3503:
and analogously for multiplication. This turns the set of such sequences into a
3291:
of reals. In fact we can add and multiply sequences componentwise; for example:
3159:
The condition of being a hyperreal field is a stronger one than that of being a
2358:
148:
42:
7198:
7154:
6845:
6702:
6389:
5897:
5231:
1733:
548:
524:
509:
143:
Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)
6134:
4869:
and it is used in the usual limit theory). If (1) also holds, U is called an
7335:
5943:
4881:) can be added as an extra axiom, as it is weaker than the axiom of choice.
4846:
3020:
If so, this integral is called the definite integral (or antiderivative) of
520:
6008:
Selected papers of
Abraham Robinson. 2: Nonstandard analysis and philosophy
5748:
with respect to the order topology on the finite hyperreals; in fact it is
3845:; the bad news is that they cannot be explicitly constructed.) We think of
1944:{\displaystyle =\operatorname {st} \left({\frac {f(x+dx)-f(x)}{dx}}\right)}
6232:, Math. Appl., vol. 510, Dordrecht: Kluwer Acad. Publ., pp. 1–95
6953:
6884:
6730:
3288:
3279:(1948) by purely algebraic techniques, using an ultrapower construction.
3268:
3225:
552:
160:
3240:, and others, infinitesimals were largely abandoned, though research in
6473:
6146:
3264:
436:
6371:
4259:
with the continuum hypothesis we can prove this field is unique up to
933:, and as the symbol ∞, used, for example, in limits of integration of
7088:
6427:
6125:
4743:
1009:, and likewise, if x is a negative infinite hyperreal number, set st(
4035:
is identified with the number 1, and any ideal containing 1 must be
6245:
Lectures on the hyperreals: an introduction to nonstandard analysis
6171:
Super-real fields: totally ordered fields with additional structure
6080:
527:
replacing such proofs with ones using other techniques such as the
4865:(an example: the complements to the finite sets, it is called the
7109:
6396:
5979:
5691:{\displaystyle \operatorname {st} (1/x)=1/\operatorname {st} (x)}
5390:{\displaystyle \operatorname {st} (x)\leq \operatorname {st} (y)}
3971:(namely, the set of the sequences that vanish in some element of
5923: – A hyperreal number that is equal to its own integer part
5839:. Note that no assumption is being made that the cardinality of
5466:{\displaystyle \operatorname {st} (x)<\operatorname {st} (y)}
380:, holds for the hyperreals just as it does for the reals; since
7113:
6400:
3956:
of hyperreals is constructed. From an algebraic point of view,
3849:
as singling out those sets of indices that "matter": We write (
3188:, but the ultrafilter itself cannot be explicitly constructed.
3148:
of the reals, which therefore has a good claim to the title of
709:
The idea of the hyperreal system is to extend the real numbers
4256:
1048:
as used by
Leibniz to define the derivative and the integral.
36:
5270:
consists of the infinitesimals and which sends every element
6579:
6284:(4th ed. ed.), New York: Dover Publications, pp.
5282:; which is to say, is infinitesimal. Put another way, every
4266:
For more information about this method of construction, see
5886:
and are identical to the ultrapowers constructed via free
3105:. Unlike the reals, the hyperreals do not form a standard
5878:. The hyperreal fields we obtain in this case are called
3228:
was put on a firm footing through the development of the
6030:
Hewitt (1948), p. 74, as reported in
Keisler (1994)
5776:) is the algebra of continuous real-valued functions on
5948:
Pages displaying short descriptions of redirect targets
4023:
and not altering its null entries. If the set on which
5278:
to a unique real number whose difference from x is in
2312:{\displaystyle =\operatorname {st} \left(2x+dx\right)}
6893:
6816:
6777:
6739:
6711:
6683:
6655:
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6543:
6510:
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6390:
Elementary
Calculus: An Approach Using Infinitesimals
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if we agree not to distinguish between two sequences
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4861:
Any family of sets that satisfies (2–4) is called a
4274:
An intuitive approach to the ultrapower construction
2856:
is then said to be integrable over a closed interval
773:
The transfer principle, however, does not mean that
7390:
7354:
7283:
7252:
7206:
7147:
7026:
6952:
6854:
6762:
6633:
6435:
6228:(2000), "An introduction to nonstandard analysis",
6215:. The classic introduction to nonstandard analysis.
5850:An important special case is where the topology on
3891:, ...) if and only if the set of natural numbers {
1097:is defined as a map which sends every ordered pair
971:), is defined as the unique closest real number to
67:. Unsourced material may be challenged and removed.
6901:
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6586:
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6301:Hatcher, William S. (1982) "Calculus is Algebra",
6230:Nonstandard analysis for the working mathematician
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4496:
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4443:
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4335:
4251:. This question turns out to be equivalent to the
4216:
4179:
4011:follows from the possibility of, given a sequence
3810:
3587:
3567:
3492:
3070:
3032:
3012:
2980:
2918:
2886:
2848:
2826:
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2529:
2503:
2462:
2430:
2342:
2311:
2255:
2149:
2059:
1943:
1858:
1796:
1773:
1721:
1701:
1681:
1652:
1527:
1507:
1484:
1461:
1432:
1397:
1359:
1339:
1316:
1290:
1167:
1144:
1124:
1089:
1066:
1028:
1001:
894:
679:
656:
500:
477:
427:
319:
299:
255:
235:
215:
179:
159:of the real numbers to include certain classes of
4877:. The existence of a nontrivial ultrafilter (the
2827:{\displaystyle \ \operatorname {st} (N\ dx)=b-a.}
1689:If so, this quotient is called the derivative of
30:"*R" and "R*" redirect here. For other uses, see
5218:Properties of infinitesimal and infinite numbers
3287:We are going to construct a hyperreal field via
921:cannot be expressed as a first-order statement.
917:.) This is possible because the nonexistence of
4812:. From the above conditions one can see that:
3263:; however it is possible to proceed using only
6341:Kleinberg, Eugene M.; Henle, James M. (2003),
3952:. With this identification, the ordered field
1175:is nonzero infinitesimal) to an infinitesimal
7125:
6412:
6281:A Short Account of the History of Mathematics
5049:means "the equivalence class of the sequence
4857:, as every set has the empty set as a subset.
4834:An intersection of any two sets belonging to
3109:, but by virtue of their order they carry an
734:over several numbers, e.g., "for any numbers
263:is said to be infinitesimal if, and only if,
8:
6106:"A definable nonstandard model of the reals"
5934: – Modern application of infinitesimals
5847:; it can in fact have the same cardinality.
5250:is isomorphic to the reals. Hence we have a
5036:
5030:
5007:
4985:
4976:
4963:
4645:should be declared zero too, no matter what
4409:
4384:
4007:; the two are equivalent. The maximality of
4003:, directly in terms of the free ultrafilter
6104:Kanovei, Vladimir; Shelah, Saharon (2004),
6010:. New Haven: Yale Univ. Press. p. 67.
4816:From two complementary sets one belongs to
4317: = 0. Thus, if for two sequences
4243:would be isomorphic as an ordered field to
3167:. It is also stronger than that of being a
3132:shows that there is a definable, countably
1663:is the same for all nonzero infinitesimals
781:have identical behavior. For instance, in *
754:. However, statements of the form "for any
346:. The transfer principle states that true
7132:
7118:
7110:
7094:
6419:
6405:
6397:
5194:is an ordinary (called standard) real and
3991:of a commutative ring by a maximal ideal,
3711:
3707:
2387:resulting in a slightly different notation
327:. The term "hyper-real" was introduced by
6895:
6894:
6892:
6818:
6817:
6815:
6779:
6778:
6776:
6741:
6740:
6738:
6713:
6712:
6710:
6685:
6684:
6682:
6657:
6656:
6654:
6578:
6577:
6575:
6545:
6544:
6542:
6512:
6511:
6509:
6484:
6483:
6481:
6456:
6455:
6453:
6316:Jerison, Meyer; Gillman, Leonard (1976),
6310:Rings of real-valued continuous functions
6124:
6079:
5708:
5668:
5651:
5637:
5563:
5490:
5428:
5402:
5352:
5326:
5199:
5179:
5153:
5133:
5113:
5087:
5079:
5077:
5054:
5028:
5013:{\displaystyle f(\{x_{n}\})=\{f(x_{n})\}}
4998:
4970:
4955:
4932:
4912:
4892:
4784:
4758:
4713:
4693:
4673:
4650:
4627:
4607:
4567:
4538:
4509:
4482:
4476:
4456:
4427:
4397:
4367:
4322:
4206:
4201:
4195:
4166:
4161:
4146:
4141:
4136:
4121:
4116:
4104:
4099:
4090:
3796:
3783:
3764:
3751:
3732:
3719:
3692:
3679:
3666:
3641:
3628:
3615:
3606:
3580:
3554:
3475:
3462:
3449:
3436:
3423:
3410:
3385:
3372:
3359:
3334:
3321:
3308:
3299:
3045:
3025:
2996:
2945:
2940:
2934:
2899:
2861:
2841:
2780:
2750:
2691:
2680:
2638:
2633:
2627:
2599:
2597:
2568:
2542:
2516:
2475:
2443:
2405:
2329:
2273:
2231:
2215:
2183:
2167:
2126:
2092:
2077:
2036:
2023:
1983:
1976:
1961:
1887:
1872:
1818:
1816:
1786:
1765:
1744:
1714:
1694:
1668:
1596:
1545:
1543:
1520:
1500:
1474:
1445:
1410:
1372:
1367:(as is commonly done) to be the function
1352:
1329:
1306:
1222:
1183:
1157:
1137:
1102:
1079:
1056:
1018:
991:
797:
669:
600:
568:
493:
458:
447:
408:
397:
312:
289:
278:
270:
268:
248:
228:
202:
194:
192:
172:
127:Learn how and when to remove this message
5734:{\displaystyle \operatorname {st} (x)=x}
4853:because then everything would belong to
4823:Any set having a subset that belongs to
4224:, and hence has the same cardinality as
2385:can be established for doing so, though
1515:is said to be differentiable at a point
138:
6381:Nonstandard Analysis and the Hyperreals
6312:. I. Trans. Amer. Math. Soc. 64, 45—99.
5965:
5946: – Line formed by the real numbers
5242:being the infinitesimals; the quotient
4907:is a real function of a real variable
2981:{\displaystyle \int _{a}^{b}(f\ dx,dx)}
4738:Now the idea is to single out a bunch
4595:(the set of all natural numbers), so:
3841:guarantees the existence of many such
1859:{\displaystyle {\frac {df(x,dx)}{dx}}}
730:" still applies. The same is true for
187:is said to be finite if, and only if,
7244:Infinitesimal strain theory (physics)
4305:In our ring of sequences one can get
2592:is infinitesimal of the same sign as
2463:{\displaystyle \int (\varepsilon )\ }
2377:is infinitesimally small compared to
7:
6169:Woodin, W. H.; Dales, H. G. (1996),
6001:
5999:
5973:
5971:
5969:
3960:allows us to define a corresponding
2470:as a map sending any ordered triple
2431:{\displaystyle \ \varepsilon (x),\ }
65:adding citations to reliable sources
6608:Set-theoretically definable numbers
5302:) is infinitesimal. This number st(
4019:inverting the non-null elements of
3595:is a certain infinitesimal number.
1804:be a non-zero infinitesimal. Then,
4415:{\displaystyle z(a)=\{i:a_{i}=0\}}
4358:. He started with the ring of the
4203:
4163:
4138:
4118:
4101:
1023:
996:
671:
641:
615:
334:The hyperreal numbers satisfy the
25:
7346:Transcendental law of homogeneity
7239:Constructive nonstandard analysis
7183:The Method of Mechanical Theorems
7170:Criticism of nonstandard analysis
5927:Influence of nonstandard analysis
5916:Constructive nonstandard analysis
5632:is finite and not infinitesimal.
3995:is a field. This is also notated
3514:. We have a natural embedding of
3224:for details). When in the 1800s
7197:
7093:
5900:
5238:, with the unique maximal ideal
4039:. In the resulting field, these
3255:. Robinson developed his theory
3116:The use of the definite article
2894:if for any nonzero infinitesimal
959:For any finite hyperreal number
687:, where st(⋅) denotes the
41:
7229:Synthetic differential geometry
4217:{\displaystyle 2^{\aleph _{0}}}
3522:by identifying the real number
2991:is independent of the choice of
2365:term. In the hyperreal system,
905:but there is no such number in
855:
830:
811:
478:{\displaystyle \sin({\pi H})=0}
428:{\displaystyle \sin({\pi n})=0}
52:needs additional citations for
6587:{\displaystyle {\mathcal {P}}}
6375:. A text using infinitesimals.
6040:Dauben, Joseph Warren (1995).
5722:
5716:
5685:
5679:
5659:
5645:
5613:
5607:
5598:
5592:
5580:
5571:
5546:
5540:
5528:
5522:
5510:
5498:
5460:
5454:
5442:
5436:
5384:
5378:
5366:
5360:
5088:
5080:
5004:
4991:
4979:
4960:
4795:
4789:
4578:
4572:
4549:
4543:
4438:
4432:
4378:
4372:
4113:
4092:
3802:
3776:
3770:
3744:
3738:
3712:
3708:
3704:
3659:
3653:
3608:
3487:
3403:
3397:
3352:
3346:
3301:
3062:
3050:
2975:
2951:
2878:
2866:
2806:
2791:
2721:
2700:
2659:
2644:
2498:
2477:
2454:
2448:
2419:
2413:
2400:For any infinitesimal function
2228:
2218:
2123:
2113:
2020:
2010:
1923:
1917:
1908:
1893:
1842:
1827:
1755:
1749:
1632:
1626:
1617:
1602:
1569:
1554:
1456:
1450:
1427:
1412:
1383:
1377:
1258:
1252:
1243:
1228:
1206:
1191:
1119:
1104:
975:; it necessarily differs from
636:
630:
621:
606:
584:
578:
508:. The transfer principle for
466:
455:
416:
405:
279:
271:
203:
195:
1:
7454:Mathematics of infinitesimals
7398:Analyse des Infiniment Petits
7234:Smooth infinitesimal analysis
6942:Plane-based geometric algebra
6318:Rings of continuous functions
6303:American Mathematical Monthly
4734:should also be declared zero.
3222:Ghosts of departed quantities
1469:will equal the infinitesimal
1301:Note that the very notation "
1051:For any real-valued function
6902:{\displaystyle \mathbb {S} }
6825:{\displaystyle \mathbb {C} }
6786:{\displaystyle \mathbb {R} }
6748:{\displaystyle \mathbb {O} }
6720:{\displaystyle \mathbb {H} }
6692:{\displaystyle \mathbb {C} }
6664:{\displaystyle \mathbb {R} }
6552:{\displaystyle \mathbb {A} }
6519:{\displaystyle \mathbb {Q} }
6491:{\displaystyle \mathbb {Z} }
6463:{\displaystyle \mathbb {N} }
4845:Finally, we do not want the
4309: = 0 with neither
4066:. Since this field contains
2438:one may define the integral
167:numbers. A hyperreal number
6173:, Oxford: Clarendon Press,
5807:is a totally ordered field
5314:, conceptually the same as
3568:{\displaystyle 7+\epsilon }
726: + 0 =
7475:
6243:Goldblatt, Robert (1998),
6202:Princeton University Press
6006:Robinson, Abraham (1979).
5319:to the nearest real number
5042:{\displaystyle \{\dots \}}
4190:it is also no larger than
4015:, constructing a sequence
3507:, which is in fact a real
3244:continued (Ehrlich 2006).
3230:(ε, δ)-definition of limit
3171:in the sense of Dales and
1774:{\displaystyle f(x)=x^{2}}
702:
307:for all positive integers
300:{\displaystyle |x|<1/n}
29:
7362:Gottfried Wilhelm Leibniz
7195:
7084:
6932:Algebra of physical space
6113:Journal of Symbolic Logic
5862:can be identified with a
5811:containing the reals. If
3588:{\displaystyle \epsilon }
1732:For example, to find the
6988:Extended complex numbers
6971:Extended natural numbers
6384:. A gentle introduction.
5870:) with the real algebra
5101:{\displaystyle |x|<a}
4708:are declared zero, then
3967:in the commutative ring
3192:From Leibniz to Robinson
2504:{\displaystyle (a,b,dx)}
1029:{\displaystyle -\infty }
1002:{\displaystyle +\infty }
785:there exists an element
680:{\displaystyle \Delta x}
338:, a rigorous version of
216:{\displaystyle |x|<n}
5874:of functions from κ to
5703:is real if and only if
5340:{\displaystyle x\leq y}
5108:for some ordinary real
4497:{\displaystyle a_{i}=0}
4298: = 0 for all
3283:Ultrapower construction
2919:{\displaystyle \ dx,\ }
2836:A real-valued function
1495:A real-valued function
1398:{\displaystyle f(x)=x,}
18:Ultrapower construction
7291:Standard part function
7044:Transcendental numbers
6903:
6880:Hyperbolic quaternions
6826:
6787:
6749:
6721:
6693:
6665:
6588:
6553:
6520:
6492:
6464:
6343:Infinitesimal Calculus
6135:10.2178/jsl/1080938834
5735:
5692:
5620:
5553:
5467:
5417:
5416:{\displaystyle x<y}
5391:
5341:
5298: – st(
5208:
5188:
5168:
5142:
5122:
5102:
5063:
5062:{\displaystyle \dots }
5043:
5014:
4941:
4921:
4901:
4802:
4773:
4728:
4702:
4682:
4659:
4639:
4616:
4585:
4556:
4527:
4504:. It is clear that if
4498:
4465:
4451:is the set of indexes
4445:
4416:
4337:
4218:
4181:
3975:), and then to define
3812:
3589:
3569:
3494:
3247:However, in the 1960s
3242:non-Archimedean fields
3200:and (more explicitly)
3072:
3034:
3014:
2982:
2920:
2888:
2850:
2828:
2765:
2736:
2696:
2613:
2586:
2585:{\displaystyle \ dx\ }
2557:
2531:
2505:
2464:
2432:
2369: ≠ 0, since
2344:
2313:
2257:
2151:
2061:
1945:
1860:
1798:
1775:
1723:
1703:
1683:
1654:
1529:
1509:
1486:
1463:
1434:
1433:{\displaystyle (x,dx)}
1399:
1361:
1347:for if one interprets
1341:
1318:
1292:
1169:
1146:
1126:
1125:{\displaystyle (x,dx)}
1091:
1068:
1030:
1003:
896:
689:standard part function
681:
658:
502:
479:
429:
321:
301:
257:
237:
217:
181:
144:
7429:Mathematical analysis
7377:Augustin-Louis Cauchy
7189:Cavalieri's principle
6976:Extended real numbers
6904:
6827:
6797:Split-complex numbers
6788:
6750:
6722:
6694:
6666:
6589:
6554:
6530:Constructible numbers
6521:
6493:
6465:
6308:Hewitt, Edwin (1948)
6198:Non-standard analysis
5984:mathworld.wolfram.com
5736:
5693:
5621:
5554:
5468:
5418:
5392:
5342:
5209:
5189:
5169:
5143:
5123:
5103:
5064:
5044:
5015:
4942:
4922:
4902:
4803:
4774:
4753:and to declare that
4729:
4703:
4683:
4660:
4640:
4617:
4586:
4557:
4528:
4499:
4466:
4446:
4417:
4338:
4235:, the quotient field
4219:
4182:
4074:at least that of the
3813:
3590:
3570:
3495:
3122:the hyperreal numbers
3097:containing the reals
3073:
3035:
3015:
3013:{\displaystyle \ dx.}
2983:
2921:
2889:
2851:
2829:
2766:
2764:{\displaystyle \ N\ }
2737:
2676:
2614:
2612:{\displaystyle \,b-a}
2587:
2558:
2556:{\displaystyle \ b\ }
2532:
2530:{\displaystyle \ a\ }
2506:
2465:
2433:
2345:
2314:
2258:
2152:
2062:
1946:
1861:
1799:
1776:
1724:
1704:
1684:
1655:
1530:
1510:
1487:
1464:
1435:
1400:
1362:
1342:
1319:
1293:
1170:
1147:
1127:
1092:
1069:
1031:
1004:
897:
682:
664:for an infinitesimal
659:
503:
480:
430:
322:
302:
258:
238:
218:
182:
142:
7434:Nonstandard analysis
7219:Nonstandard calculus
7214:Nonstandard analysis
7008:Supernatural numbers
6918:Multicomplex numbers
6891:
6875:Dual-complex numbers
6814:
6775:
6737:
6709:
6681:
6653:
6635:Composition algebras
6603:Arithmetical numbers
6574:
6541:
6508:
6480:
6452:
6320:, Berlin, New York:
6247:, Berlin, New York:
5932:Nonstandard calculus
5827:(terminology due to
5707:
5636:
5562:
5489:
5427:
5401:
5351:
5325:
5222:The finite elements
5198:
5178:
5152:
5148:will be of the form
5132:
5112:
5076:
5053:
5027:
4954:
4931:
4911:
4891:
4801:{\displaystyle z(a)}
4783:
4757:
4712:
4692:
4672:
4649:
4626:
4606:
4584:{\displaystyle z(b)}
4566:
4555:{\displaystyle z(a)}
4537:
4533:, then the union of
4526:{\displaystyle ab=0}
4508:
4475:
4455:
4444:{\displaystyle z(a)}
4426:
4366:
4321:
4253:continuum hypothesis
4194:
4089:
3924:and it turns into a
3605:
3579:
3553:
3298:
3253:nonstandard analysis
3163:strictly containing
3154:continuum hypothesis
3146:elementary extension
3044:
3024:
2995:
2933:
2898:
2887:{\displaystyle \ \ }
2860:
2840:
2779:
2749:
2626:
2596:
2567:
2541:
2515:
2474:
2442:
2404:
2328:
2272:
2166:
2076:
1960:
1871:
1815:
1785:
1743:
1713:
1693:
1667:
1542:
1519:
1499:
1473:
1462:{\displaystyle d(x)}
1444:
1409:
1371:
1351:
1328:
1305:
1182:
1156:
1136:
1101:
1078:
1055:
1017:
990:
985:extended real number
909:. (In other words, *
796:
668:
567:
545:nonstandard analysis
529:method of exhaustion
512:is a consequence of
492:
446:
396:
311:
267:
247:
227:
191:
171:
61:improve this article
7439:Field (mathematics)
7403:Elementary Calculus
7284:Individual concepts
7224:Internal set theory
6913:Split-biquaternions
6625:Eisenstein integers
6563:Closed-form numbers
5978:Weisstein, Eric W.
5167:{\displaystyle y+d}
4772:{\displaystyle a=0}
4727:{\displaystyle a+b}
4336:{\displaystyle a,b}
4313: = 0 nor
4151:
4027:vanishes is not in
3526:with the sequence (
3071:{\displaystyle \ .}
2950:
2643:
2343:{\displaystyle =2x}
1682:{\displaystyle dx.}
519:Concerns about the
358:. For example, the
354:are also valid in *
32:R* (disambiguation)
7296:Transfer principle
7160:Leibniz's notation
7071:Profinite integers
7034:Irrational numbers
6899:
6822:
6783:
6745:
6717:
6689:
6661:
6618:Gaussian rationals
6598:Computable numbers
6584:
6549:
6516:
6488:
6460:
6347:Dover Publications
6305:89: 362–370.
5980:"Hyperreal Number"
5908:Mathematics portal
5815:strictly contains
5731:
5688:
5616:
5549:
5463:
5413:
5387:
5337:
5204:
5184:
5164:
5138:
5118:
5098:
5059:
5039:
5010:
4937:
4917:
4897:
4827:, also belongs to
4798:
4769:
4724:
4698:
4678:
4655:
4638:{\displaystyle ab}
4635:
4622:is declared zero,
4612:
4581:
4552:
4523:
4494:
4461:
4441:
4412:
4333:
4214:
4177:
4137:
3808:
3585:
3565:
3490:
3068:
3030:
3010:
2978:
2936:
2916:
2884:
2846:
2824:
2761:
2732:
2629:
2609:
2582:
2553:
2527:
2501:
2460:
2428:
2340:
2309:
2253:
2147:
2057:
1941:
1856:
1797:{\displaystyle dx}
1794:
1771:
1719:
1699:
1679:
1650:
1525:
1505:
1485:{\displaystyle dx}
1482:
1459:
1430:
1395:
1357:
1340:{\displaystyle d,}
1337:
1317:{\displaystyle dx}
1314:
1288:
1168:{\displaystyle dx}
1165:
1142:
1122:
1090:{\displaystyle df}
1087:
1067:{\displaystyle f,}
1064:
1026:
999:
935:improper integrals
892:
752:transfer principle
713:to form a system *
705:Transfer principle
699:Transfer principle
677:
654:
498:
475:
425:
336:transfer principle
317:
297:
253:
233:
213:
177:
145:
76:"Hyperreal number"
7444:Real closed field
7416:
7415:
7331:Law of continuity
7321:Levi-Civita field
7306:Increment theorem
7265:Hyperreal numbers
7107:
7106:
7018:Superreal numbers
6998:Levi-Civita field
6993:Hyperreal numbers
6937:Spacetime algebra
6923:Geometric algebra
6836:Bicomplex numbers
6802:Split-quaternions
6643:Division algebras
6613:Gaussian integers
6535:Algebraic numbers
6438:definable numbers
6356:978-0-486-42886-4
6331:978-0-387-90198-5
6258:978-0-387-98464-3
6211:978-0-691-04490-3
6194:Robinson, Abraham
6180:978-0-19-853991-9
6051:978-0-691-03745-5
6017:978-0-300-02072-4
5938:Real closed field
5890:in model theory.
5856:discrete topology
5768:, also called a T
5477:We have, if both
5207:{\displaystyle d}
5187:{\displaystyle y}
5141:{\displaystyle x}
5121:{\displaystyle a}
4940:{\displaystyle f}
4920:{\displaystyle x}
4900:{\displaystyle f}
4879:ultrafilter lemma
4701:{\displaystyle b}
4681:{\displaystyle a}
4658:{\displaystyle b}
4615:{\displaystyle a}
4464:{\displaystyle i}
4261:order isomorphism
3257:nonconstructively
3161:real closed field
3049:
3033:{\displaystyle f}
3000:
2959:
2915:
2903:
2883:
2865:
2849:{\displaystyle f}
2799:
2784:
2775:number satisfying
2760:
2754:
2714:
2581:
2572:
2552:
2546:
2526:
2520:
2459:
2427:
2409:
2353:
2352:
2246:
2210:
2141:
2051:
1935:
1854:
1722:{\displaystyle x}
1702:{\displaystyle f}
1644:
1581:
1528:{\displaystyle x}
1508:{\displaystyle f}
1440:the differential
1360:{\displaystyle x}
1278:
1270:
1145:{\displaystyle x}
1074:the differential
768:first-order logic
648:
501:{\displaystyle H}
386:real closed field
350:statements about
344:law of continuity
320:{\displaystyle n}
256:{\displaystyle x}
236:{\displaystyle n}
223:for some integer
180:{\displaystyle x}
153:hyperreal numbers
137:
136:
129:
111:
16:(Redirected from
7466:
7372:Pierre de Fermat
7367:Abraham Robinson
7207:Related branches
7201:
7134:
7127:
7120:
7111:
7097:
7096:
7064:
7054:
6966:Cardinal numbers
6927:Clifford algebra
6908:
6906:
6905:
6900:
6898:
6870:Dual quaternions
6831:
6829:
6828:
6823:
6821:
6792:
6790:
6789:
6784:
6782:
6754:
6752:
6751:
6746:
6744:
6726:
6724:
6723:
6718:
6716:
6698:
6696:
6695:
6690:
6688:
6670:
6668:
6667:
6662:
6660:
6593:
6591:
6590:
6585:
6583:
6582:
6558:
6556:
6555:
6550:
6548:
6525:
6523:
6522:
6517:
6515:
6502:Rational numbers
6497:
6495:
6494:
6489:
6487:
6469:
6467:
6466:
6461:
6459:
6421:
6414:
6407:
6398:
6359:
6334:
6298:
6276:Ball, W.W. Rouse
6262:
6261:
6240:
6234:
6233:
6222:
6216:
6214:
6190:
6184:
6183:
6166:
6160:
6159:
6158:
6157:
6151:
6145:, archived from
6128:
6110:
6101:
6095:
6092:
6086:
6085:
6083:
6071:
6065:
6062:
6056:
6055:
6037:
6031:
6028:
6022:
6021:
6003:
5994:
5993:
5991:
5990:
5975:
5949:
5910:
5905:
5904:
5843:is greater than
5756:Hyperreal fields
5750:locally constant
5740:
5738:
5737:
5732:
5697:
5695:
5694:
5689:
5672:
5655:
5625:
5623:
5622:
5617:
5558:
5556:
5555:
5550:
5472:
5470:
5469:
5464:
5422:
5420:
5419:
5414:
5396:
5394:
5393:
5388:
5346:
5344:
5343:
5338:
5306:) is called the
5234:, and in fact a
5213:
5211:
5210:
5205:
5193:
5191:
5190:
5185:
5173:
5171:
5170:
5165:
5147:
5145:
5144:
5139:
5127:
5125:
5124:
5119:
5107:
5105:
5104:
5099:
5091:
5083:
5068:
5066:
5065:
5060:
5048:
5046:
5045:
5040:
5019:
5017:
5016:
5011:
5003:
5002:
4975:
4974:
4946:
4944:
4943:
4938:
4926:
4924:
4923:
4918:
4906:
4904:
4903:
4898:
4807:
4805:
4804:
4799:
4778:
4776:
4775:
4770:
4733:
4731:
4730:
4725:
4707:
4705:
4704:
4699:
4687:
4685:
4684:
4679:
4664:
4662:
4661:
4656:
4644:
4642:
4641:
4636:
4621:
4619:
4618:
4613:
4590:
4588:
4587:
4582:
4561:
4559:
4558:
4553:
4532:
4530:
4529:
4524:
4503:
4501:
4500:
4495:
4487:
4486:
4470:
4468:
4467:
4462:
4450:
4448:
4447:
4442:
4421:
4419:
4418:
4413:
4402:
4401:
4360:Cauchy sequences
4342:
4340:
4339:
4334:
4223:
4221:
4220:
4215:
4213:
4212:
4211:
4210:
4186:
4184:
4183:
4178:
4173:
4172:
4171:
4170:
4153:
4152:
4150:
4145:
4128:
4127:
4126:
4125:
4111:
4110:
4109:
4108:
4082:has cardinality
3817:
3815:
3814:
3809:
3801:
3800:
3788:
3787:
3769:
3768:
3756:
3755:
3737:
3736:
3724:
3723:
3697:
3696:
3684:
3683:
3671:
3670:
3646:
3645:
3633:
3632:
3620:
3619:
3594:
3592:
3591:
3586:
3574:
3572:
3571:
3566:
3505:commutative ring
3499:
3497:
3496:
3491:
3480:
3479:
3467:
3466:
3454:
3453:
3441:
3440:
3428:
3427:
3415:
3414:
3390:
3389:
3377:
3376:
3364:
3363:
3339:
3338:
3326:
3325:
3313:
3312:
3249:Abraham Robinson
3126:Vladimir Kanovei
3089:The hyperreals *
3077:
3075:
3074:
3069:
3047:
3039:
3037:
3036:
3031:
3019:
3017:
3016:
3011:
2998:
2987:
2985:
2984:
2979:
2957:
2949:
2944:
2925:
2923:
2922:
2917:
2913:
2901:
2893:
2891:
2890:
2885:
2881:
2863:
2855:
2853:
2852:
2847:
2833:
2831:
2830:
2825:
2797:
2782:
2770:
2768:
2767:
2762:
2758:
2752:
2741:
2739:
2738:
2733:
2728:
2724:
2712:
2695:
2690:
2642:
2637:
2618:
2616:
2615:
2610:
2591:
2589:
2588:
2583:
2579:
2570:
2562:
2560:
2559:
2554:
2550:
2544:
2536:
2534:
2533:
2528:
2524:
2518:
2510:
2508:
2507:
2502:
2469:
2467:
2466:
2461:
2457:
2437:
2435:
2434:
2429:
2425:
2407:
2349:
2347:
2346:
2341:
2318:
2316:
2315:
2310:
2308:
2304:
2262:
2260:
2259:
2254:
2252:
2248:
2247:
2245:
2237:
2236:
2235:
2216:
2211:
2209:
2201:
2184:
2156:
2154:
2153:
2148:
2146:
2142:
2140:
2132:
2131:
2130:
2093:
2066:
2064:
2063:
2058:
2056:
2052:
2050:
2042:
2041:
2040:
2028:
2027:
1988:
1987:
1977:
1950:
1948:
1947:
1942:
1940:
1936:
1934:
1926:
1888:
1865:
1863:
1862:
1857:
1855:
1853:
1845:
1819:
1809:
1808:
1803:
1801:
1800:
1795:
1780:
1778:
1777:
1772:
1770:
1769:
1728:
1726:
1725:
1720:
1708:
1706:
1705:
1700:
1688:
1686:
1685:
1680:
1659:
1657:
1656:
1651:
1649:
1645:
1643:
1635:
1597:
1582:
1580:
1572:
1546:
1535:if the quotient
1534:
1532:
1531:
1526:
1514:
1512:
1511:
1506:
1491:
1489:
1488:
1483:
1468:
1466:
1465:
1460:
1439:
1437:
1436:
1431:
1404:
1402:
1401:
1396:
1366:
1364:
1363:
1358:
1346:
1344:
1343:
1338:
1323:
1321:
1320:
1315:
1297:
1295:
1294:
1289:
1276:
1275:
1271:
1269:
1261:
1223:
1174:
1172:
1171:
1166:
1151:
1149:
1148:
1143:
1131:
1129:
1128:
1123:
1096:
1094:
1093:
1088:
1073:
1071:
1070:
1065:
1035:
1033:
1032:
1027:
1008:
1006:
1005:
1000:
901:
899:
898:
893:
686:
684:
683:
678:
663:
661:
660:
655:
653:
649:
647:
639:
601:
577:
533:Abraham Robinson
531:. In the 1960s,
507:
505:
504:
499:
484:
482:
481:
476:
465:
434:
432:
431:
426:
415:
379:
326:
324:
323:
318:
306:
304:
303:
298:
293:
282:
274:
262:
260:
259:
254:
242:
240:
239:
234:
222:
220:
219:
214:
206:
198:
186:
184:
183:
178:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
7474:
7473:
7469:
7468:
7467:
7465:
7464:
7463:
7419:
7418:
7417:
7412:
7408:Cours d'Analyse
7386:
7350:
7341:Microcontinuity
7326:Hyperfinite set
7279:
7275:Surreal numbers
7248:
7202:
7193:
7165:Integral symbol
7143:
7138:
7108:
7103:
7080:
7059:
7049:
7022:
7013:Surreal numbers
7003:Ordinal numbers
6948:
6889:
6888:
6850:
6812:
6811:
6809:
6807:Split-octonions
6773:
6772:
6764:
6758:
6735:
6734:
6707:
6706:
6679:
6678:
6675:Complex numbers
6651:
6650:
6629:
6572:
6571:
6539:
6538:
6506:
6505:
6478:
6477:
6450:
6449:
6446:Natural numbers
6431:
6425:
6366:
6357:
6340:
6332:
6322:Springer-Verlag
6315:
6296:
6274:
6271:
6269:Further reading
6266:
6265:
6259:
6249:Springer-Verlag
6242:
6241:
6237:
6224:
6223:
6219:
6212:
6192:
6191:
6187:
6181:
6168:
6167:
6163:
6155:
6153:
6149:
6108:
6103:
6102:
6098:
6093:
6089:
6073:
6072:
6068:
6063:
6059:
6052:
6039:
6038:
6034:
6029:
6025:
6018:
6005:
6004:
5997:
5988:
5986:
5977:
5976:
5967:
5962:
5947:
5906:
5899:
5896:
5864:cardinal number
5858:; in this case
5837:hyperreal field
5825:hyperreal ideal
5771:
5766:Tychonoff space
5758:
5705:
5704:
5634:
5633:
5560:
5559:
5487:
5486:
5425:
5424:
5423:does not imply
5399:
5398:
5349:
5348:
5323:
5322:
5220:
5196:
5195:
5176:
5175:
5150:
5149:
5130:
5129:
5110:
5109:
5074:
5073:
5051:
5050:
5025:
5024:
4994:
4966:
4952:
4951:
4929:
4928:
4909:
4908:
4889:
4888:
4875:axiom of choice
4781:
4780:
4779:if and only if
4755:
4754:
4710:
4709:
4690:
4689:
4670:
4669:
4647:
4646:
4624:
4623:
4604:
4603:
4564:
4563:
4535:
4534:
4506:
4505:
4478:
4473:
4472:
4453:
4452:
4424:
4423:
4393:
4364:
4363:
4319:
4318:
4297:
4276:
4202:
4197:
4192:
4191:
4162:
4157:
4132:
4117:
4112:
4100:
4095:
4087:
4086:
3912:
3903:
3890:
3883:
3876:
3869:
3862:
3855:
3835:natural numbers
3792:
3779:
3760:
3747:
3728:
3715:
3688:
3675:
3662:
3637:
3624:
3611:
3603:
3602:
3577:
3576:
3551:
3550:
3471:
3458:
3445:
3432:
3419:
3406:
3381:
3368:
3355:
3330:
3317:
3304:
3296:
3295:
3285:
3214:George Berkeley
3194:
3181:
3169:superreal field
3087:
3042:
3041:
3022:
3021:
2993:
2992:
2931:
2930:
2896:
2895:
2858:
2857:
2838:
2837:
2777:
2776:
2747:
2746:
2675:
2671:
2624:
2623:
2619:) to the value
2594:
2593:
2565:
2564:
2539:
2538:
2513:
2512:
2472:
2471:
2440:
2439:
2402:
2401:
2395:
2326:
2325:
2288:
2284:
2270:
2269:
2238:
2227:
2217:
2202:
2185:
2182:
2178:
2164:
2163:
2133:
2122:
2094:
2088:
2074:
2073:
2043:
2032:
2019:
1979:
1978:
1972:
1958:
1957:
1927:
1889:
1883:
1869:
1868:
1846:
1820:
1813:
1812:
1783:
1782:
1761:
1741:
1740:
1711:
1710:
1691:
1690:
1665:
1664:
1636:
1598:
1592:
1573:
1547:
1540:
1539:
1517:
1516:
1497:
1496:
1471:
1470:
1442:
1441:
1407:
1406:
1405:then for every
1369:
1368:
1349:
1348:
1326:
1325:
1303:
1302:
1262:
1224:
1218:
1180:
1179:
1154:
1153:
1134:
1133:
1099:
1098:
1076:
1075:
1053:
1052:
1042:
1040:Differentiation
1015:
1014:
988:
987:
927:
925:Use in analysis
794:
793:
707:
701:
666:
665:
640:
602:
596:
570:
565:
564:
490:
489:
444:
443:
442:, one also has
394:
393:
375: +
367: +
363:
360:commutative law
309:
308:
265:
264:
245:
244:
225:
224:
189:
188:
169:
168:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
7472:
7470:
7462:
7461:
7456:
7451:
7446:
7441:
7436:
7431:
7421:
7420:
7414:
7413:
7411:
7410:
7405:
7400:
7394:
7392:
7388:
7387:
7385:
7384:
7382:Leonhard Euler
7379:
7374:
7369:
7364:
7358:
7356:
7355:Mathematicians
7352:
7351:
7349:
7348:
7343:
7338:
7333:
7328:
7323:
7318:
7313:
7308:
7303:
7298:
7293:
7287:
7285:
7281:
7280:
7278:
7277:
7272:
7267:
7262:
7256:
7254:
7253:Formalizations
7250:
7249:
7247:
7246:
7241:
7236:
7231:
7226:
7221:
7216:
7210:
7208:
7204:
7203:
7196:
7194:
7192:
7191:
7186:
7179:
7172:
7167:
7162:
7157:
7151:
7149:
7145:
7144:
7141:Infinitesimals
7139:
7137:
7136:
7129:
7122:
7114:
7105:
7104:
7102:
7101:
7091:
7089:Classification
7085:
7082:
7081:
7079:
7078:
7076:Normal numbers
7073:
7068:
7046:
7041:
7036:
7030:
7028:
7024:
7023:
7021:
7020:
7015:
7010:
7005:
7000:
6995:
6990:
6985:
6984:
6983:
6973:
6968:
6962:
6960:
6958:infinitesimals
6950:
6949:
6947:
6946:
6945:
6944:
6939:
6934:
6920:
6915:
6910:
6897:
6882:
6877:
6872:
6867:
6861:
6859:
6852:
6851:
6849:
6848:
6843:
6838:
6833:
6820:
6804:
6799:
6794:
6781:
6768:
6766:
6760:
6759:
6757:
6756:
6743:
6728:
6715:
6700:
6687:
6672:
6659:
6639:
6637:
6631:
6630:
6628:
6627:
6622:
6621:
6620:
6610:
6605:
6600:
6595:
6581:
6565:
6560:
6547:
6532:
6527:
6514:
6499:
6486:
6471:
6458:
6442:
6440:
6433:
6432:
6426:
6424:
6423:
6416:
6409:
6401:
6395:
6394:
6385:
6376:
6372:Brief Calculus
6365:
6364:External links
6362:
6361:
6360:
6355:
6338:
6335:
6330:
6313:
6306:
6299:
6294:
6270:
6267:
6264:
6263:
6257:
6235:
6226:Loeb, Peter A.
6217:
6210:
6185:
6179:
6161:
6096:
6087:
6066:
6057:
6050:
6032:
6023:
6016:
5995:
5964:
5963:
5961:
5958:
5957:
5956:
5953:Surreal number
5950:
5941:
5935:
5929:
5924:
5918:
5912:
5911:
5895:
5892:
5794:factor algebra
5769:
5757:
5754:
5744:The map st is
5742:
5741:
5730:
5727:
5724:
5721:
5718:
5715:
5712:
5698:
5687:
5684:
5681:
5678:
5675:
5671:
5667:
5664:
5661:
5658:
5654:
5650:
5647:
5644:
5641:
5626:
5615:
5612:
5609:
5606:
5603:
5600:
5597:
5594:
5591:
5588:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5506:
5503:
5500:
5497:
5494:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5412:
5409:
5406:
5386:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5359:
5356:
5336:
5333:
5330:
5236:valuation ring
5219:
5216:
5203:
5183:
5163:
5160:
5157:
5137:
5117:
5097:
5094:
5090:
5086:
5082:
5058:
5038:
5035:
5032:
5021:
5020:
5009:
5006:
5001:
4997:
4993:
4990:
4987:
4984:
4981:
4978:
4973:
4969:
4965:
4962:
4959:
4936:
4916:
4896:
4867:Fréchet filter
4859:
4858:
4843:
4832:
4821:
4797:
4794:
4791:
4788:
4768:
4765:
4762:
4736:
4735:
4723:
4720:
4717:
4697:
4677:
4666:
4654:
4634:
4631:
4611:
4600:
4580:
4577:
4574:
4571:
4551:
4548:
4545:
4542:
4522:
4519:
4516:
4513:
4493:
4490:
4485:
4481:
4460:
4440:
4437:
4434:
4431:
4411:
4408:
4405:
4400:
4396:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4332:
4329:
4326:
4293:
4275:
4272:
4209:
4205:
4200:
4188:
4187:
4176:
4169:
4165:
4160:
4156:
4149:
4144:
4140:
4135:
4131:
4124:
4120:
4115:
4107:
4103:
4098:
4094:
4047:are inverses.
4031:, the product
3922:total preorder
3908:
3899:
3888:
3881:
3874:
3867:
3860:
3853:
3819:
3818:
3807:
3804:
3799:
3795:
3791:
3786:
3782:
3778:
3775:
3772:
3767:
3763:
3759:
3754:
3750:
3746:
3743:
3740:
3735:
3731:
3727:
3722:
3718:
3714:
3710:
3706:
3703:
3700:
3695:
3691:
3687:
3682:
3678:
3674:
3669:
3665:
3661:
3658:
3655:
3652:
3649:
3644:
3640:
3636:
3631:
3627:
3623:
3618:
3614:
3610:
3584:
3564:
3561:
3558:
3501:
3500:
3489:
3486:
3483:
3478:
3474:
3470:
3465:
3461:
3457:
3452:
3448:
3444:
3439:
3435:
3431:
3426:
3422:
3418:
3413:
3409:
3405:
3402:
3399:
3396:
3393:
3388:
3384:
3380:
3375:
3371:
3367:
3362:
3358:
3354:
3351:
3348:
3345:
3342:
3337:
3333:
3329:
3324:
3320:
3316:
3311:
3307:
3303:
3284:
3281:
3193:
3190:
3180:
3177:
3130:Saharon Shelah
3120:in the phrase
3111:order topology
3086:
3083:
3067:
3064:
3061:
3058:
3055:
3052:
3029:
3009:
3006:
3003:
2989:
2988:
2977:
2974:
2971:
2968:
2965:
2962:
2956:
2953:
2948:
2943:
2939:
2912:
2909:
2906:
2880:
2877:
2874:
2871:
2868:
2845:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2796:
2793:
2790:
2787:
2757:
2743:
2742:
2731:
2727:
2723:
2720:
2717:
2711:
2708:
2705:
2702:
2699:
2694:
2689:
2686:
2683:
2679:
2674:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2641:
2636:
2632:
2608:
2605:
2602:
2578:
2575:
2549:
2523:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2456:
2453:
2450:
2447:
2424:
2421:
2418:
2415:
2412:
2394:
2391:
2355:
2354:
2351:
2350:
2339:
2336:
2333:
2323:
2320:
2319:
2307:
2303:
2300:
2297:
2294:
2291:
2287:
2283:
2280:
2277:
2267:
2264:
2263:
2251:
2244:
2241:
2234:
2230:
2226:
2223:
2220:
2214:
2208:
2205:
2200:
2197:
2194:
2191:
2188:
2181:
2177:
2174:
2171:
2161:
2158:
2157:
2145:
2139:
2136:
2129:
2125:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2091:
2087:
2084:
2081:
2071:
2068:
2067:
2055:
2049:
2046:
2039:
2035:
2031:
2026:
2022:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1986:
1982:
1975:
1971:
1968:
1965:
1955:
1952:
1951:
1939:
1933:
1930:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1886:
1882:
1879:
1876:
1866:
1852:
1849:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1793:
1790:
1768:
1764:
1760:
1757:
1754:
1751:
1748:
1718:
1698:
1678:
1675:
1672:
1661:
1660:
1648:
1642:
1639:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1595:
1591:
1588:
1585:
1579:
1576:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1524:
1504:
1481:
1478:
1458:
1455:
1452:
1449:
1429:
1426:
1423:
1420:
1417:
1414:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1356:
1336:
1333:
1313:
1310:
1299:
1298:
1287:
1284:
1281:
1274:
1268:
1265:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1221:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1164:
1161:
1141:
1121:
1118:
1115:
1112:
1109:
1106:
1086:
1083:
1063:
1060:
1041:
1038:
1025:
1022:
998:
995:
926:
923:
903:
902:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
854:
851:
848:
845:
842:
839:
836:
833:
829:
826:
823:
820:
817:
814:
810:
807:
804:
801:
732:quantification
703:Main article:
700:
697:
676:
673:
652:
646:
643:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
599:
595:
592:
589:
586:
583:
580:
576:
573:
497:
474:
471:
468:
464:
461:
457:
454:
451:
424:
421:
418:
414:
411:
407:
404:
401:
316:
296:
292:
288:
285:
281:
277:
273:
252:
232:
212:
209:
205:
201:
197:
176:
135:
134:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7471:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7437:
7435:
7432:
7430:
7427:
7426:
7424:
7409:
7406:
7404:
7401:
7399:
7396:
7395:
7393:
7389:
7383:
7380:
7378:
7375:
7373:
7370:
7368:
7365:
7363:
7360:
7359:
7357:
7353:
7347:
7344:
7342:
7339:
7337:
7334:
7332:
7329:
7327:
7324:
7322:
7319:
7317:
7314:
7312:
7309:
7307:
7304:
7302:
7299:
7297:
7294:
7292:
7289:
7288:
7286:
7282:
7276:
7273:
7271:
7268:
7266:
7263:
7261:
7260:Differentials
7258:
7257:
7255:
7251:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7211:
7209:
7205:
7200:
7190:
7187:
7185:
7184:
7180:
7178:
7177:
7173:
7171:
7168:
7166:
7163:
7161:
7158:
7156:
7153:
7152:
7150:
7146:
7142:
7135:
7130:
7128:
7123:
7121:
7116:
7115:
7112:
7100:
7092:
7090:
7087:
7086:
7083:
7077:
7074:
7072:
7069:
7066:
7062:
7056:
7052:
7047:
7045:
7042:
7040:
7039:Fuzzy numbers
7037:
7035:
7032:
7031:
7029:
7025:
7019:
7016:
7014:
7011:
7009:
7006:
7004:
7001:
6999:
6996:
6994:
6991:
6989:
6986:
6982:
6979:
6978:
6977:
6974:
6972:
6969:
6967:
6964:
6963:
6961:
6959:
6955:
6951:
6943:
6940:
6938:
6935:
6933:
6930:
6929:
6928:
6924:
6921:
6919:
6916:
6914:
6911:
6886:
6883:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6862:
6860:
6858:
6853:
6847:
6844:
6842:
6841:Biquaternions
6839:
6837:
6834:
6808:
6805:
6803:
6800:
6798:
6795:
6770:
6769:
6767:
6761:
6732:
6729:
6704:
6701:
6676:
6673:
6648:
6644:
6641:
6640:
6638:
6636:
6632:
6626:
6623:
6619:
6616:
6615:
6614:
6611:
6609:
6606:
6604:
6601:
6599:
6596:
6569:
6566:
6564:
6561:
6536:
6533:
6531:
6528:
6503:
6500:
6475:
6472:
6447:
6444:
6443:
6441:
6439:
6434:
6429:
6422:
6417:
6415:
6410:
6408:
6403:
6402:
6399:
6392:
6391:
6386:
6383:
6382:
6377:
6374:
6373:
6368:
6367:
6363:
6358:
6352:
6348:
6344:
6339:
6336:
6333:
6327:
6323:
6319:
6314:
6311:
6307:
6304:
6300:
6297:
6295:0-486-20630-0
6291:
6287:
6283:
6282:
6277:
6273:
6272:
6268:
6260:
6254:
6250:
6246:
6239:
6236:
6231:
6227:
6221:
6218:
6213:
6207:
6203:
6199:
6195:
6189:
6186:
6182:
6176:
6172:
6165:
6162:
6152:on 2004-08-05
6148:
6144:
6140:
6136:
6132:
6127:
6122:
6118:
6114:
6107:
6100:
6097:
6091:
6088:
6082:
6077:
6070:
6067:
6061:
6058:
6053:
6047:
6043:
6036:
6033:
6027:
6024:
6019:
6013:
6009:
6002:
6000:
5996:
5985:
5981:
5974:
5972:
5970:
5966:
5959:
5954:
5951:
5945:
5942:
5939:
5936:
5933:
5930:
5928:
5925:
5922:
5919:
5917:
5914:
5913:
5909:
5903:
5898:
5893:
5891:
5889:
5885:
5881:
5877:
5873:
5869:
5865:
5861:
5857:
5853:
5848:
5846:
5842:
5838:
5834:
5830:
5826:
5822:
5818:
5814:
5810:
5806:
5802:
5798:
5795:
5791:
5787:
5786:maximal ideal
5783:
5779:
5775:
5772:space, and C(
5767:
5763:
5755:
5753:
5751:
5747:
5728:
5725:
5719:
5713:
5710:
5702:
5699:
5682:
5676:
5673:
5669:
5665:
5662:
5656:
5652:
5648:
5642:
5639:
5631:
5627:
5610:
5604:
5601:
5595:
5589:
5586:
5583:
5577:
5574:
5568:
5565:
5543:
5537:
5534:
5531:
5525:
5519:
5516:
5513:
5507:
5504:
5501:
5495:
5492:
5484:
5480:
5476:
5475:
5474:
5457:
5451:
5448:
5445:
5439:
5433:
5430:
5410:
5407:
5404:
5381:
5375:
5372:
5369:
5363:
5357:
5354:
5334:
5331:
5328:
5320:
5317:
5313:
5309:
5308:standard part
5305:
5301:
5297:
5293:
5289:
5285:
5281:
5277:
5273:
5269:
5265:
5261:
5257:
5253:
5249:
5245:
5241:
5237:
5233:
5229:
5225:
5217:
5215:
5201:
5181:
5161:
5158:
5155:
5135:
5115:
5095:
5092:
5084:
5070:
5056:
5033:
4999:
4995:
4988:
4982:
4971:
4967:
4957:
4950:
4949:
4948:
4934:
4914:
4894:
4885:
4882:
4880:
4876:
4872:
4868:
4864:
4856:
4852:
4849:to belong to
4848:
4844:
4841:
4837:
4833:
4830:
4826:
4822:
4819:
4815:
4814:
4813:
4811:
4792:
4786:
4766:
4763:
4760:
4752:
4748:
4745:
4741:
4721:
4718:
4715:
4695:
4675:
4667:
4652:
4632:
4629:
4609:
4601:
4598:
4597:
4596:
4594:
4575:
4569:
4546:
4540:
4520:
4517:
4514:
4511:
4491:
4488:
4483:
4479:
4458:
4435:
4429:
4406:
4403:
4398:
4394:
4390:
4387:
4381:
4375:
4369:
4361:
4357:
4352:
4350:
4346:
4330:
4327:
4324:
4316:
4312:
4308:
4303:
4301:
4296:
4292:
4288:
4283:
4281:
4273:
4271:
4269:
4264:
4262:
4258:
4254:
4250:
4246:
4242:
4238:
4234:
4229:
4227:
4207:
4198:
4174:
4167:
4158:
4154:
4147:
4142:
4133:
4129:
4122:
4105:
4096:
4085:
4084:
4083:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4048:
4046:
4042:
4038:
4034:
4030:
4026:
4022:
4018:
4014:
4010:
4006:
4002:
3998:
3994:
3990:
3986:
3982:
3978:
3974:
3970:
3966:
3963:
3962:maximal ideal
3959:
3955:
3951:
3947:
3943:
3939:
3935:
3931:
3927:
3923:
3918:
3916:
3911:
3907:
3902:
3898:
3894:
3887:
3880:
3873:
3866:
3859:
3852:
3848:
3844:
3840:
3836:
3832:
3829:
3825:
3824:partial order
3805:
3797:
3793:
3789:
3784:
3780:
3773:
3765:
3761:
3757:
3752:
3748:
3741:
3733:
3729:
3725:
3720:
3716:
3701:
3698:
3693:
3689:
3685:
3680:
3676:
3672:
3667:
3663:
3656:
3650:
3647:
3642:
3638:
3634:
3629:
3625:
3621:
3616:
3612:
3601:
3600:
3599:
3596:
3582:
3562:
3559:
3556:
3548:
3545:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3510:
3506:
3484:
3481:
3476:
3472:
3468:
3463:
3459:
3455:
3450:
3446:
3442:
3437:
3433:
3429:
3424:
3420:
3416:
3411:
3407:
3400:
3394:
3391:
3386:
3382:
3378:
3373:
3369:
3365:
3360:
3356:
3349:
3343:
3340:
3335:
3331:
3327:
3322:
3318:
3314:
3309:
3305:
3294:
3293:
3292:
3290:
3282:
3280:
3278:
3274:
3270:
3266:
3262:
3258:
3254:
3250:
3245:
3243:
3239:
3235:
3231:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3199:
3191:
3189:
3187:
3178:
3176:
3174:
3170:
3166:
3162:
3157:
3155:
3151:
3147:
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3114:
3112:
3108:
3104:
3100:
3096:
3095:ordered field
3092:
3084:
3082:
3078:
3065:
3059:
3056:
3053:
3027:
3007:
3004:
3001:
2972:
2969:
2966:
2963:
2960:
2954:
2946:
2941:
2937:
2929:
2928:
2927:
2926:the integral
2910:
2907:
2904:
2875:
2872:
2869:
2843:
2834:
2821:
2818:
2815:
2812:
2809:
2803:
2800:
2794:
2788:
2785:
2774:
2755:
2729:
2725:
2718:
2715:
2709:
2706:
2703:
2697:
2692:
2687:
2684:
2681:
2677:
2672:
2668:
2665:
2662:
2656:
2653:
2650:
2647:
2639:
2634:
2630:
2622:
2621:
2620:
2606:
2603:
2600:
2576:
2573:
2563:are real, and
2547:
2521:
2495:
2492:
2489:
2486:
2483:
2480:
2451:
2445:
2422:
2416:
2410:
2398:
2392:
2390:
2388:
2382:
2380:
2376:
2372:
2368:
2364:
2360:
2337:
2334:
2331:
2324:
2322:
2321:
2305:
2301:
2298:
2295:
2292:
2289:
2285:
2281:
2278:
2275:
2268:
2266:
2265:
2249:
2242:
2239:
2232:
2224:
2221:
2212:
2206:
2203:
2198:
2195:
2192:
2189:
2186:
2179:
2175:
2172:
2169:
2162:
2160:
2159:
2143:
2137:
2134:
2127:
2119:
2116:
2110:
2107:
2104:
2101:
2098:
2095:
2089:
2085:
2082:
2079:
2072:
2070:
2069:
2053:
2047:
2044:
2037:
2033:
2029:
2024:
2016:
2013:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1984:
1980:
1973:
1969:
1966:
1963:
1956:
1954:
1953:
1937:
1931:
1928:
1920:
1914:
1911:
1905:
1902:
1899:
1896:
1890:
1884:
1880:
1877:
1874:
1867:
1850:
1847:
1839:
1836:
1833:
1830:
1824:
1821:
1811:
1810:
1807:
1806:
1805:
1791:
1788:
1766:
1762:
1758:
1752:
1746:
1739:
1735:
1730:
1716:
1696:
1676:
1673:
1670:
1646:
1640:
1637:
1629:
1623:
1620:
1614:
1611:
1608:
1605:
1599:
1593:
1589:
1586:
1583:
1577:
1574:
1566:
1563:
1560:
1557:
1551:
1548:
1538:
1537:
1536:
1522:
1502:
1493:
1479:
1476:
1453:
1447:
1424:
1421:
1418:
1415:
1392:
1389:
1386:
1380:
1374:
1354:
1334:
1331:
1311:
1308:
1285:
1282:
1279:
1272:
1266:
1263:
1255:
1249:
1246:
1240:
1237:
1234:
1231:
1225:
1219:
1215:
1212:
1209:
1203:
1200:
1197:
1194:
1188:
1185:
1178:
1177:
1176:
1162:
1159:
1139:
1116:
1113:
1110:
1107:
1084:
1081:
1061:
1058:
1049:
1047:
1039:
1037:
1020:
1012:
993:
986:
982:
978:
974:
970:
966:
965:standard part
962:
957:
953:
951:
948: ≠
947:
943:
938:
936:
932:
924:
922:
920:
916:
912:
908:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
852:
849:
846:
843:
840:
837:
834:
831:
827:
824:
821:
818:
815:
812:
808:
805:
802:
799:
792:
791:
790:
788:
784:
780:
776:
771:
769:
765:
761:
757:
753:
749:
746: =
745:
741:
737:
733:
729:
725:
721:
716:
712:
706:
698:
696:
694:
690:
674:
650:
644:
633:
627:
624:
618:
612:
609:
603:
597:
593:
590:
587:
581:
574:
571:
562:
558:
554:
550:
546:
542:
537:
534:
530:
526:
522:
517:
515:
514:Łoś's theorem
511:
495:
488:
487:hyperintegers
472:
469:
462:
459:
452:
449:
441:
438:
422:
419:
412:
409:
402:
399:
391:
387:
383:
378:
374:
370:
366:
362:of addition,
361:
357:
353:
349:
345:
341:
337:
332:
330:
314:
294:
290:
286:
283:
275:
250:
230:
210:
207:
199:
174:
166:
165:infinitesimal
162:
158:
154:
150:
141:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
7316:Internal set
7301:Hyperinteger
7270:Dual numbers
7264:
7181:
7174:
7060:
7050:
6992:
6865:Dual numbers
6857:hypercomplex
6647:Real numbers
6388:
6379:
6370:
6345:, New York:
6342:
6317:
6280:
6244:
6238:
6229:
6220:
6197:
6188:
6170:
6164:
6154:, retrieved
6147:the original
6126:math/0311165
6116:
6112:
6099:
6090:
6069:
6060:
6041:
6035:
6026:
6007:
5987:. Retrieved
5983:
5921:Hyperinteger
5888:ultrafilters
5883:
5875:
5871:
5867:
5859:
5851:
5849:
5844:
5840:
5836:
5832:
5831:(1948)) and
5824:
5823:is called a
5820:
5816:
5812:
5808:
5804:
5800:
5796:
5792:). Then the
5789:
5781:
5777:
5773:
5761:
5759:
5743:
5700:
5629:
5485:are finite,
5482:
5478:
5318:
5315:
5311:
5303:
5299:
5295:
5294:) such that
5291:
5287:
5283:
5279:
5275:
5271:
5263:
5259:
5255:
5254:mapping, st(
5247:
5243:
5239:
5227:
5223:
5221:
5128:) hyperreal
5071:
5022:
4886:
4883:
4860:
4854:
4850:
4839:
4835:
4828:
4824:
4817:
4809:
4750:
4746:
4739:
4737:
4592:
4353:
4344:
4314:
4310:
4306:
4304:
4299:
4294:
4290:
4284:
4277:
4268:ultraproduct
4265:
4248:
4244:
4240:
4236:
4232:
4230:
4225:
4189:
4079:
4067:
4063:
4055:
4051:
4049:
4044:
4040:
4036:
4032:
4028:
4024:
4020:
4016:
4012:
4008:
4004:
4000:
3996:
3992:
3984:
3980:
3976:
3972:
3968:
3964:
3957:
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3919:
3914:
3909:
3905:
3900:
3896:
3892:
3885:
3878:
3871:
3864:
3857:
3850:
3846:
3842:
3839:Zorn's lemma
3830:
3820:
3597:
3539:
3535:
3531:
3527:
3523:
3519:
3515:
3511:
3502:
3286:
3272:
3261:model theory
3246:
3217:
3195:
3182:
3164:
3158:
3149:
3121:
3117:
3115:
3107:metric space
3098:
3090:
3088:
3079:
2990:
2835:
2773:hyperinteger
2744:
2399:
2396:
2383:
2378:
2374:
2370:
2366:
2362:
2359:Dual numbers
2356:
1731:
1662:
1494:
1300:
1152:is real and
1050:
1045:
1043:
1010:
983:) to be the
980:
976:
972:
968:
960:
958:
954:
949:
945:
941:
939:
930:
928:
918:
910:
906:
904:
786:
782:
778:
774:
772:
759:
755:
747:
743:
739:
735:
727:
723:
719:
714:
710:
708:
693:infinite sum
560:
556:
538:
518:
439:
389:
381:
376:
372:
368:
364:
355:
351:
333:
329:Edwin Hewitt
152:
146:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
7176:The Analyst
7027:Other types
6846:Bioctonions
6703:Quaternions
6119:: 159–164,
6064:Ball, p. 31
5880:ultrapowers
5252:homomorphic
4871:ultrafilter
4838:belongs to
4808:belongs to
4422:, that is,
4072:cardinality
3926:total order
3828:ultrafilter
3238:Weierstrass
3186:ultrafilter
3179:Development
3138:ω-saturated
2393:Integration
915:Archimedean
758:of numbers
510:ultrapowers
348:first-order
149:mathematics
7423:Categories
7155:Adequality
6981:Projective
6954:Infinities
6156:2004-10-13
6081:2210.07958
5989:2024-03-20
5960:References
5780:. Suppose
5746:continuous
5232:local ring
4471:for which
4060:ultrapower
4050:The field
3920:This is a
3870:, ...) ≤ (
3236:, Cauchy,
3085:Properties
1734:derivative
789:such that
563:) becomes
549:derivative
543:is called
525:Archimedes
342:heuristic
87:newspapers
7391:Textbooks
7336:Overspill
7065:solenoids
6885:Sedenions
6731:Octonions
6387:Keisler,
6378:Hermoso,
6369:Crowell,
5944:Real line
5714:
5677:
5643:
5605:
5590:
5569:
5538:
5520:
5496:
5452:
5434:
5376:
5370:≤
5358:
5332:≤
5057:…
5034:…
4847:empty set
4280:Goldblatt
4204:ℵ
4164:ℵ
4139:ℵ
4119:ℵ
4102:ℵ
4076:continuum
3987:; as the
3806:…
3790:≤
3774:∧
3758:≤
3742:∧
3726:≤
3709:⟺
3702:…
3657:≤
3651:…
3583:ϵ
3563:ϵ
3544:oscillate
3485:…
3395:…
3344:…
3289:sequences
3142:countable
3136:(meaning
3134:saturated
2938:∫
2816:−
2789:
2698:ε
2678:∑
2669:
2648:ε
2631:∫
2604:−
2452:ε
2446:∫
2411:ε
2282:
2193:⋅
2176:
2102:⋅
2086:
2030:−
1999:⋅
1970:
1912:−
1881:
1621:−
1590:
1247:−
1216:
1024:∞
1021:−
997:∞
887:…
881:ω
850:ω
825:ω
806:ω
672:Δ
642:Δ
625:−
616:Δ
594:
521:soundness
516:of 1955.
460:π
453:
410:π
403:
392:. Since
388:, so is *
340:Leibniz's
331:in 1948.
157:extension
117:July 2023
7449:Infinity
6474:Integers
6436:Sets of
6278:(1960),
6196:(1996),
6143:15104702
5894:See also
5866:κ and C(
5760:Suppose
5347:implies
5258:), from
4668:If both
4343:one has
4078:. Since
3989:quotient
3913:} is in
3895: :
3575:, where
3547:randomly
3269:topology
3259:, using
3226:calculus
3140:but not
3103:subfield
3093:form an
1738:function
1013:) to be
575:′
553:integral
541:analysis
485:for all
437:integers
435:for all
161:infinite
7459:Numbers
7148:History
7055:numbers
6887: (
6733: (
6705: (
6677: (
6649: (
6570: (
6568:Periods
6537: (
6504: (
6476: (
6448: (
6430:systems
6094:Keisler
5854:is the
5230:form a
4744:subsets
4070:it has
3833:on the
3509:algebra
3265:algebra
3234:Bolzano
3202:Leibniz
2771:is any
1736:of the
1132:(where
913:is not
155:are an
101:scholar
6855:Other
6428:Number
6353:
6328:
6292:
6255:
6208:
6177:
6141:
6048:
6014:
5829:Hewitt
5397:, but
5284:finite
5268:kernel
5266:whose
5174:where
5023:where
4863:filter
4356:Cantor
4058:is an
3277:Hewitt
3273:per se
3210:Cauchy
3198:Newton
3173:Woodin
3048:
2999:
2958:
2914:
2902:
2882:
2864:
2798:
2783:
2759:
2753:
2713:
2580:
2571:
2551:
2545:
2525:
2519:
2511:(where
2458:
2426:
2408:
1781:, let
1277:
963:, the
919:ω
787:ω
103:
96:
89:
82:
74:
7311:Monad
7063:-adic
7053:-adic
6810:Over
6771:Over
6765:types
6763:Split
6286:50–62
6150:(PDF)
6139:S2CID
6121:arXiv
6109:(PDF)
6076:arXiv
5819:then
5788:in C(
5784:is a
5764:is a
4927:then
4349:field
4255:; in
3206:Euler
3196:When
3101:as a
2745:where
967:, st(
777:and *
384:is a
108:JSTOR
94:books
7099:List
6956:and
6351:ISBN
6326:ISBN
6290:ISBN
6253:ISBN
6206:ISBN
6175:ISBN
6046:ISBN
6012:ISBN
5799:= C(
5481:and
5446:<
5408:<
5093:<
4688:and
4562:and
4287:ring
4043:and
3944:and
3932:and
3267:and
3208:and
3128:and
878:<
847:<
822:<
803:<
764:sets
738:and
551:and
284:<
208:<
163:and
80:news
6131:doi
5882:of
5770:3.5
5628:If
5310:of
5274:of
5262:to
5226:of
4887:If
4749:of
4742:of
4665:is.
4602:If
4591:is
4257:ZFC
4062:of
3979:as
3936:if
3518:in
3232:by
3150:the
3118:the
2537:and
1709:at
756:set
450:sin
400:sin
147:In
63:by
7425::
6645::
6349:,
6324:,
6288:,
6251:,
6204:,
6200:,
6137:,
6129:,
6117:69
6115:,
6111:,
5998:^
5982:.
5968:^
5835:a
5803:)/
5752:.
5711:st
5674:st
5640:st
5602:st
5587:st
5566:st
5535:st
5517:st
5493:st
5473:.
5449:st
5431:st
5373:st
5355:st
5228:*R
4345:ab
4307:ab
4302:.
4270:.
4228:.
4033:ab
3993:*R
3977:*R
3954:*R
3948:≤
3940:≤
3917:.
3904:≤
3884:,
3877:,
3863:,
3856:,
3534:,
3530:,
3218:dx
3175:.
3156:.
3144:)
3113:.
3040:on
2786:st
2666:st
2663::=
2389:.
2379:dx
2375:dx
2371:dx
2367:dx
2363:dx
2279:st
2173:st
2083:st
1967:st
1878:st
1729:.
1587:st
1492:.
1213:st
1210::=
946:2x
944:,
937:.
931:dx
770:.
748:yx
744:xy
742:,
722:,
695:.
591:st
371:=
243:.
151:,
7133:e
7126:t
7119:v
7067:)
7061:p
7057:(
7051:p
6925:/
6909:)
6896:S
6832::
6819:C
6793::
6780:R
6755:)
6742:O
6727:)
6714:H
6699:)
6686:C
6671:)
6658:R
6594:)
6580:P
6559:)
6546:A
6526:)
6513:Q
6498:)
6485:Z
6470:)
6457:N
6420:e
6413:t
6406:v
6133::
6123::
6084:.
6078::
6054:.
6020:.
5992:.
5884:R
5876:R
5872:R
5868:X
5860:X
5852:X
5845:R
5841:F
5833:F
5821:M
5817:R
5813:F
5809:F
5805:M
5801:X
5797:A
5790:X
5782:M
5778:X
5774:X
5762:X
5729:x
5726:=
5723:)
5720:x
5717:(
5701:x
5686:)
5683:x
5680:(
5670:/
5666:1
5663:=
5660:)
5657:x
5653:/
5649:1
5646:(
5630:x
5614:)
5611:y
5608:(
5599:)
5596:x
5593:(
5584:=
5581:)
5578:y
5575:x
5572:(
5547:)
5544:y
5541:(
5532:+
5529:)
5526:x
5523:(
5514:=
5511:)
5508:y
5505:+
5502:x
5499:(
5483:y
5479:x
5461:)
5458:y
5455:(
5443:)
5440:x
5437:(
5411:y
5405:x
5385:)
5382:y
5379:(
5367:)
5364:x
5361:(
5335:y
5329:x
5316:x
5312:x
5304:x
5300:x
5296:x
5292:x
5288:x
5280:S
5276:F
5272:x
5264:R
5260:F
5256:x
5248:S
5246:/
5244:F
5240:S
5224:F
5202:d
5182:y
5162:d
5159:+
5156:y
5136:x
5116:a
5096:a
5089:|
5085:x
5081:|
5037:}
5031:{
5008:}
5005:)
5000:n
4996:x
4992:(
4989:f
4986:{
4983:=
4980:)
4977:}
4972:n
4968:x
4964:{
4961:(
4958:f
4935:f
4915:x
4895:f
4855:U
4851:U
4842:.
4840:U
4836:U
4831:.
4829:U
4825:U
4820:.
4818:U
4810:U
4796:)
4793:a
4790:(
4787:z
4767:0
4764:=
4761:a
4751:N
4747:X
4740:U
4722:b
4719:+
4716:a
4696:b
4676:a
4653:b
4633:b
4630:a
4610:a
4593:N
4579:)
4576:b
4573:(
4570:z
4550:)
4547:a
4544:(
4541:z
4521:0
4518:=
4515:b
4512:a
4492:0
4489:=
4484:i
4480:a
4459:i
4439:)
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115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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