## Does Mathematics Need New Axioms? (1999)

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Venue: | American Mathematical Monthly |

Citations: | 12 - 2 self |

### BibTeX

@ARTICLE{Feferman99doesmathematics,

author = {Solomon Feferman},

title = {Does Mathematics Need New Axioms?},

journal = {American Mathematical Monthly},

year = {1999},

volume = {106},

pages = {99--111}

}

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### Abstract

this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.

### Citations

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(Show Context)
Citation Context ...d \Delta are set-theoretically definable in terms of the latter by Dedekind's result, they are not definable from them in the first-order language used for formal arithmetic. However, Godel showed in =-=[9]-=- that once we have 0, 0 ; +, and \Delta, all primitive recursive functions are definable in PA. Unlike the origin of the Dedekind-Peano axioms in a clear intuitive concept, Zermelo's axioms arose out ... |

140 | The higher infinite - Kanamori - 1994 |

93 |
The consistency of the axiom of choice and of the generalized continuum hypothesis
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(Show Context)
Citation Context ...was said by Godel until the mid 1940s, by which time he was safely ensconced at the Institute for Advanced Study in Princeton, and Hilbert was dead and gone. In the meantime, Godel had established in =-=[10]-=- his second ground-breaking result, that if ZF is consistent then it remains consistent when we add AC and GCH. Godel's method of proof for this was to produce a new cumulative hierarchy as a model of... |

79 |
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(Show Context)
Citation Context ...tuation by producing finite combinatorial statements of prima-facie mathematical interest that are independent of such S. The first example was provided by Jeff Paris and Leo Harrington who showed in =-=[22]-=- that a modified form (PH) of the finite Ramsey theorem concerning existence of homogeneous sets for certain kinds of partitions is not provable in PA. PH is recognized to be true as a simple conseque... |

70 |
The indeendence of the continuum hypothesis
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(Show Context)
Citation Context ...o, also, CH has a determinate truth value. According to Godel in [11] it is probably false. 4 ffi Thus CH should be independent of ZFC. (Indeed, this was eventually demonstrated in 1963 by Paul Cohen =-=[2]-=-.) 5 ffi And thus, in order to fix the position @ ff of 2 @ 0 in the scale of alephs, we will [no doubt] need to add new axioms to ZFC. 6 ffi These new axioms may be formulated and accepted by a direc... |

51 |
What is Cantor's continuum problem
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(Show Context)
Citation Context ...of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum probl=-=em?" [11]-=-, he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity... |

47 |
Arithmetices principia, nova methodo exposita
- Peano
(Show Context)
Citation Context ...d in the 3 proof of categoricity. The functions of one or more arguments from N to N generated by explicit and simple recursive definition are nowadays called the primitive recursive functions. Peano =-=[23]-=- made a first stab at adding axioms (to those of Dedekind) about which sets exist. He stated that every property determines a set, and then gave some closure conditions on properties. In the Peano axi... |

45 |
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(Show Context)
Citation Context ... of this work, there were objections not only to the acceptance of AC but also to the correctness of his proof of the implication. In order to meet the latter objections, Zermelo introduced axioms in =-=[31]-=- that spelled out just which principles on sets were employed in his argument. These are the axioms of: Extensionality, Empty set, Unordered pair, Power set, Union, Infinity and Separation. The latter... |

42 |
1888b, ‘Was sind und was sollen die Zahlen
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(Show Context)
Citation Context ...e reasons for their acceptance. In both cases, one started with an informal "naive" system, which was later transformed into a formal system in the precise sense of metamathematics. Dedekind=-='s axioms [3]-=- for the natural numbers N = f0; 1; 2; : : : g simply took the initial element 0 [Dedekind started with 1] and the successor operation x 7! x 0 (= x + 1) as basic, with the evident axioms that 0 is no... |

38 |
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Citation Context ...he infinitary theorem KT, a staple of graph-theoretic combinatorics, asserts the well-quasi-ordering of the embeddability relation between finite trees. Friedman's work in this respect is reported in =-=[25]-=-. 4 The systems involved and associated independent statements are more complicated to explain and would go beyond the scope of this article to do so, but at least one result is worth indicating in co... |

34 | A finite combinatorial principle which is equivalent to the l-consistency of predicative analysis - FRIEDMAN, MCALOON, et al. - 1982 |

31 |
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(Show Context)
Citation Context ...hips has been established, as witnessed by charts to be found in the recent book by Aki Kanamori, The Higher Infinite [16, p. 471], and the earlier expository article by Kanamori and Menachem Magidor =-=[17]. A rough distinctio-=-n is made between "small" large cardinals, and "large" large cardinals, according to whether they are weaker or stronger, in some logical measure or other, than measurable cardinal... |

29 |
Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlangen der Mengenlehre”, Fundamenta Mathematicae 16
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(Show Context)
Citation Context ...iom of Separation avoids, by applying P only to elements of a "pre-existing" set a. What justifies that, but not the more general, contradictory concept of set? An answer was first offered b=-=y Zermelo [32]-=- in 1930, in terms of what has since come to be called the cumulative hierarchy of sets. In this picture, sets are conceived of as being built up from below in stages, starting with the urelements at ... |

28 | Finite functions and the necessary use of large cardinals
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- 1998
(Show Context)
Citation Context ...ematically perspicuous finite combinatorial statements OE whose proof requires the existence of many Mahlo axioms or even stronger axioms of infinity and has come up with various candidates for that (=-=[7]-=- contains the latest work in this direction). From the point of view of metamathematics, this kind of result is of the same character as the earlier work just mentioned; that is, for certain very stro... |

27 | Heijenoort. From Frege to Gödel: A Source Book - van - 1967 |

24 |
dass jede Menge wohlgeordnet werden kann
- Beweis
(Show Context)
Citation Context ...t is a "Law of Thought"; then he sought a proof of it on the basis of a more evident principle, but failed to come up with anything satisfactory. Such a principle was first offered in 1904 b=-=y Zermelo [30]-=- in the form of the Axiom of Choice (AC). Zermelo proved that AC implies WO; in fact, they are equivalent, but Zermelo argued that AC is evident in a way that WO is not. Following publication of this ... |

22 | Kritische Untersuchungen über die Grundlagen der Analysis, Veit - Weyl, Kontinuum - 1918 |

17 |
Believing the axioms
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- 1988
(Show Context)
Citation Context ...her, Penelope Maddy, in two interesting articles called "Believing the axioms", analyzed the various kinds of arguments for these and other kinds of strong axioms and summarized the evidence=-= for them [20]-=-. Broadly speaking, the arguments are classified as being based on intrinsic or extrinsic reasons. The above-mentioned reflection principles are examples of intrinsic reasons, but these do not take us... |

16 |
Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics
- Feferman
- 1993
(Show Context)
Citation Context ...alysis. As a result of these studies, I have come to conjecture that practically all scientifically applicable mathematics can be formalized in systems reducible to PA, or, as I have sloganized it in =-=[4]-=-: a little bit goes a long way. Against this, I have learned of a couple of cases in some approaches to the foundations of quantum field theory where it appears one must go beyond the resources of PA;... |

13 |
Measurable cardinals and constructible sets
- Scott
(Show Context)
Citation Context ...Ulam, in 1930. Ulam called an uncountable cardinalsmeasurable if there exists a two-valued -additive measure on (). Not much was known about the strength of this until 1961, when Dana Scott proved in =-=[24]-=- that the existence (MC) of measurable cardinals implies V 6= L, so MC then became a viable possibility to settle CH. A few years later, Alfred Tarski with his students William Hanf and H. Jerome Keis... |

12 |
Incompactness in languages with infinitely long expressions
- Hanf
(Show Context)
Citation Context ...istence (MC) of measurable cardinals implies V 6= L, so MC then became a viable possibility to settle CH. A few years later, Alfred Tarski with his students William Hanf and H. Jerome Keisler proved (=-=[15]-=-, [18]) that ifsis a measurable cardinal then it is very large, since Vssatisfies the axioms of Mahlo type and other powerful axioms of infinity. Their work led further to a notion of strongly compact... |

10 |
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- 1960
(Show Context)
Citation Context ...atever closure property P one recognizes to be satisfied by the universe V of all sets, there will exist arbitrarily largesfor which Vssatisfies P . Formal versions of this, introduced by Azriel Levy =-=[19] and Paul -=-Bernays [1], are called Reflection Principles in set theory. They are behind Godel's reason for saying that we are led to new axioms, such as those of Mahlo type, "without arbitrariness" and... |

10 |
Hilbert’s first problem: the continuum hypothesis. In Mathematical developments arising from Hilbert problems (Proc
- Martin
- 1974
(Show Context)
Citation Context ...d new axioms to settle CH has not been realized, what about the origins of his program in the incompleteness results for number theory? As we saw, throughout his life Godel said we would 2 Cf. Martin =-=[21]-=-; the situation reported there in 1976 remains unchanged to date. 12 need new, ever-stronger set-theoretical axioms to settle open arithmetical problems of even the simplest, purely universal, form---... |

9 |
From accessible to inaccessible cardinals
- Keisler, Tarski
- 1964
(Show Context)
Citation Context ...e (MC) of measurable cardinals implies V 6= L, so MC then became a viable possibility to settle CH. A few years later, Alfred Tarski with his students William Hanf and H. Jerome Keisler proved ([15], =-=[18]-=-) that ifsis a measurable cardinal then it is very large, since Vssatisfies the axioms of Mahlo type and other powerful axioms of infinity. Their work led further to a notion of strongly compact cardi... |

9 |
Kritische Untersuchungen über die Grundlagen der Analysis
- Kontinuum
- 1918
(Show Context)
Citation Context ... much as the laws of physics formulated in mathematical terms are highly idealized models of aspects of physical reality. (Hermann Weyl raised just such questions in his 1918 monograph Das Kontinuum, =-=[28]-=-.) But even if we grant some kind of independent existence, abstract or physical, to the continuum, in order to formulate CH we need to refer to arbitrary subsets of the continuum and possible mapping... |

9 |
Large cardinal axioms and independence: the continuum problem revisited
- Woodin
- 1994
(Show Context)
Citation Context ...ng to settle them, given that present axioms are insufficient. At the beginning of this 5 For an opposite point of view and beautiful exposition of the need for new axioms in that respect, cf. Woodin =-=[29]-=-. 15 piece I promised to tell you my own views of these matters. By now, you have probably guessed what these are, but let me say them out loud: I am convinced that the Continuum Hypothesis is an inhe... |

6 |
Zur Frage der Unendlichkeitsschemata in der axiomatische Mengenlehre
- Bernays
- 1961
(Show Context)
Citation Context ...y P one recognizes to be satisfied by the universe V of all sets, there will exist arbitrarily largesfor which Vssatisfies P . Formal versions of this, introduced by Azriel Levy [19] and Paul Bernays =-=[1], are called Re-=-flection Principles in set theory. They are behind Godel's reason for saying that we are led to new axioms, such as those of Mahlo type, "without arbitrariness" and as a "natural contin... |

4 |
1996]: ‘Gödel’s program for new axioms
- FEFERMAN
(Show Context)
Citation Context ...nt ways for over thirty years; during the last year I have arrived at what I think is the most satisfactory general formulation of that idea, in what I call the unfolding of a schematic formal system =-=[5]. And this-=- returns in an essential respect to the original "naive" schematic formulation of principles such as induction in number theory and separation in set theory, in their use of the pre-theoreti... |

4 | Some problems and results relevant to the foundations of set theory - Tarski - 1962 |

3 |
What is Cantor’s Continuum Problem? (revised version). See Benacerraf
- Gödel
- 1983
(Show Context)
Citation Context ..., or which at least would make it very plausible that the hypothesis stating the existence of such cardinals is consistent with familiar axiom systems of set theory. [26, p. 134] In his 1964 revision =-=[12]-=- of his 1947 article, Godel seconded this view of Tarski's in full, but then added: However, [the new axioms] are supported by rather strong argument from analogy... ([12, p. 264, ftn. 20], italics mi... |

3 | The Higher In nite - Kanamori - 1994 |

2 | Axiom schemata of strong in nity in axiomatic set theory, Paci c - Levy - 1960 |

2 | Hilbert's rst problem: The Continuum Hypothesis - Martin - 1976 |

1 |
The unfolding of non-finitist arithmetic (in preparation
- Feferman, Strahm
(Show Context)
Citation Context ...tions is total and thus can be added to our language. There are already some definitive results for specific systems on what can be obtained by the unfolding process, in joint work with Thomas Strahm =-=[6]-=-, with a host of new and interesting problems waiting to be tackled. But that's another story for another occasion. ACKNOWLEDGMENTS Text of an invited AMS-MAA lecture, Joint Annual Meeting, San Diego,... |

1 | The unfolding of non- nitist arithmetic (in preparation - Feferman, Strahm |

1 | Incompactness in languages with in nitely long expressions, Fund - Hanf - 1964 |