## Does Mathematics Need New Axioms? (1999)

Venue: | American Mathematical Monthly |

Citations: | 11 - 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}

}

### OpenURL

### 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.

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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... |

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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 ... |

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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;... |

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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... |

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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... |

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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---... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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 |

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