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Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... 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 f ..."
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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.
A Note on Applicability of the Incompleteness Theorem to Human Mind
"... We shall present some relations between consistency and reflection principles which explain why is Godel's incompleteness theorem wrongly used to argue that thinking machines are impossible. 1 Introduction Since its publishing Godel's incompleteness theorem attracted a lot of attention among philoso ..."
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We shall present some relations between consistency and reflection principles which explain why is Godel's incompleteness theorem wrongly used to argue that thinking machines are impossible. 1 Introduction Since its publishing Godel's incompleteness theorem attracted a lot of attention among philosophers. In 1959 Lucas [8] presented an argument that this theorem implies that human thinking is essentially different from what any machine can do. This means that the ultimate goal of artificial intelligence cannot be achieved. The argument is roughly the following. A machine (nowadays we would rather say "a computer") behaves according to fixed rules (a program), hence we can view it as a formal system. Applying Godel's theorem to this system we get a true sentence which is unprovable in the system. Thus the machine does not know that the sentence is true while we can see that it is true. The spectrum of attitudes of various people to this argument was nicely characterized by Hofstadter [...
Presentation to the panel, “Does mathematics need new axioms?”
"... The point of departure for this panel is a somewhat controversial paper that I published in the American Mathematical Monthly under the title “Does mathematics need new axioms? ” [4]. The paper itself was based on a lecture that I gave in 1997 to a joint session of the American Mathematical Society ..."
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The point of departure for this panel is a somewhat controversial paper that I published in the American Mathematical Monthly under the title “Does mathematics need new axioms? ” [4]. The paper itself was based on a lecture that I gave in 1997 to a joint session of the American Mathematical Society and the Mathematical Association of America, and it was thus written for a general mathematical audience. Basically, it was intended as an assessment of Gödel’s program for new axioms that he had advanced most prominently in his 1947 paper for the Monthly, entitled “What is Cantor’s continuum problem? ” [7]. My paper aimed to be an assessment of that program in the light of research in mathematical logic in the intervening years, beginning in the 1960s, but especially in more recent years. In my presentation here I shall be following [4] in its main points, though enlarging on some of them. Some passages are even taken almost verbatim from that paper where convenient, though of course all expository background material that was necessary there for a general audience is omitted. 1 For a logical audience I have written before about
Are There Absolutely Unsolvable Problems? Gödel’s Dichotomy
 PHILOSOPHIA MATHEMATICA
, 2006
"... This is a critical analysis of the first part of Gödel’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Gödel’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of ..."
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This is a critical analysis of the first part of Gödel’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Gödel’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. Either... the human mind... infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems.