Bulletin of the American Mathematical Society

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Society, 1914
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Page 74 - ... mathematics, creates not only the numbers one and two, but also all finite ordinal numbers, inasmuch as one of the elements of the two-oneness may be thought of as a new two-oneness, which process may be repeated indefinitely; this gives rise still further to the smallest infinite ordinal...
Page 74 - From the present point of view of intuitionism therefore all mathematical sets of units which are entitled to that name can be developed out of the basal intuition, and this can only be done by combining a finite number of times the two operations: "to create a finite ordinal number...
Page 73 - This neointuitionism considers the falling apart of the moments of life into qualitatively different parts, to be reunited only while remaining separated by time, as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare...
Page 78 - In this form the axiom permits only the introduction of such sets as are subsets of sets previously introduced; if one wishes to operate with other sets, their existence must be explicitly postulated. Since however in order to accomplish anything at all the existence of a certain collection of sets will have to be postulated at the outset, the only valid argument that can be brought against the introduction of a new set is that it leads to contradictions; indeed the only modifications that the discovery...
Page 76 - In the domain of finite sets in which the formalist axioms have an interpretation perfectly clear to the intuitionists, unreservedly agreed to by them, the two tendencies differ solely in their method, not in their results; this becomes quite different however in the domain of infinite or transfinite sets, where, mainly by the application of the axiom of inclusion, quoted above, the formalist introduces various concepts, entirely meaningless to the intuitionist, such as for instance "the set whose...
Page 71 - It is true that from certain relations among mathematical entities, which we assume as axioms, we deduce other relations according to fixed laws, in the conviction that in this way we derive truths from truths by logical reasoning, but this non-mathematical conviction of truth or legitimacy has no exactness whatever and is nothing but a vague sensation of delight arising from the knowledge of the efficacy of the projection into nature of these relations and laws of reasoning. For the formalist therefore...
Page 75 - ... relation of a set to its elements." With reference to this relation various axioms are postulated, suggested by the practice with natural finite sets; the principal of these are: "a set is determined by its elements"; for any two mathematical objects it is decided whether or not one of them is contained in the other one as an element"; "to every set belongs another set containing as its elements nothing but the subsets of the given set"; the axiom of selection : "a set which is split into subsets...
Page 73 - This neo-intuitionism considers the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time as the fundamental phenomenon of the human intellect, passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare two-oneness.
Page 133 - Five applications for membership were received. A committee was appointed to audit the accounts of the treasurer for the current year. A list of nominations for officers and other members of the council was prepared and ordered printed on the official ballot for the annual election at the December meeting.
Page 74 - ... argument. I confess that I have not studied the question as to whether he can find further cultural support to meet all the objections of opponents of Intuitionism. It would seem, however, that he would have to drop his insistence that in construction of sets (to quote Brouwer [4; p. 86]) "neither ordinary language nor any symbolic language can have any other role than that of serving as a non-mathematical auxiliary...

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