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Is the Continuum Hypothesis a definite mathematical problem?
"... [t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is gr ..."
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[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516517 [in Gödel 1990, 178]
1 Logic, Mathematics and Conceptual Structuralism
"... Abstract. Conceptual structuralism is a nonrealist philosophy of mathematics according to which the objects of mathematical thought are humanly conceived “idealworld” structures. Basic conceptions of structures, such as those of the natural numbers, the continuum, and sets in the cumulative hierar ..."
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Abstract. Conceptual structuralism is a nonrealist philosophy of mathematics according to which the objects of mathematical thought are humanly conceived “idealworld” structures. Basic conceptions of structures, such as those of the natural numbers, the continuum, and sets in the cumulative hierarchy, differ in their degree of clarity. One may speak of what is true in a given conception, but that notion of truth may be partial. Mathematics proceeds from such basic conceptions by reflective expansion and carefully reasoned argument, the last of which is analyzed in logical terms. The main questions for the role of logic here is whether there are principled demarcations on its use. It is claimed that in the case of a completely clear conception, such as that of the natural numbers, the logical notions are just those of firstorder classical logic and hence that that is the appropriate vehicle for reasoning. At the other extreme, in the case of set theory, where each set is conceived of as a definite totality but the universe of “all ” sets is an indefinite totality, it is proposed that the appropriate logic is semiintuitionistic in which classical logic applies only to (set) bounded formulas. Certain subsystems of classical set theory in which extensive parts of mathematics can be formalized are reducible to
1 The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem1
"... Cantor’s continuum problem is simply the question: How many points are there on a straight line in Euclidean space? In other terms, the question is: How many different sets of integers do there exist?... [t]he analysis of the phrase “how many” leads unambiguously to quite a definite meaning for [th ..."
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Cantor’s continuum problem is simply the question: How many points are there on a straight line in Euclidean space? In other terms, the question is: How many different sets of integers do there exist?... [t]he analysis of the phrase “how many” leads unambiguously to quite a definite meaning for [this] question... namely, to find out which one of the א’s is the number of points on a straight line.