New PDF release: A brief history of mathematics;: An authorized translation

By Karl Fink

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Additional info for A brief history of mathematics;: An authorized translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik,

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Math is analytic after all. But we don’t know what follows from its being so and we will have to study the subanalytic to see what (logically) entails what just as hard as before. It is known that HP does not follow (a word I will not surrender) from the conjunction of two of its strongest consequences: the (interesting) statements that nothing precedes zero and that precedes is a one–one relation. If HP is analytic, then it is strictly stronger (another non-negotiable term) than some of its strong consequences.

There are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new terms also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property. ([1906], p. 144) The term “indefinite extensibility” is due to Michael Dummett, however, who extended Russell’s idea as follows: An indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it.

1993], p. 441) It has become standard to use the term ‘definite’ for those concepts that are not indefinitely extensible. The ordinal numbers provide perhaps the clearest example of an indefinitely extensible collection. e. a set of ordinals). e. e. either the successor of the greatest ordinal in the collection in question, or the supremum of the collection in question). As a result, there seems to be a sense in which we can never collect together all of the ordinals into a definite totality, since we could repeat this reasoning on such a collection to obtain an ordinal that is not in such a collection of all ordinals – contradiction (this is essentially just the reasoning behind the Burali-Forti paradox).

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A brief history of mathematics;: An authorized translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik, by Karl Fink

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