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4 ideas
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
Full Idea: 'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.1) | |
A reaction: On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before. |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
Full Idea: The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4) |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
Full Idea: By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.3) |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
Full Idea: The 'continuum' is the cardinality of the powerset of a denumerably infinite set. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2) |