Combining Texts

Ideas for 'fragments/reports', 'Foundations without Foundationalism' and 'Papers of 1906'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

4 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
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)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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)