display all the ideas for this combination of texts
2 ideas
| 8454 | The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc. [Orenstein] |
| Full Idea: The 'natural' numbers are the whole numbers 1, 2, 3 and so on. The 'rational' numbers consist of the natural numbers plus the fractions. The 'real' numbers include the others, plus numbers such a pi and root-2, which cannot be expressed as fractions. | |
| From: Alex Orenstein (W.V. Quine [2002], Ch.2) | |
| A reaction: The 'irrational' numbers involved entities such as root-minus-1. Philosophical discussions in ontology tend to focus on the existence of the real numbers. |
| 17877 | The number series is primitive, not the result of some set theoretic axioms [Almog] |
| Full Idea: On Skolem's account, to 'get' the natural numbers - that primal structure - do not 'look for it' as the satisfier of some abstract (set-theoretic) axiomatic essence; start with that primitive structure. | |
| From: Joseph Almog (Nature Without Essence [2010], 12) | |
| A reaction: [Skolem 1922 and 1923] Almog says the numbers are just 0,1,2,3,4..., and not some underlying axioms. That makes it sound as if they have nothing in common, and that the successor relation is a coincidence. |