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Single Idea 8454

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number ]

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.

Gist of Idea

The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc.

Source

Alex Orenstein (W.V. Quine [2002], Ch.2)

Book Ref

Orenstein,Alex: 'W.V. Quine' [Princeton 2002], p.27

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.

The 14 ideas from 'W.V. Quine'

8452 Traditionally, universal sentences had existential import, but were later treated as conditional claims [Orenstein]
8454 The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc. [Orenstein]
8457 The Principle of Conservatism says we should violate the minimum number of background beliefs [Orenstein]
8458 Just individuals in Nominalism; add sets for Extensionalism; add properties, concepts etc for Intensionalism [Orenstein]
8471 Three ways for 'Socrates is human' to be true are nominalist, platonist, or Montague's way [Orenstein]
8465 Mereology has been exploited by some nominalists to achieve the effects of set theory [Orenstein]
8474 Unlike elementary logic, set theory is not complete [Orenstein]
8472 Sentential logic is consistent (no contradictions) and complete (entirely provable) [Orenstein]
8476 Axiomatization simply picks from among the true sentences a few to play a special role [Orenstein]
8473 The logicists held that is-a-member-of is a logical constant, making set theory part of logic [Orenstein]
8475 The substitution view of quantification says a sentence is true when there is a substitution instance [Orenstein]
8477 People presume meanings exist because they confuse meaning and reference [Orenstein]
8484 If two people believe the same proposition, this implies the existence of propositions [Orenstein]
8480 S4: 'poss that poss that p' implies 'poss that p'; S5: 'poss that nec that p' implies 'nec that p' [Orenstein]