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Single Idea 17615

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis ]

Full Idea

One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals.

Gist of Idea

Every infinite set of reals is either countable or of the same size as the full set of reals

Source

Penelope Maddy (Defending the Axioms [2011], 2.4 n40)

Book Ref

Maddy,Penelope: 'Defending the Axioms' [OUP 2013], p.56


The 7 ideas from 'Defending the Axioms'

The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
If two mathematical themes coincide, that suggest a single deep truth [Maddy]