Proof and certainty · Scope

What do Gödel's incompleteness theorems tell us about the limits of mathematical knowledge?

In 1931 Kurt Gödel showed that any consistent formal system rich enough to describe ordinary arithmetic contains true statements it cannot prove, and cannot prove its own consistency. This question asks what that means for the hope that mathematics could be made completely certain.

Claims

Gödel's results show that there are limits to what any single set of rules can prove, so mathematical knowledge can never be captured by one complete, mechanical system.
Some natural questions have turned out to be undecidable from the usual axioms, which shows the limits are not only theoretical.

Counterclaims

The theorems themselves are proved mathematics, so they are an achievement of mathematical knowledge as much as a limit on it.
Almost all the mathematics used in schools, science and engineering is unaffected; the limits apply at the edges, not to whether 2 + 2 = 4.

Real-life situations from mathematics

The continuum hypothesis

Georg Cantor asked whether there is a size of infinity between that of the whole numbers and that of the real numbers. Gödel showed in 1940 that the usual axioms of set theory cannot disprove that there is none, and Paul Cohen showed in 1963 that they cannot prove it either. The question cannot be settled from those axioms at all.

Hilbert's programme

In the 1920s David Hilbert hoped to put all of mathematics on axioms that could be shown, by finite methods, never to lead to a contradiction. Gödel's second theorem showed that this goal, as Hilbert framed it, could not be reached.

Check dates and figures in a reliable source before you use them, and cite that source.

Use this in your TOK work

Essay. Use it carefully for titles about limits of knowledge or certainty: state the result accurately and do not claim it makes all mathematics uncertain.

Exhibition. A printed self-referential sentence such as 'This sentence cannot be proved' makes a thought-provoking object for a prompt about limits or about language.

Link it to the prescribed title or the exhibition prompt you are working on, in your own words. See using maths examples in your essay and choosing exhibition objects.

Themes and study heading

Knowledge and the knower Knowledge and language Scope

What mathematics is about, and where its edges lie.