How many real numbers exist? New proof moves closer to an answer (quantamagazine.org)

Article summary

Mathematicians David Asperó and Ralf Schindler have made a breakthrough in understanding the sizes of infinity, proving that one axiom implies another and bringing them closer to answering how many real numbers exist. Their work unites two rival axioms that have been posited as competing foundations for infinite mathematics. The result strengthens the case against the continuum hypothesis, a 1878 conjecture about the strata of infinities. This development is seen as a significant step forward in the field of mathematics.

Main themes

• sizes of infinity
• continuum hypothesis
• axiom of choice
• forcing technique
• non-standard models
• foundations of mathematics
• cardinality of real numbers

What commenters say

• The diagonalization argument is a key concept in understanding the sizes of infinity, but some commenters struggle to see its relevance to the forcing process.
• Forcing is a technique used to construct new models of sets, but its underlying philosophy and technical details are not well understood by all commenters.
• The axiom of choice is a crucial component of many mathematical constructions, but its elimination can lead to different and sometimes counterintuitive results.
• Some commenters argue that the concept of size is not a necessary property of a set, and that it may not be rational to speak of size as a universal property.
• The relationship between the cardinality of the set of real numbers and the cardinality of the natural numbers is still not fully understood, with some commenters suggesting that the continuum hypothesis may be independent of the standard axioms of mathematics.
• Different axioms and mathematical constructions can lead to varying conclusions about the nature of infinity and the real numbers, highlighting the complexity and nuance of the subject.
• The use of non-standard models of arithmetic, such as those constructed using ultrafilters, can provide insight into the nature of the natural numbers and the limitations of first-order logic.