Last Friday I attended the Löb lecture. This is an quadrennial lecture in honour of Martin Löb who founded the logic group here in the University of Leeds after fleeing Nazi Germany as a teenager. See Wikipedia for more information: https://en.wikipedia.org/wiki/Martin_L%C3%B6b
The lecture was given Justin Moore (Cornell). Logic is far from my comfort zone and I don't usually attend logic seminars, but this time I gave it a go because I understood the title: "What makes the continuum ℵ₂?"
This is about how many real numbers there are. Cantor proved that there are more real numbers than natural numbers: |ℝ| > |ℕ|. The Continuum Hypothesis is that there is nothing in between: every uncountable subset of the real numbers has the same cardinality as ℝ.
The cardinality of the natural numbers is denoted ℵ₀ (aleph_0). The next biggest cardinalities are ℵ₁ , ℵ₂ , ℵ₃ , ... So the continuum hypothesis is |ℝ| = ℵ₁. To be precise, this is assuming the standard set theory of mathematics (ZFC, which includes the Axiom of Choice).
The Continuum Hypothesis (CH) is independent of ZFC, so we need arguments outside standard mathematics to decide whether it holds or not. The Justin Moore's argument is that CH has some strange consequences. The only one I understood is that is implies the existence of a bijection ℝ→ℝ that is not monotone on any uncountable set (but I don't have any intuition about this). Assuming CH is false, the next best thing is |ℝ| = ℵ₂ and we can use this to prove some nice theorems ... here is where it got too technical for me. Since we want to prove stuff, Justin Moore argued that we should use |ℝ| = ℵ₂.
#logic #SetTheory #ContinuumHypothesis
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