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Sakai Lab
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A set theory lab that explores, from the axioms, what can be proved about infinity.
AboutAbout the lab
Thinking about infinity from the axioms
The Sakai Lab studies axiomatic set theory, a branch of the foundations of mathematics.
ZFC, the standard axiom system of set theory, allows almost all of ordinary mathematics to be developed, but it is known that many statements concerning infinity, such as the continuum hypothesis, can be neither proved nor disproved in ZFC.
The lab studies what statements can be proved in axiom systems obtained by adding large cardinal axioms, forcing axioms, and reflection principles to ZFC.
Key terms
- ZFC
- The standard axiom system of set theory, consisting of the Zermelo–Fraenkel axioms together with the axiom of choice
- Continuum Hypothesis
- The statement that there is no infinite set whose size lies strictly between that of the set of natural numbers and that of the set of real numbers
- Large Cardinal Axiom
- An axiom asserting the existence of extremely large infinite sizes (cardinals)
ResearchResearch themes
The lab studies what can be proved in axiom systems obtained by adding various axioms to ZFC. In particular, it is interested in which statements of infinitary combinatorics and cardinal arithmetic can be proved in systems obtained by adding large cardinal axioms, forcing axioms, and reflection principles.
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MembersFaculty
- ProfessorHiroshi SakaiPublications ↗
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Cantor's theorem visual” Electro blob / CC BY-SA 4.0 (cropped)
- Forcing Axioms, Large Cardinal Axioms, and Infinity: “Von Neumann universe 4” Alan U. Kennington / Public domain (cropped)
- Related lab (Kato Lab): “Orange Genus 2 Surface” Gelada / CC BY-SA 3.0 (cropped)
- Related lab (Matsui Lab): “Illustration of Yang Baxter Equation” AHusain314 / Public domain (cropped)
- Related lab (Shiraishi Lab): “Weierstrass p function” Claudio Rocchini / CC BY-SA 4.0 (cropped)
- Related lab (Hayashi Lab): “Lorenz attractor and its transients” Yapparina / CC0 (cropped)
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