Editing Alfred Tarski (section)

==Work in mathematics==
Tarski's mathematical interests were exceptionally broad. His collected papers run to about 2,500 pages, most of them on mathematics, not logic. For a concise survey of Tarski's mathematical and logical accomplishments by his former student Solomon Feferman, see "Interludes I&ndash;VI" in Feferman and Feferman.<ref>[[#F-F|Feferman & Feferman]], pp. 43-52, 69-75, 109-123, 189-195, 277-287, 334-342</ref>

Tarski's first paper, published when he was 19 years old, was on [[set theory]], a subject to which he returned throughout his life.<ref name="mathshistory.st-andrews.ac.uk">{{Cite web |url=https://mathshistory.st-andrews.ac.uk/Biographies/Tarski/#:~:text=Tarski%27s%20first%20paper%20was%20published%20in%201921%20when,submitted%20his%20doctoral%20thesis%20for%20examination%20in%201923. |title=Alfred Tarski |website=mathshistory.st-andrews.ac.uk |access-date=28 April 2023}}</ref> In 1924, he and [[Stefan Banach]] proved that, if one accepts the [[Axiom of Choice]], a [[Ball (mathematics)|ball]] can be cut into a finite number of pieces, and then reassembled into a ball of larger size, or alternatively it can be reassembled into two balls whose sizes each equal that of the original one. This result is now called the [[Banach–Tarski paradox]].<ref>{{Cite arXiv |title=The Banach-Tarski Paradox |author=Katie Buchhorn |date=8 August 2012 |class=math.HO |eprint=2108.05714 }}</ref>

In ''A decision method for elementary algebra and geometry'', Tarski showed, by the method of [[quantifier elimination]], that the [[first-order theory]] of the [[real number]]s under addition and multiplication is [[Decidability (logic)|decidable]]. (While this result appeared only in 1948, it dates back to 1930 and was mentioned in Tarski (1931).) This is a very curious result, because [[Alonzo Church]] proved in 1936 that [[Peano arithmetic]] (the theory of [[natural number]]s) is ''not'' decidable. Peano arithmetic is also incomplete by [[Gödel's incompleteness theorem]]. In his 1953 ''Undecidable theories'', Tarski et al. showed that many mathematical systems, including [[lattice theory]], abstract [[projective geometry]], and [[closure algebra]]s, are all undecidable. The theory of [[Abelian group]]s is decidable, but that of non-Abelian groups is not.

While teaching at the Stefan Żeromski Gimnazjum in the 1920s and 30s, Tarski often taught [[geometry]].{{sfn|McFarland|McFarland|Smith|2014|loc=Section 9.2: Teaching geometry, pp. 179–184}} Using some ideas of [[Mario Pieri]], in 1926 Tarski devised an original [[axiomatization]] for plane [[Euclidean geometry]], one considerably more concise than [[Hilbert's axioms|Hilbert's]].<ref>{{Cite web |url=https://ceur-ws.org/Vol-1785/F2.pdf |title=Tarski's Geometry and the Euclidean Plane in Mizar |website=ceur-ws.org |author=Adam Grabowski |access-date=28 April 2023}}</ref> [[Tarski's axioms]] form a first-order theory devoid of set theory, whose individuals are [[Point (geometry)|points]], and having only two primitive [[finitary relation|relations]]. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers.

In 1929 he showed that much of Euclidean [[solid geometry]] could be recast as a second-order theory whose individuals are ''spheres'' (a [[primitive notion]]), a single primitive binary relation "is contained in", and two axioms that, among other things, imply that containment [[partial order|partially orders]] the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of [[mereology]] far easier to exposit than [[Lesniewski]]'s variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry.<ref>{{Cite journal |title=Tarski's System of Geometry |journal=The Bulletin of Symbolic Logic |doi=10.2307/421089 |jstor=421089 |s2cid=18551419 |last1=Tarski |first1=Alfred |last2=Givant |first2=Steven |year=1999 |volume=5 |issue=2 |pages=175–214 }}</ref>

''Cardinal Algebras'' studied algebras whose models include the arithmetic of [[cardinal number]]s. ''Ordinal Algebras'' sets out an algebra for the additive theory of [[order type]]s. Cardinal, but not ordinal, addition commutes.

In 1941, Tarski published an important paper on [[binary relation]]s, which began the work on [[relation algebra]] and its [[metamathematics]] that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work of [[Roger Lyndon]]) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express most [[axiomatic set theory]] and [[Peano arithmetic]]. For an introduction to [[relation algebra]], see Maddux (2006). In the late 1940s, Tarski and his students devised [[cylindric algebra]]s, which are to [[first-order logic]] what the [[two-element Boolean algebra]] is to classical [[sentential logic]]. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985).<ref>{{Cite news |url=https://goodmancoaching.nl/tarskis-convention-t-and-inductive-definition/ |title=Tarski's convention-T and inductive definition? |newspaper=Goodmancoaching |date=22 May 2022 |access-date=28 April 2023}}</ref>