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144-551: Born in Muncie, Indiana , and raised on a farm outside Mathews, Lawvere received his undergraduate degree in mathematics from Indiana University . Lawvere studied continuum mechanics and kinetic theory as an undergraduate with Clifford Truesdell . He learned of category theory while teaching a course on functional analysis for Truesdell, specifically from a problem in John L. Kelley 's textbook General Topology . Lawvere found it

288-489: A 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} to b 1 , … , b n {\displaystyle b_{1},\dots ,b_{n}} respectively, then a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} and b 1 , … , b n {\displaystyle b_{1},\dots ,b_{n}} realise

432-416: A 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} . This is called the complete (n-)type realised by a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} over A . If there is an automorphism of M {\displaystyle {\mathcal {M}}} that is constant on A and sends

576-480: A 2 {\displaystyle a_{1}=a_{2}} or a 2 < a 1 {\displaystyle a_{2}<a_{1}} . Over the subset Z ⊆ R {\displaystyle \mathbb {Z} \subseteq \mathbb {R} } of integers, the 1-type of a non-integer real number a depends on its value rounded down to the nearest integer. More generally, whenever M {\displaystyle {\mathcal {M}}}

720-480: A n {\displaystyle a_{1},\dots ,a_{n}} of a structure M {\displaystyle {\mathcal {M}}} and a subset A of M {\displaystyle {\mathcal {M}}} , one can consider the set of all first-order formulas φ ( x 1 , … , x n ) {\displaystyle \varphi (x_{1},\dots ,x_{n})} with parameters in A that are satisfied by

864-408: A ) {\displaystyle \varphi (a)} is true. In this way, one can study definable groups and fields in general structures, for instance, which has been important in geometric stability theory. One can even go one step further, and move beyond immediate substructures. Given a mathematical structure, there are very often associated structures which can be constructed as a quotient of part of

1008-406: A and b are connected by the order automorphism that shifts all numbers by b-a . The complete 2-type over the empty set realised by a pair of numbers a 1 , a 2 {\displaystyle a_{1},a_{2}} depends on their order: either a 1 < a 2 {\displaystyle a_{1}<a_{2}} , a 1 =

1152-514: A graduate or professional degree. 20.4% had no degree. 53.2% of Muncie residents were employed, working a mean of 34.0 hours per week. The median gross rent in Muncie was $ 714 and the homeownership rate was 50.0%. 4,491 housing units were vacant at an average density of 162.7 per square mile (62.8/km ). From its early days as a regional trading center for the surrounding agricultural community to its first wave of industrial development brought on by

1296-408: A Grothendieck topology can be entirely described as an endomorphism of the subobject representor, and Tierney showed that the conditions it needs to satisfy are just idempotence and the preservation of finite intersections. These "topologies" are important in both algebraic geometry and model theory because they determine the subtoposes as sheaf-categories. Dalhousie University in 1969 set up

1440-468: A certain type over A is called type-definable over A . For an algebraic example, suppose M {\displaystyle M} is an algebraically closed field . The theory has quantifier elimination . This allows us to show that a type is determined exactly by the polynomial equations it contains. Thus the set of complete n {\displaystyle n} -types over a subfield A {\displaystyle A} corresponds to

1584-399: A curve. In general, definable sets without quantifiers are easy to describe, while definable sets involving possibly nested quantifiers can be much more complicated. This makes quantifier elimination a crucial tool for analysing definable sets: A theory T has quantifier elimination if every first-order formula φ( x 1 , ..., x n ) over its signature is equivalent modulo T to

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1728-510: A first-order formula ψ( x 1 , ..., x n ) without quantifiers, i.e. ∀ x 1 … ∀ x n ( ϕ ( x 1 , … , x n ) ↔ ψ ( x 1 , … , x n ) ) {\displaystyle \forall x_{1}\dots \forall x_{n}(\phi (x_{1},\dots ,x_{n})\leftrightarrow \psi (x_{1},\dots ,x_{n}))} holds in all models of T . If

1872-452: A formula of the following form: where ψ is quantifier free. A theory that is not model-complete may have a model completion , which is a related model-complete theory that is not, in general, an extension of the original theory. A more general notion is that of a model companion . In every structure, every finite subset { a 1 , … , a n } {\displaystyle \{a_{1},\dots ,a_{n}\}}

2016-496: A group of 15 Killam -supported researchers with Lawvere at the head; but in 1971 it terminated the group. Lawvere was controversial for his political opinions, for example, his opposition to the 1970 use of the War Measures Act , and for teaching the history of mathematics without permission. But in 1995 Dalhousie hosted the celebration of 50 years of category theory with Lawvere and Saunders Mac Lane present. Lawvere ran

2160-661: A long illness at the age of 85. Lawvere completed his Ph.D. at Columbia in 1963 with Eilenberg. His dissertation introduced the category of categories as a framework for the semantics of algebraic theories . From 1964 to 1967 at the Forschungsinstitut für Mathematik at the ETH in Zürich he worked on the category of categories and was especially influenced by Pierre Gabriel 's seminars at Oberwolfach on Grothendieck 's foundation of algebraic geometry . He then taught at

2304-443: A model fluctuated in the history of the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively: where universal algebra stands for mathematical structures and logic for logical theories; and where logical formulas are to definable sets what equations are to varieties over a field. Nonetheless, the interplay of classes of models and the sets definable in them has been crucial to

2448-484: A model of T which is itself a model of T is an elementary substructure. There is a useful criterion for testing whether a substructure is an elementary substructure, called the Tarski–Vaught test . It follows from this criterion that a theory T is model-complete if and only if every first-order formula φ( x 1 , ..., x n ) over its signature is equivalent modulo T to an existential first-order formula, i.e.

2592-839: A new board was appointed by Ball State's Board of Trustees. In 2021, following the Fall of Kabul and the U.S. withdrawal from the War in Afghanistan , several Afghan refugees arrived in Camp Atterbury , near Edinburgh, Indiana . Munsonian members of the nationwide organization, Afghan Women's and Kids' Education and Necessities (AWAKEN) formed the Muncie Afghan Refugee Resettlement Committee (MARRC) to help Afghan refugees resettle in Muncie. Afghan refugees began arriving in Muncie soon after. As

2736-514: A new city hall, a new public library, and a new high school. The city's gasworks also began operations in the late 1870s. The Muncie Star was founded in 1899 and the Muncie Evening Press was founded in 1905. A new public library , which was a Carnegie library project, was dedicated on January 1, 1904, and served as the main branch of the city's public library system. The forerunner to Ball State University also arrived at

2880-491: A new life. The black population in Muncie grew from 3.7% in 1890 to 5.6% in 1920 to 13.2% in 2013. Muncie's Riverside and Normal City neighborhoods had imposed sundown town rules by 1904, meaning "that unknown negroes who are found on the streets after nightfall are liable to be severely dealt with." In 1920, the city had the fifth-largest black population in Indiana, with 2,054 black residents. Two major parades were held by

3024-413: A passage from Quantifiers and Sheaves (1970): When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing. Doing this for "set theory" requires taking account of the experience that the main pairs of opposing tendencies in mathematics take

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3168-586: A promising framework for simple rigorous axioms for the physical ideas of Truesdell and Walter Noll . Truesdell supported Lawvere's application to study further with Samuel Eilenberg , a founder of category theory, at Columbia University in 1960. Before completing the Ph.D. Lawvere spent a year in Berkeley as an informal student of model theory and set theory , following lectures by Alfred Tarski and Dana Scott . In his first teaching position at Reed College he

3312-528: A seminar in Perugia, Italy (1972–1974) and especially worked on various kinds of enriched category . For example, a metric space can be regarded as an enriched category. From 1974 until his retirement in 2000 he was professor of mathematics at University at Buffalo , often collaborating with Stephen Schanuel . In 1977 he was elected to the Martin professorship in mathematics for five years, which made possible

3456-424: A structure are the isolated types, then the structure is called atomic . On the other hand, no structure realises every type over every parameter set; if one takes all of M {\displaystyle {\mathcal {M}}} as the parameter set, then every 1-type over M {\displaystyle {\mathcal {M}}} realised in M {\displaystyle {\mathcal {M}}}

3600-445: A structure is also called the theory of that structure . It's a consequence of Gödel's completeness theorem (not to be confused with his incompleteness theorems ) that a theory has a model if and only if it is consistent , i.e. no contradiction is proved by the theory. Therefore, model theorists often use "consistent" as a synonym for "satisfiable". A signature or language is a set of non-logical symbols such that each symbol

3744-548: A student at Indiana University from 1955 to January 1960. I liked experimental physics but did not appreciate the imprecise reasoning in some theoretical courses. So I decided to study mathematics first. Truesdell was at the Mathematics Department but he had a great knowledge in Engineering Physics. He took charge of my education there. ... in 1955 (and subsequently) had advised me on pursuing

3888-404: A theory does not have quantifier elimination, one can add additional symbols to its signature so that it does. Axiomatisability and quantifier elimination results for specific theories, especially in algebra, were among the early landmark results of model theory. But often instead of quantifier elimination a weaker property suffices: A theory T is called model-complete if every substructure of

4032-557: A third major study that resulted in two books by Theodore Caplow, Middletown Families (1982) and All Faithful People (1983). Caplow returned to Muncie in 1998 to begin another study, Middletown IV, which became part of a Public Broadcasting Service documentary titled " The First Measured Century ", released in December 2000. The Ball State Center for Middletown Studies continues to survey and analyze social change in Muncie. A database of Middletown surveys conducted between 1978 and 1997

4176-733: A two-year or four-year college degree, which is below the national average. The first decade of the 21st century saw a cultural shift toward local businesses and economic empowerment, boosted by the Muncie Downtown Development Partnership and the residents, patrons, and business owners of the downtown community. In 2007, Muncie was rated the most affordable college town in America by real estate company Coldwell Banker . In 2015, Forbes ranked Muncie 27th among small places for business and careers and 18th for cost of doing business. First Merchants Corporation

4320-431: A σ'-theory, and one can extend it (in more than one way) to a complete σ'-theory. The terms reduct and expansion are sometimes applied to this relation as well. The compactness theorem states that a set of sentences S is satisfiable if every finite subset of S is satisfiable. The analogous statement with consistent instead of satisfiable is trivial, since every proof can have only a finite number of antecedents used in

4464-437: Is Hermann Grassmann with his Ausdehnungslehre . The category theorist William Lawvere was a committed Marxist-Leninist ; at one point he gave a talk called " Applying Marxism-Leninism-Mao Tse-Thung Thought to Mathematics & Science ". According to Anders Kock's obituary, in 1971: the [Dalhousie] university administration refused to renew the contract with [Lawvere], due to his political activities in protesting against

William Lawvere - Misplaced Pages Continue

4608-472: Is a 1-type over Z ⊆ R {\displaystyle \mathbb {Z} \subseteq \mathbb {R} } that is not realised in the real number line R {\displaystyle \mathbb {R} } . A subset of M n {\displaystyle {\mathcal {M}}^{n}} that can be expressed as exactly those elements of M n {\displaystyle {\mathcal {M}}^{n}} realising

4752-458: Is complete . The set of complete n -types over A is often written as S n M ( A ) {\displaystyle S_{n}^{\mathcal {M}}(A)} . If A is the empty set, then the type space only depends on the theory T {\displaystyle T} of M {\displaystyle {\mathcal {M}}} . The notation S n ( T ) {\displaystyle S_{n}(T)}

4896-458: Is syntactic in nature, in contrast to model theory, which is semantic in nature. The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic . This page focuses on finitary first order model theory of infinite structures. The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within

5040-559: Is a finite union of points and intervals. Particularly important are those definable sets that are also substructures, i. e. contain all constants and are closed under function application. For instance, one can study the definable subgroups of a certain group. However, there is no need to limit oneself to substructures in the same signature. Since formulas with n free variables define subsets of M n {\displaystyle {\mathcal {M}}^{n}} , n -ary relations can also be definable. Functions are definable if

5184-405: Is a map f : A → B between the domains which can be written as an isomorphism of A {\displaystyle {\mathcal {A}}} with a substructure of B {\displaystyle {\mathcal {B}}} . If it can be written as an isomorphism with an elementary substructure, it is called an elementary embedding. Every embedding is an injective homomorphism, but

5328-408: Is a model of a theory exactly when the superstructure is a model. Example: While the field of algebraic numbers Q ¯ {\displaystyle {\overline {\mathbb {Q} }}} is an elementary substructure of the field of complex numbers C {\displaystyle \mathbb {C} } , the rational field Q {\displaystyle \mathbb {Q} }

5472-456: Is a sentence and A {\displaystyle {\mathcal {A}}} an elementary substructure of B {\displaystyle {\mathcal {B}}} , then A ⊨ φ {\displaystyle {\mathcal {A}}\models \varphi } if and only if B ⊨ φ {\displaystyle {\mathcal {B}}\models \varphi } . Thus, an elementary substructure

5616-432: Is a structure and A a subset of M {\displaystyle {\mathcal {M}}} , a (partial) n-type over A is a set of formulas p with at most n free variables that are realised in an elementary extension N {\displaystyle {\mathcal {N}}} of M {\displaystyle {\mathcal {M}}} . If p contains every such formula or its negation, then p

5760-404: Is a structure that models T {\displaystyle T} . A substructure A {\displaystyle {\mathcal {A}}} of a σ-structure B {\displaystyle {\mathcal {B}}} is a subset of its domain, closed under all functions in its signature σ, which is regarded as a σ-structure by restricting all functions and relations in σ to

5904-419: Is a subset of Q 2 {\displaystyle \mathbb {Q} ^{2}} ), one obtains a structure ( Q , σ o r ) {\displaystyle (\mathbb {Q} ,\sigma _{or})} . A structure N {\displaystyle {\mathcal {N}}} is said to model a set of first-order sentences T {\displaystyle T} in

William Lawvere - Misplaced Pages Continue

6048-535: Is a unary (= 1-ary) function symbol, and < {\displaystyle <} is a binary relation symbol. Then, when these symbols are interpreted to correspond with their usual meaning on Q {\displaystyle \mathbb {Q} } (so that e.g. + {\displaystyle +} is a function from Q 2 {\displaystyle \mathbb {Q} ^{2}} to Q {\displaystyle \mathbb {Q} } and < {\displaystyle <}

6192-603: Is available online from the Association of Religion Data Archives (ARDA). Due to the extensive information collected from the Middletown studies during the twentieth century, Muncie is said to be one of the most studied cities of its size in the United States. In addition to being called a "typical American city", as the result of the Middletown studies , Muncie is known as Magic City or Magic Muncie, as well as

6336-606: Is based in Muncie, and the first Scotty's Brewhouse location opened in the city in 1996. As of July 2022 (compare to October 2014), the largest employers in the city were: The David Owsley Museum of Art collection, which includes over 11,000 works, has been in the Fine Arts Building on the Ball State University campus since 1935. The Horizon Convention Center, located downtown, offers 47,000 square feet (4,400 m ) of exhibition space and houses

6480-486: Is called isolated . Since the real numbers R {\displaystyle \mathbb {R} } are Archimedean , there is no real number larger than every integer. However, a compactness argument shows that there is an elementary extension of the real number line in which there is an element larger than any integer. Therefore, the set of formulas { n < x | n ∈ Z } {\displaystyle \{n<x|n\in \mathbb {Z} \}}

6624-497: Is called a (first-order) theory , which takes the sentences in the set as its axioms. A theory is satisfiable if it has a model M ⊨ T {\displaystyle {\mathcal {M}}\models T} , i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T {\displaystyle T} . A complete theory is a theory that contains every sentence or its negation. The complete theory of all sentences satisfied by

6768-506: Is commonly used for the set of types over the empty set consistent with T {\displaystyle T} . If there is a single formula φ {\displaystyle \varphi } such that the theory of M {\displaystyle {\mathcal {M}}} implies φ → ψ {\displaystyle \varphi \rightarrow \psi } for every formula ψ {\displaystyle \psi } in p , then p

6912-439: Is definable by a quantifier-free formula in one variable. Quantifier-free formulas in one variable express Boolean combinations of polynomial equations in one variable, and since a nontrivial polynomial equation in one variable has only a finite number of solutions, the theory of algebraically closed fields is strongly minimal. On the other hand, the field R {\displaystyle \mathbb {R} } of real numbers

7056-552: Is definable with parameters: Simply use the formula Since we can negate this formula, every cofinite subset (which includes all but finitely many elements of the domain) is also always definable. This leads to the concept of a minimal structure . A structure M {\displaystyle {\mathcal {M}}} is called minimal if every subset A ⊆ M {\displaystyle A\subseteq {\mathcal {M}}} definable with parameters from M {\displaystyle {\mathcal {M}}}

7200-451: Is either a constant symbol, or a function or relation symbol with a specified arity . Note that in some literature, constant symbols are considered as function symbols with zero arity, and hence are omitted. A structure is a set M {\displaystyle M} together with interpretations of each of the symbols of the signature as relations and functions on M {\displaystyle M} (not to be confused with

7344-501: Is either finite or cofinite. The corresponding concept at the level of theories is called strong minimality : A theory T is called strongly minimal if every model of T is minimal. A structure is called strongly minimal if the theory of that structure is strongly minimal. Equivalently, a structure is strongly minimal if every elementary extension is minimal. Since the theory of algebraically closed fields has quantifier elimination, every definable subset of an algebraically closed field

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7488-419: Is isolated by a formula of the form a = x for an a ∈ M {\displaystyle a\in {\mathcal {M}}} . However, any proper elementary extension of M {\displaystyle {\mathcal {M}}} contains an element that is not in M {\displaystyle {\mathcal {M}}} . Therefore, a weaker notion has been introduced that captures

7632-717: Is located in East Central Indiana , about 50 miles (80 km) northeast of Indianapolis . At the 2020 United States Census , the city's population was 65,195, down from 70,085 in the 2010 Census. It is the principal city of the Muncie Metropolitan Statistical Area , which encompasses all of Delaware County. The city is also included in the Indianapolis–Carmel–Muncie, IN Combined Statistical Area . The Lenape ( Delaware ) people, led by Buckongahelas arrived in

7776-479: Is not minimal: Consider, for instance, the definable set This defines the subset of non-negative real numbers, which is neither finite nor cofinite. One can in fact use φ {\displaystyle \varphi } to define arbitrary intervals on the real number line. It turns out that these suffice to represent every definable subset of R {\displaystyle \mathbb {R} } . This generalisation of minimality has been very useful in

7920-413: Is not, as we can express "There is a square root of 2" as a first-order sentence satisfied by C {\displaystyle \mathbb {C} } but not by Q {\displaystyle \mathbb {Q} } . An embedding of a σ-structure A {\displaystyle {\mathcal {A}}} into another σ-structure B {\displaystyle {\mathcal {B}}}

8064-591: Is the Löwenheim-Skolem theorem . According to the Löwenheim-Skolem Theorem, every infinite structure in a countable signature has a countable elementary substructure. Conversely, for any infinite cardinal κ every infinite structure in a countable signature that is of cardinality less than κ can be elementarily embedded in another structure of cardinality κ (There is a straightforward generalisation to uncountable signatures). In particular,

8208-453: The Boolean connectives ¬ , ∧ , ∨ , → {\displaystyle \neg ,\land ,\lor ,\rightarrow } and prefixing of quantifiers ∀ v {\displaystyle \forall v} or ∃ v {\displaystyle \exists v} . A sentence is a formula in which each occurrence of a variable is in

8352-799: The Chesapeake and Ohio Railway ) reached Muncie in 1903. The Chicago, Indiana, and Eastern Railroad (acquired by a subsidiary of the Pennsylvania Railroad system) and the Chicago and Southeastern (sometimes called the Central Indiana Railroad) also served the city. In addition to the railroads, Muncie's roads connected to nearby towns and an electric interurban system, which arrived in the early 1900s, linked it to smaller towns and larger cities, including Indianapolis and Fort Wayne, Indiana , and Dayton, Ohio . With

8496-654: The Indiana gas boom in the mid-1880s, Muncie has retained its ties to an industrial economy, and to a lesser extent its agricultural roots. In addition, the arrival of the forerunner to Ball State in the early twentieth century contributed to Muncie's development as an educational center, while Ball Memorial Hospital, established in 1929, led to the city's reputation as a healthcare center for east-central Indiana. Muncie's major industrial development included glass manufacturing, iron and steel mills, and automobile manufacturing and auto parts factories. Among its early major employers

8640-480: The Ku Klux Klan in 1922 and 1924, with the mayor and police chief attending the 1924 parade. The first black police officer in Muncie was hired in 1899, and the city's first black police chief took his position in 1995. The first black teacher in the city was hired in 1952, and the first black principal was appointed in 1956. According to Hurley Goodall , the first serious black political candidate in Muncie

8784-467: The University of Chicago in 1967. He mentions that he and Mac Lane co-taught a course on mechanics , which led him to consider the justification of older intuitive methods in geometry , eventually coining the term " synthetic differential geometry " This course was based on Mackey's book Mathematical Foundations of Quantum Mechanics , indicating Mackey's influence on category theory . Further in

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8928-568: The White River , including Munsee Town, near the site of present-day Muncie. Contrary to popular legend, the city's early name of Munsee Town is derived from the "Munsee" clan of Lenape people, the white settlers' name for a group of Native Americans whose village was once situated along the White River. There is no evidence that a mythological Chief Munsee ever existed. ("Munsee" means a member of or one of their languages . ) In 1818,

9072-533: The census of 2000, there were 67,430 people, 27,322 households, and 14,589 families residing in the city. The population density was 2,788.2 inhabitants per square mile (1,076.5/km ). There were 30,205 housing units at an average density of 1,248.9 per square mile (482.2/km ). The racial makeup of the city was 83.72% White , 12.97% African American , 0.27% Native American , 0.79% Asian , 0.09% Pacific Islander , 0.67% from other races , and 1.49% from two or more races. Hispanic or Latino of any race were 1.44% of

9216-422: The rational numbers , regarded as a structure in the signature {+,0} can be expanded to a field with the signature {×,+,1,0} or to an ordered group with the signature {+,0,<}. Similarly, if σ' is a signature that extends another signature σ, then a complete σ'-theory can be restricted to σ by intersecting the set of its sentences with the set of σ-formulas. Conversely, a complete σ-theory can be regarded as

9360-687: The 1880s. It is home to Ball State University . As a result of the Middletown studies , sociological research that was first conducted in the 1920s, Muncie is said to be one of the most studied United States cities of its size. The area was first settled in the 1790s by the Lenape (Delaware) people, who were forced west from their tribal lands in the Mid-Atlantic region (all of New Jersey , southeastern New York , eastern Pennsylvania , and northern Delaware ) to new lands in present-day Ohio and eastern Indiana . The Lenape founded several towns along

9504-551: The 1920s, Robert and Helen Lynd led a team of sociologists in a study of a typical middle-American community. The Lynds chose Muncie as the locale for their field research, although they never specifically identified it as " Middletown " the fictional name of the town in their study. Muncie received national attention after the publication of their book, Middletown: A Study in Contemporary American Culture (1929). The Lynds returned to Muncie to re-observe

9648-636: The 1950s and 1960s, Muncie has continued as an education center in the state and emerged as a regional health center. As enrollment at Ball State increased, new buildings were erected on the college's campus. Ball Memorial Hospital also expanded its facilities. However, by the 1960s, industrial trends had shifted. Beginning in the 1970s several manufacturing plants closed or moved elsewhere, while others adapted to industrial changes and remained in Muncie. Ball Corporation, for example, closed its Muncie glass manufacturing facilities in 1962 and its corporate headquarters relocated to Broomfield, Colorado in 1998. Muncie

9792-631: The 1970s, the subject has been shaped decisively by Saharon Shelah 's stability theory . Compared to other areas of mathematical logic such as proof theory , model theory is often less concerned with formal rigour and closer in spirit to classical mathematics. This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane" . The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory

9936-496: The AWAKEN organization wanted to continue work directly in Afghanistan, the temporary MARRC organization was closed. However, the presence of 130 Afghan refugees in Muncie in late 2022 compelled continued work in medical advocacy, employment, legal aid, and financial and reading education. A new organization, Refugee Alliance of Delaware County (RADC), was formed to welcome any refugees in Muncie and help them in establishing and building

10080-542: The Delaware County seat and platted in 1827. On January 13, 1845, Indiana's governor signed legislation passed by the Indiana General Assembly to shorten the town's name to Muncie. Soon, a network of roads connected Muncie to nearby towns, adjacent counties, and to other parts of Indiana. The Indianapolis and Bellefontaine Railroad , the first to arrive in Muncie in 1852, provided the town and

10224-510: The Friendly City. During World War II, the city's manufacturers once again turned their efforts to wartime production. Ball State and Muncie's airport also trained pilots for the U.S. Navy . The postwar era was another period of expansion for Muncie, with continued growth and development of industries, construction of new homes, schools, and businesses. A population boom brought further development, especially from 1946 to 1965. Since

10368-623: The Indiana State Normal School, Eastern Division, the forerunner to Ball State University, opened in 1918. It was named Ball Teachers College in 1922, Ball State Teachers College in 1929, and Ball State University in 1965. Beginning in the late nineteenth century, in tandem with the gas boom, Muncie developed an active cultural arts community, which included music and art clubs, women's clubs, self-improvements clubs, and other social clubs. Hoosier artist J. Ottis Adams , who came to Muncie in 1876, later formed an art school in

10512-541: The Löwenheim-Skolem Theorem implies that any theory in a countable signature with infinite models has a countable model as well as arbitrarily large models. In a certain sense made precise by Lindström's theorem , first-order logic is the most expressive logic for which both the Löwenheim–Skolem theorem and the compactness theorem hold. In model theory, definable sets are important objects of study. For instance, in N {\displaystyle \mathbb {N} }

10656-587: The Muncie Ballet and the Muncie Symphony Orchestra are prominent in the city's arts community. Model theory In mathematical logic , model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure ), and their models (those structures in which the statements of the theory hold). The aspects investigated include

10800-585: The Muncie Children's Museum. The city also has a large group of independent art galleries. Three of the city's largest performing arts centers belong to Ball State University: the 3,581-seat Emens Auditorium , the 600-seat Sursa Performance Hall, and the 410-seat University Theatre. Downtown performing arts spaces include the Muncie Civic Theatre and Canan Commons, an outdoor amphitheater and greenspace that opened in 2011. In addition,

10944-408: The U.S. headquarters for Brevini Wind, an Italian-based company that manufactures gearboxes for wind turbines. In 2011, locomotive maker Progress Rail (a subsidiary of Caterpillar Inc ) opened in a former Westinghouse facility that had been vacant since 1998. The local economy is a controversial topic among Munsonians. While many older unemployed or underemployed residents strongly identify with

11088-675: The University of Chicago, working with Mac Lane , and at the City University of New York Graduate Center (CUNY), working with Alex Heller . Lawvere's Chicago lectures on categorical dynamics were a further step toward topos theory and his CUNY lectures on hyperdoctrines advanced categorical logic especially using his 1963 discovery that existential and universal quantifiers can be characterized as special cases of adjoint functors . Back in Zürich for 1968 and 1969 he proposed elementary (first-order) axioms for toposes generalizing

11232-760: The Vietnam war and against the War Measures Act proclaimed by Trudeau, suspending civil liberties under the pretext of danger of terrorism. As per the obituary on the Communist Party of Canada (Marxist–Leninist) site: More than 1,000 students rallied in the lobby of the Dal Student Union Building to oppose the arbitrary dismissal of Professor Lawvere. He saw his political commitments as related to his mathematical work in sometimes surprising and unexpected ways: for instance, here's

11376-410: The age of 18. The poverty rate for the town was 17.3% higher than that of the state. 18.6% of the population was disabled and 8.7% had no healthcare coverage. 34.3% of the population had attained a high school or equivalent degree, 19.7% had attended college but received no degree, 8.9% had attained an Associate's degree or higher, 14.0% had attained a Bachelor's degree or higher, and 11.6% had

11520-402: The area in the 1790s, founding several villages, including one known as Munsee Town, along the White River . The trading post , renamed Muncietown, was selected as the Delaware County seat and platted in 1827. Its name was officially shortened to Muncie in 1845 and incorporated as a city in 1865. Muncie developed as a manufacturing and industrial center, especially after the Indiana gas boom of

11664-452: The area's native tribes ceded their lands to the federal government under the terms of the Treaty of St. Mary's and agreed to move farther west by 1821. New settlers began to arrive in what became Delaware County, Indiana , c.  1820 , shortly before the area's public lands were formally opened for purchase. The small trading village of Munsee Town, renamed Muncietown, was selected as

11808-488: The arrival of the auto manufacturing and the related auto parts industry after the turn of the twentieth century, Muncie's industrial and commercial development increased as the population grew. During World War I, local manufacturers joined others around the county in converting their factories to production of war materiel . In the 1920s Muncie continued its rise as an automobile-manufacturing center, primarily due to its heavy industry and skilled labor force. During this time,

11952-404: The average family size was 2.86. In the city, the age distribution of the population shows 19.8% under the age of 18, 24.6% from 18 to 24, 24.2% from 25 to 44, 18.3% from 45 to 64, and 13.2% who were 65 years of age or older. The median age was 29 years. For every 100 females, there were 89.9 males. For every 100 females age 18 and over, there were 86.5 males. The median income for a household in

12096-415: The bibliography. Additionally, he connects various mathematical concepts to Hegel's Dialectic and Lenin's theory of knowledge in other parts of his work. Muncie, Indiana Muncie ( / ˈ m ʌ n s i / MUN -see ) is an incorporated city and the seat of Delaware County, Indiana , United States. Previously known as Buckongahelas Town, named after the prominent Delaware Chief, it

12240-447: The city was $ 26,613, and the median income for a family was $ 36,398. Males had a median income of $ 30,445 versus $ 21,872 for females. The per capita income for the city was $ 15,814. About 14.3% of families and 23.1% of the population were below the poverty line , including 24.2% of those under age 18 and 9.7% of those age 65 or over. As of the census of 2010, there were 70,085 people, 27,722 households, and 13,928 families residing in

12384-538: The city with fellow artist William Forsyth . Although their school closed with a year or two, other art groups were established, most notably the Art Students' League (1892) and the Muncie Art Association (1905). By the early twentieth century several railroads served Muncie, which helped to establish the city as a transportation hub. The Cincinnati, Richmond and Muncie Railroad (later known as

12528-617: The city's economic development. As in many mid-sized cities in the Rust Belt , deindustrialization , which began in the 1960s, impacted Muncie's economy. Several manufacturing plants closed or moved elsewhere. From 2001 to 2011, Muncie lost thousands of jobs as the city continued transitioning from a blue-collar workforce to a white-collar service economy primarily based on health care, education, and retail. Muncie has attracted some new manufacturers, while older factories have been converted to other industrial uses. In 2009, Muncie became

12672-731: The city's gas-boom period was the Ball Brothers Glass Manufacturing Company, which was renamed the Ball Corporation in 1969. The Ball brothers , who were searching for a new site for their glass manufacturing business that was closer to an abundant natural-gas supply, built a new glass-making foundry in Muncie, beginning its glass production on March 1, 1888. In 1889 the company relocated its metal manufacturing operations to Muncie. In addition to several other glass factories, Muncie attracted iron and steel mills. Kitselman Steel & Wire Company

12816-423: The city. The population density was 2,576.7 inhabitants per square mile (994.9/km ). There were 31,958 housing units at an average density of 1,174.9 per square mile (453.6/km ). The racial makeup of the city was 84.0% White , 10.9% African American , 0.3% Native American , 1.2% Asian , 0.1% Pacific Islander , 0.8% from other races , and 2.8% from two or more races. Hispanic or Latino of any race were 2.3% of

12960-469: The city. The population density was 2,577.0 inhabitants per square mile (995.0/km ). There were 31,183 housing units at an average density of 1,129.8 per square mile (436.2/km ). The racial makeup of the city was 78.0% White , 11.6% African American , 0.3% Native American or Alaskan Native , 1.5% Asian , 0.1% Native Hawaiian or Pacific Islander , 2.1% from other races and 6.5% were from two or more races. Hispanic and Latino of any race were 4.2% of

13104-435: The commercial trading center for local farmers. The Indiana gas boom of the 1880s ushered in a new era of prosperity to Muncie. Abundant supplies of natural gas attracted new businesses, industries, and additional residents to the city. Although agriculture continued to be an economic factor in the region, industry dominated the city's development for the next 100 years. One of the major manufacturers that arrived early in

13248-520: The community also became a center of Ku Klux Klan activity. Muncie's Klan membership was estimated at 3,500 in the early 1920s. Scandals within the Klan's leadership, divisions among its members, and some violent confrontations with their opponents damaged the organization's reputation. Increasing hostility toward the Klan's political activities, beliefs, and values also divided the Muncie community, before its popularity and membership significantly declined by

13392-705: The community during the Depression , which resulted in a sequel, Middletown in Transition: A Study in Cultural Conflicts (1937). The Lynds' Middletown study, which was funded by the Rockefeler Institute of Social and Religious Research, was intended to study "the interwoven trends that are the life of a small American city." The Lynds were only the first to conduct a series of studies in Muncie. The National Science Foundation funded

13536-543: The concept of the Grothendieck topos (see History of topos theory ) and worked with the algebraic topologist Myles Tierney to clarify and apply this theory. Tierney discovered major simplifications in the description of Grothendieck "topologies" . Anders Kock later found further simplifications so that a topos can be described as a category with products and equalizers in which the notions of map space and subobject are representable. Lawvere had pointed out that

13680-405: The converse holds only if the signature contains no relation symbols, such as in groups or fields. A field or a vector space can be regarded as a (commutative) group by simply ignoring some of its structure. The corresponding notion in model theory is that of a reduct of a structure to a subset of the original signature. The opposite relation is called an expansion - e.g. the (additive) group of

13824-446: The definitions mentioned here are parameter-free , that is, the defining formulas don't mention any fixed domain elements. However, one can also consider definitions with parameters from the model . For instance, in R {\displaystyle \mathbb {R} } , the formula uses the parameter π {\displaystyle \pi } from R {\displaystyle \mathbb {R} } to define

13968-759: The development of continuum physics. William Lawvere has also proposed formalizations in category theory , categorical logic and topos theory of concepts which are motivated from philosophy , notably in Georg Hegel 's Science of Logic (see there for more). This includes for instance definitions of concepts found there such as objective and subjective logic, abstract general, concrete general, concrete particular, unity of opposites , Aufhebung , being, becoming, space and quantity, cohesion, intensive and extensive quantity ... and so on. In his work "Categories of Space and Quantity" from The Space of Mathematics (1992), William Lawvere expresses his belief that

14112-526: The development of model theory throughout its history. For instance, while stability was originally introduced to classify theories by their numbers of models in a given cardinality , stability theory proved crucial to understanding the geometry of definable sets. A first-order formula is built out of atomic formulas such as R ( f ( x , y ) , z ) {\displaystyle R(f(x,y),z)} or y = x + 1 {\displaystyle y=x+1} by means of

14256-506: The early 1960s: I felt a strong need to learn more set theory and logic from experts in that field, still of course with the aim of clarifying the foundations of category theory and of physics. The title of the early text "Toposes of laws of motion", which is often cited as the text introducing synthetic differential geometry , clearly witnesses the origin and motivation of these ideas in classical mechanics . In an interview, William F. Lawvere reflects on his time as an assistant professor at

14400-632: The end of the decade. Muncie residents also made it through the challenges of the Great Depression , with the Ball brothers continuing their role as major benefactors to the community by donating funds for construction of new facilities at Ball State and Ball Memorial Hospital . (The hospital, which opened in 1929, later affiliated with Indiana University Health . ) The Works Progress Administration (WPA) also provided jobs such as road grading, city sewer improvements, and bridge construction. In

14544-440: The equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the semiring of natural numbers N {\displaystyle {\mathcal {N}}} , viewed as a structure with binary functions for addition and multiplication and constants for 0 and 1 of the natural numbers, for example, an element n {\displaystyle n} satisfies

14688-521: The form of adjoint functors, and frees us of the mathematically irrelevant traces (∈) left behind by the process of accumulating (∪) the power set (P) at each stage of a metaphysical "construction". In the earlier sections of the paper, he discusses the "unity of opposites" between logic and geometry. He clarifies that his discussion of contradiction, ideology, and opposition is rooted in the Marxist tradition , referencing Mao's " On Contradiction " (1937) in

14832-679: The formal notion of an " interpretation " of one structure in another). Example: A common signature for ordered rings is σ o r = ( 0 , 1 , + , × , − , < ) {\displaystyle \sigma _{or}=(0,1,+,\times ,-,<)} , where 0 {\displaystyle 0} and 1 {\displaystyle 1} are 0-ary function symbols (also known as constant symbols), + {\displaystyle +} and × {\displaystyle \times } are binary (= 2-ary) function symbols, − {\displaystyle -}

14976-530: The formula φ {\displaystyle \varphi } if and only if n {\displaystyle n} is a prime number. The formula ψ {\displaystyle \psi } similarly defines irreducibility . Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth" , for the satisfaction relation ⊨ {\displaystyle \models } , so that one easily proves: A set T {\displaystyle T} of sentences

15120-490: The formula defines the subset of prime numbers, while the formula defines the subset of even numbers. In a similar way, formulas with n free variables define subsets of M n {\displaystyle {\mathcal {M}}^{n}} . For example, in a field, the formula defines the curve of all ( x , y ) {\displaystyle (x,y)} such that y = x 2 {\displaystyle y=x^{2}} . Both of

15264-528: The foundations of continuum mechanics , inspired by Truesdell's teachings, Noll's axiomatizations, and his own efforts in 1958 to categorize topological dynamics. A more detailed review of these ideas and their relation to physics can be found in the introduction to the book collection Categories in Continuum Physics , which is the proceedings of a meeting organized by Lawvere in 1982. In his 1997 talk "Toposes of Laws of Motion", Lawvere remarks on

15408-500: The foundations of continuum physics as inspired by Truesdell, Noll, and others. He notes that while the mathematical apparatus used in this field is powerful, it often does not fit the phenomena well. Lawvere questions whether the problems and necessary axioms could be stated more directly and clearly, potentially leading to a simpler yet rigorous account. These questions led him to apply the topos method in his 1967 Chicago lectures on categorical dynamics. He realized that further work on

15552-403: The function graph is a definable relation, and constants a ∈ M {\displaystyle a\in {\mathcal {M}}} are definable if there is a formula φ ( x ) {\displaystyle \varphi (x)} such that a is the only element of M {\displaystyle {\mathcal {M}}} such that φ (

15696-425: The gas boom ended in Indiana c.  1910 , Muncie was well established as an industrial town and a commercial center for east-central Indiana, especially with several railroad lines connecting it to larger cities and the arrival of automobile industry manufacturing after 1900. Numerous civic developments also occurred as a result of the city's growth during the 1870s, 1880s, and 1890s, when Muncie citizens built

15840-464: The given language if each sentence in T {\displaystyle T} is true in N {\displaystyle {\mathcal {N}}} with respect to the interpretation of the signature previously specified for N {\displaystyle {\mathcal {N}}} . (Again, not to be confused with the formal notion of an " interpretation " of one structure in another) A model of T {\displaystyle T}

15984-396: The interpreted structures to the language of the original structure. Thus one can show that if a structure M {\displaystyle {\mathcal {M}}} interprets another whose theory is undecidable, then M {\displaystyle {\mathcal {M}}} itself is undecidable. For a sequence of elements a 1 , … ,

16128-670: The interview, he discusses the origins of synthetic differential geometry, noting that the idea for the joint course on mechanics came from a suggestion by Chandra. This course was the first in a series, and Mac Lane later gave a talk on the Hamilton-Jacobi equation at the Naval Academy in 1970, which was published in The American Mathematical Monthly . He explains that he began applying Grothendieck topos theory, learned from Gabriel, to simplify

16272-482: The longstanding program of infinitesimal calculus , continuum mechanics , and differential geometry , which aims to reconstruct the world from the infinitely small. He acknowledges the skepticism around this idea but emphasizes its fruitful outcomes over the past 300 years. He believes that recent developments have positioned mathematicians to make this program more explicit, focusing on how continuum physics can be mathematically constructed from "simple ingredients". In

16416-497: The manufacturing identity of the city, newer residents identify with the city's shift towards educational and health services. Contention is greatest among residents living in the once-industrialized sections of the city's south side, as much of the economic growth over that last few decades has taken place on Muncie's north side. The city also struggles to retain college graduates. Despite Ball State's presence, only 32.2 percent of Delaware County's working-age adults (ages 25–64) hold

16560-460: The meeting on "Categories in Continuum Physics" in 1982. Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations of continuum physics and in the synthetic differential geometry that had evolved from the spatial part of Lawvere's categorical dynamics program. Lawvere continued to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications. He

16704-425: The model theory of ordered structures. A densely totally ordered structure M {\displaystyle {\mathcal {M}}} in a signature including a symbol for the order relation is called o-minimal if every subset A ⊆ M {\displaystyle A\subseteq {\mathcal {M}}} definable with parameters from M {\displaystyle {\mathcal {M}}}

16848-453: The notion of topos was necessary to achieve his goals. His time spent with Berkeley logicians in 1961-62, listening to experts on foundations, also influenced his approach. Lawvere highlights that several books on simplified topos theory , including the recent and accessible text by MacLane and Moerdijk, along with three excellent books on synthetic differential geometry , provide a solid foundation for further work in functional analysis and

16992-454: The number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski , who first used the term "Theory of Models" in publication in 1954. Since

17136-409: The original structure via an equivalence relation. An important example is a quotient group of a group. One might say that to understand the full structure one must understand these quotients. When the equivalence relation is definable, we can give the previous sentence a precise meaning. We say that these structures are interpretable . A key fact is that one can translate sentences from the language of

17280-411: The population. There were 26,692 households, of which 16.2% had children under the age of 18 living with them, 29.6% were married couples living together, 35.7% had a female householder with no husband present, 26.2% had a male householder with no wife present, and 8.5% were non-families. 61.9% of all households were made up of individuals. The average household size was 2.44 and the average family size

17424-408: The population. There were 27,322 households, out of which 23.7% had children under the age of 18 living with them, 36.4% were married couples living together, 13.0% had a female householder with no husband present, and 46.6% were non-families. 34.1% of all households were made up of individuals, and 11.8% had someone living alone who was 65 years of age or older. The average household size was 2.24 and

17568-441: The population. There were 27,722 households, of which 23.9% had children under the age of 18 living with them, 31.5% were married couples living together, 14.1% had a female householder with no husband present, 4.6% had a male householder with no wife present, and 49.8% were non-families. 35.2% of all households were made up of individuals, and 10.9% had someone living alone who was 65 years of age or older. The average household size

17712-475: The proof. The completeness theorem allows us to transfer this to satisfiability. However, there are also several direct (semantic) proofs of the compactness theorem. As a corollary (i.e., its contrapositive), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in model theory, where the words "by compactness" are commonplace. Another cornerstone of first-order model theory

17856-410: The same complete type over A . The real number line R {\displaystyle \mathbb {R} } , viewed as a structure with only the order relation {<}, will serve as a running example in this section. Every element a ∈ R {\displaystyle a\in \mathbb {R} } satisfies the same 1-type over the empty set. This is clear since any two real numbers

18000-496: The same talk, Lawvere mentions that the essential spaces required for functional analysis and physical field theories can be found in any topos with an appropriate object (T). In his 2000 article "Comments on the Development of Topos Theory", Lawvere discusses his motivation for simplifying and generalizing Grothendieck's concept of topos. He explains that his interest stemmed from his earlier studies in physics, particularly

18144-534: The scope of a corresponding quantifier. Examples for formulas are φ {\displaystyle \varphi } (or φ ( x ) {\displaystyle \varphi (x)} to indicate x {\displaystyle x} is the unbound variable in φ {\displaystyle \varphi } ) and ψ {\displaystyle \psi } (or ψ ( x ) {\displaystyle \psi (x)} ), defined as follows: (Note that

18288-408: The set of prime ideals of the polynomial ring A [ x 1 , … , x n ] {\displaystyle A[x_{1},\ldots ,x_{n}]} , and the type-definable sets are exactly the affine varieties. While not every type is realised in every structure, every structure realises its isolated types. If the only types over the empty set that are realised in

18432-504: The state house until Carl Kizer Jr.'s appointment to the board in 1993. The school board became majority black for the first time after the 2008 election. According to the 2010 census, Muncie has a total area of 27.392 square miles (70.94 km ), of which 27.2 square miles (70.45 km ),or 99.3%, is land and 0.192 square miles (0.50 km ),or 0.7%, is water. Muncie has a humid continental climate (Köppen climate classification: Dfa ), experiencing four distinct seasons. As of

18576-422: The study of continuum mechanics and kinetic theory. In Summer 1958, I studied Topological Dynamics with George Whaples, with the agenda of understanding as much as possible in categorical terms. ... Categories would clearly be important for simplifying the foundations of continuum physics. I concluded that I would make category theory a central line of my study. Then in the same interview (page 11) he said about

18720-482: The subset. This generalises the analogous concepts from algebra; for instance, a subgroup is a substructure in the signature with multiplication and inverse. A substructure is said to be elementary if for any first-order formula φ {\displaystyle \varphi } and any elements a 1 , ..., a n of A {\displaystyle {\mathcal {A}}} , In particular, if φ {\displaystyle \varphi }

18864-418: The surrounding area with access to larger markets for its agricultural production, as well as a faster means of transporting people and goods into and out of the area. Muncie incorporated as a town on December 6, 1854, and became an incorporated city in 1865. John Brady was elected as the city's first mayor. Muncie's early utility companies also date to the mid-1860s, including the city's waterworks , which

19008-635: The technical advancements made by category theorists will significantly benefit dialectical philosophy in the coming decades and century. He argues that these advancements will provide precise mathematical models for age-old philosophical distinctions, such as general versus particular, objective versus subjective, and being versus becoming. He emphasizes that mathematicians need to engage with these philosophical questions to make mathematics and other sciences more accessible and useful. This, he notes, will require philosophers to learn mathematics and mathematicians to learn philosophy. A precursor to this undertaking

19152-451: The theory of a structure has quantifier elimination, every set definable in a structure is definable by a quantifier-free formula over the same parameters as the original definition. For example, the theory of algebraically closed fields in the signature σ ring = (×,+,−,0,1) has quantifier elimination. This means that in an algebraically closed field, every formula is equivalent to a Boolean combination of equations between polynomials. If

19296-457: The turn of the twentieth century. Eastern Indiana Normal School opened in 1899, but it closed after two years. Several subsequent efforts to establish a private college in Muncie during the late nineteenth and early twentieth centuries also failed, but one proved to be very successful. After the Ball brothers bought the school property and its vacant buildings and donated them to the State of Indiana,

19440-462: Was English with 95.5% speaking it at home, 1.7% spoke Spanish at home, 1.4% spoke an Asian or Pacific Islander language at home, 1.3% spoke another Indo-European language at home, and 0.1% spoke some other language. 2.6% of the population were foreign born . The median household income in Muncie was $ 34,602, 38.4% less than the median average for the state of Indiana . 30.2% of the population were in poverty, including 32.6% of residents under

19584-452: Was professor emeritus of mathematics and adjunct professor emeritus of philosophy at Buffalo. A central motivation for Lawvere's work is the search for a good mathematical (rigorous) foundations of physics , specifically of (classical) continuum mechanics (or at least some kinematical aspects thereof , Lawvere does not seem to mention Hamiltonians , Lagrangians or action functionals). In an interview (page 8) he recalled: I had been

19728-438: Was 2.22 and the average family size was 2.85. The median age in the city was 28.1 years. 17.8% of residents were under the age of 18; 27.5% were between the ages of 18 and 24; 21.4% were from 25 to 44; 20.2% were from 45 to 64; and 13% were 65 years of age or older. The gender makeup of the city was 47.5% male and 52.5% female. As of the census of 2020, there were 65,194 people, 26,692 households and 6,179 families residing in

19872-438: Was 3.09. Just over half (50.1%) of the population had never been married, 29.6% were married and not separated , 5.7% were widowed , 13.1% were divorced , and 1.6% were separated. The median age of the city was 28.8. 4.4% of residents were under the age of 5, 16.2% were under 18 years, 83.8% were age 18 or older, and 14.0% were age 65 or older. 5.8% of the population were veterans . The most common language spoken at home

20016-511: Was Wayne Brooks, who ran for the Republican nomination for mayor in 1934. Ray Armstrong was elected as the first black member of the city council in 1951, and Alice McIntosh was elected as the first black female member of the city council in 1983. Goodall was the first black person elected to the city's school board and later to represent the area in the state house. No other black person served on Muncie's school board after Goodall's election to

20160-490: Was also home to other manufacturing operations, including Warner Gear (a division of BorgWarner ), Delco Remy , General Motors , Ontario Corporation, A. E. Boyce Company, and Westinghouse Electric , among others. In 2017, the Muncie Community Schools system was declared a "distressed political subdivision", and put in direct control of the state government. In 2018, the school district was reformed and

20304-525: Was established in 1865. After the American Civil War , two factors helped Muncie attract new commercial and industrial development: the arrival of additional railroads from the late 1890s to the early 1900s and the discovery of abundant supplies of natural gas in the area. Prior to the discovery of nearby natural-gas wells and the beginning of the gas boom in Muncie in 1886, the region was primarily an agricultural area, with Muncie serving as

20448-715: Was instructed to devise courses in calculus and abstract algebra from a foundational perspective. He tried to use the then current axiomatic set theory but found it unworkable for undergraduates, so he instead developed the first axioms for the more relevant composition of mappings of sets. He later streamlined those axioms into the Elementary Theory of the Category of Sets (1964), which became an ingredient (the constant case) of elementary topos theory . Lawvere died on January 23, 2023, in Chapel Hill, N.C., after

20592-669: Was the Ball Corporation, established by the Ball brothers of Buffalo, New York , who opened a glass factory in Muncie in 1888. Other notable manufacturers in addition to the Ball Corporation with operations in Muncie have included BorgWarner , The Broderick Company (aformer division of Harsco ), Dayton-Walther Corporation, Delco Remy , General Motors , New Venture Gear , Hemingray Glass Company , Ontario Corporation, A. E. Boyce Company, Indiana Steel and Wire, and Westinghouse Electric . Changing industrial trends caused shifts in

20736-554: Was the largest employer in Indiana in 1900 with 11,000 employees; it later became Indiana Steel & Wire. Others included Republic Iron and Steel Company and the Midland Steel Company. (Midland became Inland Steel Company and later moved to Gary, Indiana .) Indiana Bridge Company was also a major employer. By the time the natural gas supply from the Trenton Gas Field had significantly declined and

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