Cuny virtual set theory seminar april 17th, 2020 corey switzer cuny specializing wide trees virtual set theory150. Almost all the standard large cardinal notions are known to be consistent with the gch. It is a remarkable empirical fact that for any natural statement in the language of set theory. Contribute to 9beachjech settheorysolutions development by creating an account on github.
Large cardinals are rightly viewed as some of the most interesting axioms of contemporary set theory. I will state and give supporting evidence for several theses. These concepts were extended to that of ksubtle, kalmost ineffable, and kineffable cardinals in ba75, where a highly developed theory is presented. It turns out that numerous existing concepts in descriptive set theory and analysis have intuitive and informative geometric restatements, and the geometric point of. Its not intended as a textbook, but it shows whats there. Cardinals with such properties are, as the name suggests, generally very large for example, bigger than the least. In order for a true believer to really know mount everest, he must slowly and painfully trudge up its forbidding side, climbing the rocks amid the snow and the slush, with his confidence waning and his skepticism growing as to the possibility of ever scaling the height. The notes are intended for a student who has taken one or two semesters of graduatelevel set theory, but may have little or no prior exposure to ultrapowers and iteration.
The higher infinite large cardinals in set theory from. Set theory and its place in the foundations of mathematics. They can shed more light on the behaviour of the large cardinals, provide lower consistency bounds on axioms of interest and also provide. Large cardinals arrange themselves naturally into a more or less linear order of size and consistency strength, and provide a convenient yardstick to measure the consistency strength of various other assertions that are unprovable from zfc. Finally, some considerations will be made on how constructive one can get when using large cardinal notions, and whether the constructive approach to a large cardinal hierarchy could avoid some. Indeed, most are provably preserved by the canonical forcing of the gch. Usually the birth of set theory is traced back to his 1874 paper cantor 1874, in which cantor shows that there are at least two sizes of infinity and derives liouvilles theorem which states that there are infinitely many transcendental numbers. Early work was motivated by gambling and recreational games such as chess, hence the \game in game theory. Partition relations for cardinals, volume 106 1st edition. This means that 1,2,3 is a set but 1,1,3 is not because 1 appears twice in the second collection. Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with zfc citation needed. Finite functions and the necessary use of large cardinals. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. One should note that this does not represent the entire large cardinal discussion to date.
Introduction in this paper my main goal is to elaborate on some ideas from a talk which i gave at the workshop on \set theory and the philosophy of mathematics held at the university of pennsylvania in october 2010. Partition calculus 1 introduction partition calculus is the area of set theory that deals with ramsey theory. These canonical models can then be used in various di. Introduction to large cardinals the aim of this talk is to.
The rst of these is the expansion of g odels class lof. These axioms are strong enough to imply the consistency of zfc. These hypotheses posit cardinals that prescribe their own transcendence over smaller cardinals and provide a sup structure for the analysis of strong propositions. To show a set is denumerable, it is sufficient to produce a bijection from. Large cardinals arrange themselves naturally into a more or less linear order of size and consistency strength, and provide a convenient yardstick to measure the consistency strength. And just to finish with a small point, large cardinals dont always have to be inaccessible. Professor suncica canic named a fellow of the american mathematical society.
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. This alone assures the subject of a place prominent in human culture. One must be careful when understanding the power set axiom. Set theory has discovered an entire cosmos of settheoretical universes, connected by forcing or large cardinal embeddings, like lines in a constellation. Independence and large cardinals stanford encyclopedia of. Rodriguess specific answer gets at the general issue. Introduction to set theory, second edition revised and expanded, by karel hrbacuk and thomas jech. The approach to measure theory here is inspired by the text. Cardinal and ordinal numbers math 6300 klaus kaiser april 9, 2007. The first time i discovered large cardinals in jechs 2000 book set theory, i was amazed. Introduction to set theory notes introduction to set theory, second edition revised and expanded, by karel hrbacuk and thomas jech, dekker 1984 copies of the classnotes are on the internet in pdf format as given below. But even more, set theory is the milieu in which mathematics takes place today. Jan 18, 2017 set theory started as a purely mathematical subject, brought into life by george cantor. The ultrapower construction the study of ultrapowers originates in model theory, although it has found applications both in algebra and in analysis.
Classes are collections that sometimes are too large to be sets, while on the other hand,allsetsareclasses. Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. Pdf the evolution of large cardinal axioms in set theory. We sketch a brief development of large cardinals as they apply to determinacy results and to inner model. Erdos cardinal, inaccessible cardinal, indescribable cardinal, jack silver, large cardinal, list of large cardinal properties, mahlo cardinal, measurable cardinal, ramsey cardinal, shrewd cardinal, strongly compact cardinal, supercompact cardinal, weakly compact cardinal, zero sharp. Welch school of mathematics, university of bristol, england, isaac newton institute for mathematical sciences, cambridge, england p. The multiverse view in set theory, singapore 2011joel david hamkins, new york. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. Studies in logic and the foundations of mathematics, vol. Elements of set theory by sidneyfelder in the words of georgcantor 18451918,the creator of set theory and the modern theory of the in. Set theory now exhibits a categorytheoretic nature. Introduction this paper presents an extended set theory xst and proves its consistency relative to the classical zermelofraenkel set theory with the axiom of choice zfc and an axiom asserting the existence of arbitrarily large inaccessible cardinals also known as grothendiecks axiom of universes.
Meanwhile, woodin cardinals are not upwardly reflecting, and indeed, there are numerous large cardinals that do not necessarily exhibit your upward reflecting property, including measurable cardinals, strong cardinals, strongly compact cardinals, supercompact cardinals, and many others. Again, this is established using the dual techniques of inner and outer model theory only now large cardinals enter the mix. In fact, it is a basic idea in the construction of universes to make judicious choices of which subsets of a set to include in uand which to leave out. Here n is the set of all natural numbers and on is the class of all ordinal numbers. They can dive headfirst into paradoxes, proving, say, that a threedimensional sphere can be decomposed into pieces that, when put back together, form two spheres identical to the original. This means both strengthenings of ramseys theorem for sets of natural numbers like the carlson.
Basic set theory isbn 9780486420790 pdf epub azriel levy. These results provide a reinterpretation of certain large cardinals axioms as extensions of known facts about functions on n. Cantors theory of ordinal and cardinal numbers topics. On the one hand, there are the integers, the rational numbers, the real numbers, and the complex numbers. Kanamori and others published the evolution of large cardinal axioms in set theory find, read and cite all the research you need on. The proposition that such cardinals exist cannot be proved in the most. Despite its subtitle, it contains very nice presentations of a lot of general settheoretic background before getting to the large cardinals. The first part of this advancedlevel text covers pure set theory, and the second deals with applications and advanced topics point set topology, real spaces, boolean algebras, infinite combinatorics and large cardinals. In constructive contexts, the ordinals are not linearly ordered. Large cardinals in set theory from their beginnings springer monographs in mathematics at. The field has occupied friedman since his teenage years, when he first read bertrand russells introduction to mathematical. However, it is accurate to say that it is mainly exploited in set theory. Northholland publishing company, amsterdam and london, and american elsevier publishing company, inc. Drake, set theory, an introduction to large cardinals.
Contains introductory material as well as some advanced topics. A conceptual overview of the method is described at the end of the introduction. The main result is that the lowenheim number of the logic with the hartigquantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly mahlo, the first weakly compact, the first ramsey, the first measurable and the first supercompact cardinals. From this simple beginning, an increasingly complex and useful. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. An exploration of the axioms of set theory that best serve the practice of mathematics outside of set theo. Copies of the classnotes are on the internet in pdf format as given below. Kunen, and a number of basic facts were proved there. Cantors theory of ordinal and cardinal numbers topics i. Introduction the main question i want to discuss in this talk is \when can a wide aronszajn tree be specialized by a forcing which does not add reals. Introduction the results presented here establish unexpected formal relationships between the functions on n and the functions on on. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Set theorists can construct proofs using large cardinals, which deal with higher levels of infinity and are too large to be proven to exist within zfc.
Introduction there are numerous mathematical generalizations of the natural numbers. In set theory, zermelofraenkel set theory, named after mathematicians ernst zermelo and. An introduction to large cardinals, northholland, 1974. For the smaller large cardinals, one may alternatively appeal to the canonical inner models, which have the large cardinals and the gch. But it quickly became clear that the framework had much broader application. Learn about the people and activities that make uc berkeley one of the best places in the world for advanced research, graduate and undergraduate study in mathematics.
An introduction to large cardinals already mentioned by ali kare deserves your attention even if youre not particularly interested in large cardinals. A large cardinal is a cardinal number that is larger than can be proven to exist in the ambient set theory, usually zf or zfc. However, it is a really good text to get ones hands on the basic concepts in. Cantors theory of ordinal and cardinal numbers ordinals measure the length or shape of a set cardinals measure the size of a set in correlation sense. For the variablezonlyreferstoobjectsin uandnot subsetsofxthathappennottobeinu. Game theory is a mathematical framework for analyzing con ict and cooperation. Download for offline reading, highlight, bookmark or take notes while you read set theory and logic. Finite functions and the necessary use of large cardinals 805 in clear terms how one uses large cardinals in an essential and completely natural way in the integers. A large cardinal is just a very big set, after all, but i did not realize that the existence of such a set. This site contains pdfs built from the source latex files of the most recent version of the open logic text complete pdfs.
Large cardinals and the iterative conception of set. Set theory and logic is the result of a course of lectures for advanced undergraduates, developed at oberlin college for the purpose of introducing students to the conceptual foundations of mathematics. Smaller large cardinals usually imply the existence of embeddings1 for miniuniverses of set theory of size. V is a forcing extension of the constructible universe l, v. Introduction the subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of r. Discrete mathematicsset theory wikibooks, open books. Cardinal numbers page 6 of 14 14 examples of denumerable sets abound. A large cardinal axiom is an axiom stating that there exists a cardinal or perhaps many of them with some specified large cardinal property. My understanding is that large cardinals are ordered by consistency strength, but how does this correlate with their size cardinality.
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