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### Ordinal notations based on a weakly Mahlo cardinal

Michael Rathjen
1990 Archive for Mathematical Logic
Pohlers (see  ) extended the O-hierarchy of functions by defining a doubly indexed hierarchy 2~. O0el3, which is also based on 0-inaccessible cardinals.  ...  Indeed, it is possible to develop the notation system on the basis of recursively large ordinals by replacing each occurrence of a cardinal notion by its "recursive analogue".  ...  definition of the set T(M) and the relations < and = NV on T(M), which can be converted into a simultaneous primitive recursive definition of ~ and -~.  ...

### On Randomness and Infinity [chapter]

Grégory Lafitte
2002 Foundations of Information Technology in the Era of Network and Mobile Computing
We present alternative definitions based on infinite time machines and set theory and explain how and why randomness is strongly linked to strong axioms of infinity.  ...  We discuss the nature of randomness and different ways of obtaining satisfactory definitions of randomness after reviewing previous attempts at producing a non-algorithmical definition.  ...  ., 1983] of Yuri Gurevich, Saharan Shelah and himself. We are also greatly indebted to Jacques Mazoyer for his advice and his never-failing enthusiasm.  ...

### Page 1546 of Mathematical Reviews Vol. 45, Issue 6 [page]

1973 Mathematical Reviews
By a notation system based on 7 is meant one of the form (7) =<¢,, D,> where D,, is a set of ordinals and ¢, is one-one map of D, into T,.  ...  Ordinal notation systems that have been used in proof theory have either been suggested by proof-theoretic reduction procedures or induced in a canonical way on terms corresponding to a system of ordinal  ...

### Page 3380 of Mathematical Reviews Vol. , Issue 88g [page]

1988 Mathematical Reviews
This system is based, aside from certain elementary basic functions (the Veblen functions ¢,), on the order functions J, of strongly inaccessible cardinals and on generalizations of the collapsing functions  ...  As for decreasing F-sequences, termination can be shown and their length can be calculated in terms of a hierarchy of ordinal functions defined by recursion on dilators.  ...

### Ordinal Notation [article]

Dmytro Taranovsky
2018 arXiv   pre-print
We introduce a framework for ordinal notation systems, present a family of strong yet simple systems, and give many examples of ordinals in these systems.  ...  While much of the material is conjectural, we include systems with conjectured strength beyond second order arithmetic (and plausibly beyond ZFC), and prove well-foundedness for some weakened versions.  ...  choice of the notation for the least inaccessible cardinal).  ...

### The Landscape of Large Cardinals [article]

Rohan Srivastava
2022 arXiv   pre-print
By a large cardinal, we mean any cardinal κ whose existence is strong enough of an assumption to prove the consistency of ZFC.  ...  The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory.  ...  Finally, Lucas Strammello helped me through a semester each of set theory and theory of computation, explaining concepts whenever I found myself lost (which was quite frequently).  ...

### Categorical large cardinals and the tension between categoricity and set-theoretic reflection [article]

Joel David Hamkins, Hans Robin Solberg
2022 arXiv   pre-print
Thus we mount an analysis of the categorical large cardinals.  ...  addition to ZFC_2 either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory.  ...  For any cardinal κ, let us use the boldface successor notation κ to denote the next inaccessible cardinal above κ. Theorem 9.  ...

### Extending the Language of Set Theory [article]

Dmytro Taranovsky
2016 arXiv   pre-print
Finally, we introduce and axiomatize a powerful extension to set theory.  ...  We discuss the problems of incompleteness and inexpressibility.  ...  Some variations of fixed point logic are described in  . A Hierarchy of Large Cardinals The replacement schema can be viewed as a statement that the class of ordinals, Ord, is inaccessible.  ...

### Inaccessibility and Subinaccessibility. In two parts. Part I [article]

A. Kiselev
2011 arXiv   pre-print
The work presents the first part of second edition of the previous edition of 2000 under the same title containing the proof (in ZF) of the nonexistence of inaccessible cardinals, now enriched and improved  ...  This part contains the apparatus of subinaccessible cardinals and its basic tools -- theories of reduced formula spectra and matrices, disseminators and others -- which are used in this proof and is set  ...  Weakly inaccessible cardinals become strongly inaccessible in Gödel constructive class L; let us remind that it is the class of values of Gödel constructive function F defined on the class of all ordinals  ...

### SET THEORY FROM CANTOR TO COHEN [chapter]

Akihiro Kanamori
2009 Philosophy of Mathematics
What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been  ...  With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength.  ...  ACKNOWLEDGEMENTS This is a revision with significant changes of the author's The Mathematical Development of Set Theory from Cantor to Cohen, The Bulletin of Symbolic Logic, volume 2, 1996, pages 1-71,  ...

### Set Theory from Cantor to Cohen [chapter]

Akihiro Kanamori
2012 Handbook of the History of Logic
What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been  ...  With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength.  ...  ACKNOWLEDGEMENTS This is a revision with significant changes of the author's The Mathematical Development of Set Theory from Cantor to Cohen, The Bulletin of Symbolic Logic, volume 2, 1996, pages 1-71,  ...

### Strong Statements of Analysis

A. R. D. Mathias
2000 Bulletin of the London Mathematical Society
of which the only known proofs use strong set-theoretical concepts.  ...  Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses, and of apparently classical statements provable in ZFC  ...  For further information on large cardinals see, e.g.,  or  . 3: Strongly inaccessible cardinals and Statement A Strongly inaccessible cardinals cannot be proved to exist in ZF C, assuming that  ...

E.I. Gordon