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Ordinal notations based on a weakly Mahlo cardinal

1990
*
Archive for Mathematical Logic
*

Pohlers (see [11] ) extended the O-

doi:10.1007/bf01651328
fatcat:yima7tzr6fe4dg7bvctdcs2wfy
*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 -~. ...##
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On Randomness and Infinity
[chapter]

2002
*
Foundations of Information Technology in the Era of Network and Mobile Computing
*

We present alternative definitions

doi:10.1007/978-0-387-35608-2_23
dblp:conf/ifipTCS/Lafitte02
fatcat:iiio5gobirf3bpg3ie2t6vzpfq
*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. ...##
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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*...##
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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. ...##
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Ordinal Notation
[article]

2018
*
arXiv
*
pre-print

We introduce

arXiv:1610.04633v3
fatcat:hl5i7zjgonhg5nyvwteh7r6ibu
*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*). ...##
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The Landscape of Large Cardinals
[article]

2022
*
arXiv
*
pre-print

By

arXiv:2205.01787v1
fatcat:eiezuj25afeqfggxalpauodkeu
*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). ...##
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Categorical large cardinals and the tension between categoricity and set-theoretic reflection
[article]

2022
*
arXiv
*
pre-print

Thus we mount an analysis

arXiv:2009.07164v2
fatcat:crfpzugnlfep7m4ic7nzpz5vca
*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]

2016
*
arXiv
*
pre-print

Finally, we introduce and axiomatize

arXiv:math/0504375v2
fatcat:slhhmaa6erhjjgyc5xp4lnnfcq
*a*powerful extension to set theory. ... We discuss the problems*of*incompleteness and inexpressibility. ... Some variations*of*fixed point logic are described in [5] .*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]

2011
*
arXiv
*
pre-print

The work presents the first part

arXiv:1010.1956v4
fatcat:zeieuisddrcxzhv34eqi3rspya
*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*...##
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SET THEORY FROM CANTOR TO COHEN
[chapter]

2009
*
Philosophy of Mathematics
*

What follows is an account

doi:10.1016/b978-0-444-51555-1.50014-6
fatcat:vnj2vdlpofdp7pg37tyiemdkiq
*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, ...##
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Set Theory from Cantor to Cohen
[chapter]

2012
*
Handbook of the History of Logic
*

What follows is an account

doi:10.1016/b978-0-444-51621-3.50001-3
fatcat:ogdmpb6mlvgxbdvoh3jqq6umqe
*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, ...##
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Strong Statements of Analysis

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., [6] or [10] . 3: Strongly

*inaccessible*

*cardinals*and Statement

*A*Strongly

*inaccessible*

*cardinals*cannot be proved to exist in ZF C, assuming that ...

##
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Some remarks about nonstandard methods in analysis. I

2019
*
Vladikavkaz mathematical journal
*

The presentation

doi:10.23671/vnc.2019.21.44619
fatcat:tixkzp5kzvbxvfiu4jzfzxvhze
*of*the forcing method is preceded by*a*brief review*of*C.Godels result*on*the compatibility*of*the Axiom*of*Choice and the Continuum Hypothesis with Zermelo--Fraenkels axiomatics. ... This article contains*a*review*of*infinitesimal analysis and the original method*of*forcing. ...*A*. Lyubetskiy, who taught me the method*of*forcing, to my friends and colleagues*A*. G. Kusraev and S. S. ...##
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Page 2593 of Mathematical Reviews Vol. , Issue 96e
[page]

1996
*
Mathematical Reviews
*

For this, some notions

*of*recursively large*ordinals*, which are modelled upon notions*of*large*cardinals*such as recursively*inaccessible**ordinals*and recursively Mahlo*ordinals*, are suggested. ... “The large*cardinal*analogue for*a*suitable*notation*system resides below the first Ramsey*cardinal*, and is compatible with V = L.” ...##
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Sets in types, types in sets
[chapter]

1997
*
Lecture Notes in Computer Science
*

More precisely, we actually construct two families

doi:10.1007/bfb0014566
fatcat:r2wt56ltwbealgf4bwjbqo3a6q
*of*encodings, relating the number*of*universes in the type theory with the number*of**inaccessible**cardinals*in the set theory. ... The main result is that both*hierarchies**of*logical formalisms interleave w.r.t. expressive power and thus are essentially equivalent. ... The idea*of*investigating this topic was given to me by Christine Paulin-Mohring and Martin Ho man. ...
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