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On the consistency and independence of some set-theoretical axioms

Alexander Abian, Samuel LaMacchia
1978 Notre Dame Journal of Formal Logic  
In this paper by means of simple models it is shown that the five set-theoretical axioms of Extensionality, Replacement, Power-Set, Sum-Set, and Choice are consistent and that each of the axioms of Extensionality  ...  , Replacement, and Power-Set is independent from the remaining four axioms.  ...  is a set. (3) Axiom of Power-Set The set of the subsets of any set exists. (4) Axiom of Sum-Set The set of the elements of the elements of any set exists. (5) Axiom of Choice There exists a choice function  ... 
doi:10.1305/ndjfl/1093888220 fatcat:ulcqzp2byfbshgozggfl6zdbgu

Page 521 of The American Mathematical Monthly Vol. 54, Issue 9 [page]

1947 The American Mathematical Monthly  
There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole discipline, and furnishing such powerful methods for solving given problems (and even solving  ...  , inductively by studying its “success,” that is, its fruitfulness in consequences and in particular in “veri- fiable” consequences, #.e., consequences demonstrable without the new axiom, whose proofs  ... 

Remarks on Second-Order Consequence

Ignacio Jané
2010 THEORIA : an International Journal for Theory, History and Fundations of Science  
We discuss the dependence of canonical second-order consequence on set theory and raise doubts on the assumption that canonical consequence is a definite relation.  ...  In the canonical, or standard, version, a model is just an ordinary structure and the (monadic) second-order variables are meant to range over all subsets of its domain.  ...  A large amount of set-theoretical propositions which are known to be independent of the usual set theory ZFC (Zermelo-Fraenkel with the axiom of choice) are precisely about the contents of the power set  ... 
doi:10.1387/theoria.421 doaj:0fd8f87adda94489bca490f8727ddb5d fatcat:efbtngjivjhobkvioh7llx76ya

What is Cantor's Continuum Problem?

Kurt Gödel
1947 The American mathematical monthly  
power of the set of integers or of the whole continuum.  ...  as to their consequences for proposi- tions referring to limited domains of sets (such as the continuum hypothesis) are contained in the axioms depending on the concept of set. 18 See E.  ... 
doi:10.1080/00029890.1947.11991877 fatcat:h4uyur3sf5a6pb4od635r7lp3a

Believing the Axioms. I

Penelope Maddy
1988 Journal of Symbolic Logic (JSL)  
Thus some mathematicians will stand by the truth of any consequence of ZFC, but dismiss additional axioms and their consequences as metaphysical rot.  ...  the power-set principle gains some confirmatory support.  ... 
doi:10.2307/2274520 fatcat:tuzzxqkutne3fhreox7dhsx6dm

Remarks on Intended Models of Mathematical Theories

Jerzy Pogonowski
2011 Lingua Posnaniensis  
Some emphasis is put on the role of extremal axioms in the characterization of such models. the notion of the intended model seems to be of some interest for linguists, too. this is one of the reasons  ...  pp. 83-100. this note is a summary of the talk given on june 10, 2010 at the university of opole during a meeting of The Group of Logic, Language and Information. we limit ourselves to some major points  ...  Furthermore, for set theorist, any advantage that V = L has in terms of power can be obtained with more powerful axioms of the same rough type that accommodate measurable cardinals and the like - e.g.,  ... 
doi:10.2478/v10122-011-0015-4 fatcat:hizsisix2nd6xchvjw7omp7a24

Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity [article]

Minseong Kim
2016 arXiv   pre-print
The paper then exposes some consequences that are in the scope of the computational complexity theory.  ...  This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC).  ...  Then, the cardinality of the power set of the set X is 2 ℵ k . Representing the power set of the set X as the set P(X), X P(X) as |X| < |P(X)|.  ... 
arXiv:1203.0494v2 fatcat:no27u5zggvb3fhmuwbocxesfsa

Believing the axioms. I

Penelope Maddy
1988 Journal of Symbolic Logic (JSL)  
Conversely, the axiomatization of set theory has led to the consideration of axiom candidates that no one finds obvious, not even their staunchest supporters.  ...  Again, the more sophisticated might prefer to say that the axioms are "laws of logic" or "implicit definitions" or "conceptual truths" or some such thing.Unfortunately, heartwarming answers along these  ...  Thus some mathematicians will stand by the truth of any consequence of ZFC, but dismiss additional axioms and their consequences as metaphysical rot.  ... 
doi:10.1017/s0022481200028425 fatcat:k6lbbfkk4vekvbhd7qamczsxcq

Page 415 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 62, Issue 4 [page]

1956 American Mathematical Society. Bulletin of the American Mathematical Society  
The generalized continuum hypothesis, some consequences of it, and some propositions equiva- lent to it are also discussed. Chapter six deals with the well-ordering and the cardinal number 415  ...  The fourth chapter is concerned with the arithmetic of powers without the aid of the axiom of choice, and then in the fifth chapter this axiom is used to develop the arithmetic of cardinal numbers.  ... 

Foundational implications of the Inner Model Hypothesis

Tatiana Arrigoni, Sy-David Friedman
2012 Annals of Pure and Applied Logic  
The purpose of this paper is to illustrate the hypothesis, and discuss it with respect to the current debate on the consequences of independence results in set theory.  ...  The Inner Model Hypothesis (IMH) is a new axiomatic approach in set theory formulated by Sy-D. Friedman.  ...  Acknowledgement The first author is supported by Provincia Autonoma di Trento, Italy (Bando Post-Doc 2007).  ... 
doi:10.1016/j.apal.2012.01.009 fatcat:yk2avi6w3rbtje5a2lco3ohl3y

Do Not Claim Too Much: Second-order Logic and First-order Logic

STEWART SHAPIRO
1999 Philosophia Mathematica  
Let AR be the conjunction of the standard Peano axioms, including the  ...  In each interpretation, the property or set variables range over the entire powerset of the domain d, the binary relation variables range over the powerset of d 2 , etc.  ...  On the set-theoretic approach, the presupposition is registered with the power-set axiom, stating that every set has a (unique) powerset.  ... 
doi:10.1093/philmat/7.1.42 fatcat:m2spad6v3rb4hnt6jzpqo4jscy

Second-Order Logic and Set Theory

Jouko Väänänen
2015 Philosophy Compass  
We argue that we should think of first order set theory as a very high order logic.  ...  We take it upon ourselves in this paper to compare the two approaches, second order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses.  ...  Often in set theory one applies the Axiom of Choice to an auxiliary larger set obtained by means of the Power Set Axiom, or even a combination of the Power Set Axiom and the Replacement Axiom.  ... 
doi:10.1111/phc3.12229 fatcat:2vq3iweyxzckbfbhtooe73hs5a

Page 392 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 67, Issue 4 [page]

1961 American Mathematical Society. Bulletin of the American Mathematical Society  
The set-theoretic axioms, added to the elementary logic of truth functions and quantification, are three: the usual extensionality axiom, the axiom that {x, y} exists for all x and y, and the axiom that  ...  Thus the possibility. is preserved, even if some infinite classes are eventually added for purposes beyond number theory, of maintaining something like the old distinction between sets and other classes  ... 

Page 4646 of Mathematical Reviews Vol. , Issue 80M [page]

1980 Mathematical Reviews  
of sets of cardinality <«, for some fixed x.  ...  As the author states, “We do not need the power set axiom, replacement, or extensionality, and so the result is applicable to more predicative or intensional notions of set than the usual one.”  ... 

Page 609 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
The authors prove, by means of Easton-type forcing, that CU is consistent with ZF together with some consequences of the axiom of choice.  ...  A table indicates the different ways of defining each of the large cardinal properties considered, its relationship with the axiom of constructibility and some of the consequences of the existence of cardinals  ... 
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