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On the consistency and independence of some set-theoretical axioms
1978
Notre Dame Journal of Formal Logic
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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
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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
2010
THEORIA : an International Journal for Theory, History and Fundations of Science
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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?
1947
The American mathematical monthly
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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
1988
Journal of Symbolic Logic (JSL)
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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
2011
Lingua Posnaniensis
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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]
2016
arXiv
pre-print
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2012 pre-print arXiv:1203.0494v1 fatcat:f7ulf7hxondqxnu7wdvygmkdka
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
1988
Journal of Symbolic Logic (JSL)
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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
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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
2012
Annals of Pure and Applied Logic
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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
1999
Philosophia Mathematica
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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
2015
Philosophy Compass
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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
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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
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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
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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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