Models of ZF-Set Theory

Felgner, Ulrich

doi:10.1007/bfb0061160

Cited by 62 publications

(29 citation statements)

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“…This lemma is similar to the continuity lemma given by Feigner (1971) page 133. We give here only an indication of the form Feigner's proof takes in our situation.…”

Section: Large Dedekind-finite Setssupporting

confidence: 60%

“…This however makes no essential difference to the proof; indeed Feigner's continuity lemma for the Halpern-Levy model (Feigner (1971) page 133) is of this more general form.…”

Section: (I) Jv Is a Transitive Model Ofzf(k) (Ii) M N And M[g] Hmentioning

confidence: 98%

“…The model we shall construct is a generalization of the model used by Halpern and Levy to show that the Boolean prime ideal theorem does not imply the axiom of choice. A treatment of the Halpern-Levy model is given by Feigner (1971) …”

Section: Large Dedekind-finite Setsmentioning

confidence: 99%

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Independence results concerning Dedekindfinite sets

Monro

1975

J. Aust. Math. Soc.

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A Dedekind-finite set is one not equinumerous with any of its proper subsets; it is well known that the axiom of choice implies that all such sets are finite. In this paper we show that in the absence of the axiom of choice it is possible to construct Dedekind-finite sets which are large, in the sense that they can be mapped onto large ordinals; we extend the result to proper classes. It is also shown that the axiom of choice for countable sets is not implied by the assumption that all Dedekind-finite sets are finite.

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“…This lemma is similar to the continuity lemma given by Feigner (1971) page 133. We give here only an indication of the form Feigner's proof takes in our situation.…”

Section: Large Dedekind-finite Setssupporting

confidence: 60%

“…This however makes no essential difference to the proof; indeed Feigner's continuity lemma for the Halpern-Levy model (Feigner (1971) page 133) is of this more general form.…”

Section: (I) Jv Is a Transitive Model Ofzf(k) (Ii) M N And M[g] Hmentioning

confidence: 98%

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Independence results concerning Dedekindfinite sets

Monro

1975

J. Aust. Math. Soc.

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“…But ¬ lim n→∞ Φ G (ξn) = IR Φ G (x) and thus G is not sequentially continuous at the point represented by x. ¾ Remark 4.2 The use of QF-AC 0,1 in the proof of '→' in the proposition above is unavoidable already for X = Y = IR since in this case the implication is known to be unprovable even in Zermelo-Fraenkel set theory ZF, see [16], [15] and [12].…”

Section: φ Is Continuous If It Is Continuous At Every Xmentioning

confidence: 99%

Foundational and Mathematical Uses of Higher Types

Kohlenbach¹

1999

BRICS

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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are proof-theoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on non-collapsing hierarchies (Φ n -WKL + , Ψ n -WKL + ) of principles which generalize (and for n = 0 coincide with) the so-called 'weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X → Y between Polish spaces X, Y , the more flexible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for 1 to be equivalent to the existence of a continuous modulus of pointwise continuity. For the direct representation of F the existence of such a modulus is independent even of full arithmetic in all finite types E-PA ω plus quantifierfree choice, as we show using a priority construction due to L. Harrington. The usual WKL-based proofs of properties of F given in reverse mathematics make use of the enrichment provided by codes of F , and WKL does not seem to be sufficient to obtain similar results for the direct representation of F in our setting. However, it turns out that Ψ 1 -WKL + is sufficient. Our conservation results for (Φ n -WKL + , Ψ n -WKL + ) are proved via a new elimination result for a strong non-standard principle of uniform Σ 0 1 -boundedness which we introduced in 1996 and which implies the WKL-extensions studied in this paper.

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“…2 The proof will be presented in §111. Limited formulas and terms are defined as in [1]. A condition is a finite set of statements of the form n e S al or -i n e S aj .…”

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confidence: 99%