scholarly journals GENERIC LARGE CARDINALS AS AXIOMS

2019 ◽  
Vol 13 (2) ◽  
pp. 375-387
Author(s):  
MONROE ESKEW
Keyword(s):  
Continuum Hypothesis ◽  
Large Cardinals ◽  
Large Cardinal ◽  
The Continuum

AbstractWe argue against Foreman’s proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.

The Roles of Set Theories in Mathematics

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Set Theory ◽  
Gauge Theories ◽  
Elementary Theory ◽  
Continuum Hypothesis ◽  
Large Cardinals ◽  
The Continuum

What mathematicians know and use about sets varies across branches of mathematics but rarely includes such fundamental aspects of Zermelo–Fraenkel (ZF) set theory as the iterative hierarchy. All mathematicians know and use the axioms of the Elementary Theory of the Category of Sets (ETCS), though few know ETCS or any set theory by name. The chapter depicts the iterative hierarchy of ZF and constructibility as gauge theories. Since gauge theories are prominently used in physics, so these are used in work on the continuum hypothesis, large cardinals, and provability in arithmetic. But mathematicians outside logic avoid these gauges and work with structures only up to isomorphism, as does ETCS.

TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

2010 ◽  
Vol 3 (1) ◽  
pp. 71-92 ◽  
Author(s):  
ZACH WEBER
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Paraconsistent Logic ◽  
Peano Arithmetic ◽  
Large Cardinals ◽  
Cardinal Numbers ◽  
Naive Set Theory ◽  
Cantor’S Theorem ◽  
The Axiom Of Choice ◽  
The Continuum

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

scholarly journals BOUNDING BY CANONICAL FUNCTIONS, WITH CH

2003 ◽  
Vol 03 (02) ◽  
pp. 193-215 ◽  
Author(s):  
PAUL LARSON ◽  
SAHARON SHELAH
Keyword(s):  
Continuum Hypothesis ◽  
Large Cardinals ◽  
Canonical Function ◽  
The Continuum

We show that the members of a certain class of semi-proper iterations do not add countable sets of ordinals. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from ω1 to ω1 is bounded on a club by a canonical function for an ordinal less than ω2.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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