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14 - Limitations of Martin’s Axiom and Countable Supports

from Part II - What is New in Set Theory

Published online by Cambridge University Press:  28 September 2020

Mirna Džamonja
Mirna Džamonja
Affiliation:
Institute for the History and Philosophy of Science and Technology, Paris
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Summary

A very well known early instance that shows a limitation of Martin’s Axiom is Richard Laver’s proof of the consistency of the Borel Conjecture [65]. Laver proved that this problem, which seems along the lines of those that have been solved using Martin’s Axiom, on cardinal invariants of the continuum, cannot be answered using Martin’s Axiom. A subset X of R is said to have strong measure zero if for every sequence (εn)n of positive real numbers, there is a sequence of intervals (In)n such that lg(In) < εn and X ⊆ ∪n<ω In. It is clear that all countable subsets of R have strong measure 0 and the Borel Conjecture from [8] states that the only strong measure zero sets are the countable ones.

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Fast Track to Forcing , pp. 85 - 89
Publisher: Cambridge University Press
Print publication year: 2020

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