Is Luzin Hypothesis Consistent with Set Theory?

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In summary, the conversation discussed the weak continuum hypothesis and the Luzin Hypothesis, and the question of where to find Bukovsky's paper or a proof of the consistency of the Luzin Hypothesis. A brief search led to references on Luzin spaces, Martin's Axiom, and the Rasiowa-Sikorski lemma, with the suggestion to start with those references.
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xouper
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Regarding the weak continuum hypothesis (B.F. Jones):

[tex]{\displaystyle 2^ {\aleph_{0}} < 2^ {\aleph_{1}}}[/tex]

and the Luzin Hypothesis:

[tex]{\displaystyle 2^ {\aleph_{0}} = 2^ {\aleph_{1}}}[/tex]

Where can I find Bukovsky's paper that shows the Luzin Hypothesis is consistent with set theory? Or perhaps someone can repeat the proof here (or summarize it)?
 
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FAQ: Is Luzin Hypothesis Consistent with Set Theory?

1. What is the Luzin Hypothesis?

The Luzin Hypothesis, also known as the Luzin's S Hypothesis, is a conjecture in set theory proposed by Russian mathematician Nikolai Luzin in 1913. It states that every set of real numbers can be decomposed into a countable union of sets of measure zero.

2. What is the significance of the Luzin Hypothesis?

The Luzin Hypothesis has significant implications in the study of measure theory and real analysis. It essentially states that every set of real numbers is "small" in terms of measure, and thus, the majority of real numbers are "non-measurable". This has led to further investigations and developments in the field of descriptive set theory.

3. What is the relationship between the Luzin Hypothesis and the Continuum Hypothesis?

The Luzin Hypothesis is closely related to the Continuum Hypothesis, which is one of the most famous unsolved problems in mathematics. The Continuum Hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers. It has been shown that if the Continuum Hypothesis is assumed to be true, then the Luzin Hypothesis is also true.

4. What is the weak Continuum Hypothesis?

The weak Continuum Hypothesis, also known as the weak CH, is a weakened version of the Continuum Hypothesis. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers, except for sets of measure zero. In other words, it allows for the existence of some non-measurable sets.

5. What is the current status of the Luzin Hypothesis and weak CH?

Both the Luzin Hypothesis and the weak CH are still open problems in mathematics. While there have been significant developments and progress in the field of set theory, these conjectures remain unsolved. It is a topic of ongoing research and debate among mathematicians, and their resolutions could have a significant impact on the understanding of the structure of the real numbers.

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