The most difficult equation in mathematics

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SUMMARY

The discussion centers around the complexity of the equation $$2^{\aleph_0}=\aleph_k$$ and its implications in set theory, particularly regarding the continuum hypothesis. Participants highlight that the problem of determining k for a given m is challenging and has been deemed unsolvable using standard axioms of set theory, as established by Cohen's theorem. The conversation also touches on the subjective nature of measuring mathematical difficulty and references other complex problems such as those listed in Hilbert's problems and the P vs NP problem.

PREREQUISITES
  • Understanding of set theory, particularly aleph numbers and the continuum hypothesis.
  • Familiarity with Cohen's theorem and its implications on the solvability of mathematical problems.
  • Basic knowledge of mathematical logic and its applications in theoretical mathematics.
  • Awareness of significant mathematical problems, including Hilbert's problems and the P vs NP problem.
NEXT STEPS
  • Research the implications of Cohen's theorem on the continuum hypothesis and its undecidability.
  • Explore Hilbert's problems and their significance in the field of mathematics.
  • Study the P vs NP problem and its relevance to computational complexity theory.
  • Investigate the role of mathematical logic in understanding foundational concepts in set theory.
USEFUL FOR

Mathematicians, theoretical physicists, and students of advanced mathematics interested in the complexities of set theory and the nature of mathematical problems.

  • #61
Here's one that I think is hard:

math = ?
 
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  • #62
math = math
 
  • #63
Not necessarily equation, but counting problems are always terribly hard. Here's one that's really difficult: How many DISTINCT valid (within the rules of the game) endings of a standard 9x9 Sudoku game are there?
 

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