Abstract
In our study of transcendental numbers, we reviewed how Cantor defined two infinite cardinal numbers, ℵ0 and ℵp for two kinds of infinite sets, sets that are countable and sets that are uncountable. If N is the infinite set of all natural numbers, then we define N̄ as the cardinal number of N. In this way we avoid confusing a set with its cardinal number. In a similar fashion we let R stand for the set of all real numbers. Then R̄ is its cardinal number and is frequently shown as simply c. We know that N̄ = ℵ 0. Cantor believed that c = ℵ1 but could not prove it. We also know that the question has been asked whether there are any infinite cardinals between ℵ0 and c, and the answer is undecidable.
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End Notes
E. Kamke, Theory of Sets (New York: Dover Publications, 1950), p. 47.
Rucker, pp. 48–50.
Dauben, p. 232.
Rucker, p. 276.
Ibid., pp. 281–285.
Muir, p. 237.
Dauben, p. 285.
Ibid., p. 243.
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© 1994 Calvin C. Clawson
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Clawson, C.C. (1994). Really Big. In: The Mathematical Traveler. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-6014-6_13
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DOI: https://doi.org/10.1007/978-1-4899-6014-6_13
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