Abstract
This chapter explores the structure \(V(\omega +\omega )\) and provides a construction of the real numbers.
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Notes
- 1.
See Zermelo [2].
- 2.
The entries of the super-duper-numbers form a model of Zermelo’s set theory Z. For a helpful discussion see Uzquiano [1] (http://www.jstor.org/stable/421182).
- 3.
Here is another way of expressing the same idea. If \(A\) is a part of \(\omega \) and \(j\) is an entry of \(\omega \), let \(A\cap j\) be the list whose entries are, exactly, the entries of \(A\) smaller than (i.e., listed by) \(j\). Now suppose \(A\) and \(B\) are non-cofinite parts of \(\omega \). Then \(A<B\) if and only if, for some number \(j\), \(B{\setminus }A\) lists \(j\) and \((A\cap j)=(B\cap j)\).
- 4.
Yes, this is possible. Note that \(\emptyset <\{2\}<\{1,2\}\).
References
Uzquiano, G. (1999). Models of second-order Zermelo set theory. Bulletin of Symbolic Logic, 5, 289–302.
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche. Fundamenta Mathematicae, 16, 29–47.
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Pollard, S. (2014). Zermelian Lists. In: A Mathematical Prelude to the Philosophy of Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-05816-0_4
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DOI: https://doi.org/10.1007/978-3-319-05816-0_4
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