• Gaisi Takeuti
  • Wilson M. Zaring
Part of the Graduate Texts in Mathematics book series (GTM, volume 1)


Definition 3.1 is incomplete in that the variable x does not occur in a = b, consequently we have not made clear which of the infinitely many formulas, equivalent under alphabetic change of variable, we intend a = b to be an abbreviation for. This problem is easily resolved by specifying that x is the first variable an our list
$$ {x_0},\,{x_1}, \ldots $$
that is distinct from a and from b. Having thereby shown that we can specify a particular formula we will not bother to do so either here or in similar definitions to follow.


Basic Property Induction Hypothesis Publishing Company Intuitive Idea Similar Definition 
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  1. 1.
    Essays on the Foundations of Mathematics. Amsterdam: North-Holland Publishing Company, 1962, pp. 115–131.Google Scholar
  2. 2.
    See Quine, Van Orman: Set Theory and its Logic. Cambridge: Mass. Harvard University Press 1969, 30f.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1971

Authors and Affiliations

  • Gaisi Takeuti
    • 1
  • Wilson M. Zaring
    • 1
  1. 1.University of IllinoisUSA

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