• Raymond M. Smullyan
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 43)


Trees shall play an important role throughout this work, so we shall commence with some pertinent definitions:
By an unordered tree, 𝒯, we shall mean a collection of the following items:
  1. (1)

    A set S of elements called points.

  2. (2)

    A function, \(\ell\), which assigns to each point x a positive integer \(\ell\)(x) called the level of x.

  3. (3)

    A relation xRy defined in S, which we read “x is a predecessor of y” or “y is successor of x”. This relation must obey the following conditions:

  4. C1:

    There is a unique point a1 of level 1. This point we call the origin of the tree.

  5. C2:

    Every point other than the origin has a unique predecessor.

  6. C3:

    For any points x, y, if y is a successor of x, then \(\ell \left( y \right) = \ell \left( x \right) + 1\)



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Copyright information

© Springer-Verlag Berlin · Heidelberg 1968

Authors and Affiliations

  • Raymond M. Smullyan
    • 1
  1. 1.Lehman CollegeCity University of New YorkUSA

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