# Choice functions and well-orderings over the infinite binary tree

Open Mathematics (2010)

• Volume: 8, Issue: 4, page 662-682
• ISSN: 2391-5455

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## Abstract

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We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded order on the infinite binary tree by showing that every infinite binary tree with a well-founded order has an undecidable MSO-theory.

## How to cite

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Arnaud Carayol, et al. "Choice functions and well-orderings over the infinite binary tree." Open Mathematics 8.4 (2010): 662-682. <http://eudml.org/doc/269264>.

@article{ArnaudCarayol2010,
abstract = {We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded order on the infinite binary tree by showing that every infinite binary tree with a well-founded order has an undecidable MSO-theory.},
author = {Arnaud Carayol, Christof Löding, Damian Niwinski, Igor Walukiewicz},
journal = {Open Mathematics},
keywords = {Monadic second-order logic; Definability; Choice function; monadic second-order logic; definability; choice function},
language = {eng},
number = {4},
pages = {662-682},
title = {Choice functions and well-orderings over the infinite binary tree},
url = {http://eudml.org/doc/269264},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Arnaud Carayol
AU - Christof Löding
AU - Damian Niwinski
AU - Igor Walukiewicz
TI - Choice functions and well-orderings over the infinite binary tree
JO - Open Mathematics
PY - 2010
VL - 8
IS - 4
SP - 662
EP - 682
AB - We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded order on the infinite binary tree by showing that every infinite binary tree with a well-founded order has an undecidable MSO-theory.
LA - eng
KW - Monadic second-order logic; Definability; Choice function; monadic second-order logic; definability; choice function
UR - http://eudml.org/doc/269264
ER -

## References

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