Abstract
We introduce axioms LFAN and CFAN , where the former follows from the law of excluded middle and the latter follows from the axiom of countable choice. Then we show that Brouwer's fan theorem is constructively equivalent to LFAN + CFAN . This decomposition of the fan theorem into a logical axiom and a function existence axiom contributes to the programme of constructive reverse mathematics.
Highlights
The objective of constructive reverse mathematics is to classify theorems by means of logical axioms and function existence axioms
A decomposition of the weak Konig lemma can be found in Ishihara [3]
We even give a proof of it, because there is a slight difference between CWKL and the choice axiom used in Ishihara [3]
Summary
Assume LLPO and let T be an infinite tree. CWKL Every Π01 -spread has an infinite path. There exists an infinite path α of T, which is an infinite path of A Berger and Ishihara [1] have shown that the statement ‘every infinite tree with at most one infinite path has an infinite path’ is constructively equivalent to FAN. See Schwichtenberg [5] for a more formal proof of this result This characterisation of FAN, together with Proposition 2, gives rise to the definition of the axioms LFAN and CFAN. Proof The fact that LLPO implies LFAN follows from Lemma 1. Let T be an infinite tree with at most one infinite path ∀α ∃n ∀β 0 ∗ αn ∈/ T ∨ 1 ∗ βn ∈/ T
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
