[FOM] Finite binary trees and ordinals

[Peter Smith]

The following seem 'natural' conditions to put on a well-ordering < 
of finite binary trees: 

1. If A is a proper part of B, then A < B. 
2  If T(A) is a tree containing A as a part, and T(B) is a properly 
formed tree that results from substituting B for A, then if A < B then 
T(A) < T(B). 

These are simply the thoughts that a part should precede the bigger 
whole which contains it, and that two wholes which differ in just one 
part should be ordered to reflect the ordering of those differing 
parts. 

Now, it certainly is the case that there is a well-ordering that 
respects 1 and 2, a well-ordering of the order-type of the ordinals 
less than epsilon_0 (see e.g. http://www.ml.kva.se/preprints/meta/ 
JervellMon_Jun_11_10_58_31.rdf.html -- the idea goes back to 
Ackermann). 

But a natural question arises: What are the minimum equally 
'natural' conditions that, if added to 1 and 2, ENSURE that the 
well-ordering is of the order-type of the ordinals less than 
epsilon_0? 

As I say, that seems a natural question, but I can't find any
treatment of it. Any thoughts and/or pointers gratefully 
received! 

-- 
Dr Peter Smith: Faculty of Philosophy, University of Cambridge

Gödel's Theorems: http://www.godelbook.net 
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