MoL201113: Baartse, Willem M. (2011) Finding the phase transition for Friedman's long finite sequences. [Report]

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Abstract
The phase transition that we will focus on is about fast growing sequences, which were discovered by Friedman. PA can prove that the length of these sequences remains finite, but IΣ_2 cannot. IΣ_2 is the subsystem of PA where the induction scheme is limited to induction of Σ_2 formulas. This system can prove the totality of a function if and only if it is multiple recursive. The multiple recursive functions are the functions that can be defined using the elementary functions and nested recursion schemes with some finite number of variables. So it can prove the totality of the primitive recursive functions. It can also prove the totality of the Ackermann function which is double recursive. The multiple recursive functions are the same as the < ω^{ω^ω} recursive functions. Hence, IΣ_2 proves the totality of the Hardy functions H_α for α < ω^{ω^ω} but does not prove the totality of H_{ω^{ω^ω}}. In section 4, which follows section 5 from [4], we will first show that the growth of the length of these sequences is ω^{ω^ω} recursive (this is a direct consequence of a theorem from [14]), which implies that IΣ_3 (and thus PA) can prove that the length of these sequences remains finite. Then, using lemma 3.2.5 it will be shown that this growth is so fast that it eventually dominates every < ω^{ω^ω} recursive function. From theorem 3.3.3 it then follows that IΣ_2 cannot prove that these sequences remain of finite length. Section 5 is based on an article with Weiermann. In section 5.1 we show that if the parameter function grows very slowly, it is easy to find an upper bound on the length of the sequences. In section 5.2 we make a con struction which shows that for a slowly growing function f the length of the sequences doesn’t grow significantly slower than for the identity function. To nicely characterize the phase transition we use that IΣ_2 proves the totality of H_α iff α < ω^{ω^ω}. This known fact is proved in section 3.3 using a theorem from [1] and the notion of ordinal recursion. In the last section we study a similar phase transition which seems easier to characterize. It is again about sequences but now an extra condition is introduced which makes the sequences grow so fast that PA is no longer able to prove that they remain finite. The phase transition here is very similar to the one in section 5.
Item Type:  Report 

Report Nr:  MoL201113 
Series Name:  Master of Logic Thesis (MoL) Series 
Year:  2011 
Date Deposited:  12 Oct 2016 14:38 
Last Modified:  12 Oct 2016 14:38 
URI:  https://eprints.illc.uva.nl/id/eprint/858 
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