Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  prove the existence of a limitwise monotonic function g : N → N \{ 0} such that, for any � 0 function f : N → N \{ 0} ,R anf �= Ran g. Relativising this result  deduce the existence of an η-like computable linear ordering A such that, for any � 0 function F : Q → N \{ 0} ,a ndη-like B of order type { F( q)| q ∈ Q}, BA.  prove directly that, for any computable A which is either (i) strongly η-like or (ii) η-like with no strongly η-like interval, there exists 0 � -limitwise monotonic G : Q → N \{ 0} such that A has order type {G(q) | q ∈ Q}. In so doing  provide an alternative proof to the fact that, for every η-like computable linear ordering A with no strongly η-like interval, there exists computable B ∼ A with � 0 block relation.  also use our results to prove the existence of an η-like computable linear ordering which is � 0 categorical but not � 0 categorical.
['Charles M. Harris']
I am writing a research paper on a idea On limitwise monotonicity and maximal block functions
[{"from": "human", "value": "I am writing a research paper on a idea On limitwise monotonicity and maximal block functions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  prove the existence of a limitwise monotonic function g : N \u2192 N \\{ 0} such that, for any \ufffd 0 function f : N \u2192 N \\{ 0} ,R anf \ufffd= Ran g. Relativising this result  deduce the existence of an \u03b7-like computable linear ordering A such that, for any \ufffd 0 function F : Q \u2192 N \\{ 0} ,a nd\u03b7-like B of order type { F( q)| q \u2208 Q}, BA.  prove directly that, for any computable A which is either (i) strongly \u03b7-like or (ii) \u03b7-like with no strongly \u03b7-like interval, there exists 0 \ufffd -limitwise monotonic G : Q \u2192 N \\{ 0} such that A has order type {G(q) | q \u2208 Q}. In so doing  provide an alternative proof to the fact that, for every \u03b7-like computable linear ordering A with no strongly \u03b7-like interval, there exists computable B \u223c A with \ufffd 0 block relation.  also use our results to prove the existence of an \u03b7-like computable linear ordering which is \ufffd 0 categorical but not \ufffd 0 categorical."}]