Recent content by Deedlit

Hi SSequence, I haven't taken the time yet to read over the whole thread, but let me just address the initial question: what about Q? Is there some condition that eliminates Q from the list of available linear orders? Edit: Looking further, you seem to have answered the initial question...

Yes, you are right; I am mistaken.

You could say: Given a computable well-ordering ##\preceq## on ##\mathbb{N}## of order type ##\alpha##, let ##address## be the unique order-preserving bijection from ##\alpha## to ##(\mathbb{N},\preceq)##, and let ##position## be its inverse function.

Okay, I believe the answer is no, there is no negative example to your first question. At least one of the value functions will have arbitrarily large values, and we can use that value function to construct the successor function. Just go through the values of the unbounded value function one...

I believe I understand the first question, but I can see perhaps why the question has not received that much attention. First, very long questions and answers tend to be skipped over by a lot of people; I tend to write long answers on MSE, and I typically don't get that many upvotes; my short...

That's a great idea, but I don't think it quite works as is. \lim_{m\rightarrow\infty}bb_m(n) is the largest value of k such that k \le n and there is a program of size n or less that halts in k steps. So for example \lim_{m\rightarrow\infty}bb_m(n) \le n. We can use the definition bb_m(n)...

Hmm, that makes sense. This is the argument I had in mind: 1. Halt is recursively enumerable (trivial) 2. Halt and BB are Turing equivalent (not hard to argue) 3. a set Turing equivalent to a recursively enumerable set is recursively enumerable I thought that that last statement was true...

BB is recursively enumerable, I think. (Halt is the classic example of a recursively enumerable set.)

The Gamma function is not the only analytic function with those properties; you need to add the condition that the positive real part of the function is log convex, i.e. that log f(x) is a convex function. Note that two analytic functions that satisfy those two properties will not have to agree...

It seems like you've hit the nail on the head: a graph can be represented as a union of k bipartite graphs if and only if it is 2^k-colorable. For one direction, say a graph is a union of k bipartite graphs. For each bipartite graph we can 2-color it. Then for the original graph, for each...

It's not a question of philosophy, since it is a concrete problem with a definite answer. Your argument seems to assume that whenever there are two possibilities the chances are always 50/50, but that is certainly not the case. There are pages on the web where you can test the problem...

Well, the purpose of using a big ordinal like ω1CK is to have some of these details handled for you. At least, that is how I have seen the "big ordinals" used; I haven't seen a "normal function method" that uses big ordinals like ω1CK to stratify the levels of the Veblen hierarchy, so I guess...

The roots of rational polynomials are known as the algebraic numbers; this is a countable set. So, if a random number generator selected a real number uniformly between 0 and 1, say, it would select a transcendental number with probability 100%.

You can use the Principle of Inclusion-Exclusion. Define A = the event that you get A,K,Q of spades B = the event that you get A,K,Q of hearts C = the event that you get A,K,Q of diamonds D = the event that you get A,K,Q of clubs then P(A \cup B \cup C \cup D) = P(A) + P(B) + P(C) + P(D) -...

I know I'm coming into this thread quite late, but here is my two cents: Like SSequence said, the introduction of uncountable ordinals into the ordinal notations is for convenience, not for necessity. One way to think about Ω that may be easier to digest is that it is a formal symbol that...