Recent content by Stoney Pete

How would you go about listing the computable real numbers in binary terms?

I see that I can't get my array for all possible EA's and their outputs for all inputs properly aligned... I trust you will nevertheless understand its meaning.

Thanks for the comments, they were helpful. When I began with the enumeration of all possible algorithms, I was indeed supposing that an algorithm need not halt on every input in order to be an algorithm. This seems to be the standard way of thinking about algorithms. Hence the standard phrasing...

It seems to me, on second thought, that the detour through the self-halting problem is not really necessary. For if we have H, then we automatically have D as well. Then, on the assumption that d is the number of D on the list of all A's, we can ask is H(d,n) is defined for any n from ℕ. And...

Hi everybody, I've been trying to understand the various proofs for the unsolvabilty of the Halting Problem, and I was wondering wether the following somewhat informal reconstruction of one the usual proof strategies is by and large correct. I'd like to get your opinion on this, not just...

Let me see if I understand this correctly... Sdiag basically says about itself that it is not on the presumed list of subsets of ℕ. It does this because its definition refers to all sets on the list. Thus, assuming the enumerability of P(ℕ), Sdiag also refers to itself. Assuming its place on...

Okay, that's interesting... I'm trying to understand it. Some questions: -When you write "n ∈ Sdiag iff n ∉ Sn" does this mean for example that 2 ∈ Sdiag iff 2 ∉ S2? So the index n is intended to have the same value as the variable n? -With the formula Φ(x) that defines a set I guess you mean...

Yes, I see your point. But what is the role of diagonalization in Gödel's proof? In some way I don't understand yet, diagonalization also underlies the construction of the self-referential sentence G in the language of arithmetic which states 'G is unprovable'. And this diagonalization, I would...

Okay, I start to see your point. I guess we can't define V using just the language of Peano arithmetic. We need some set theory to define V. I have to mull this over... Thanks for your input.

Well, I am trying to understand the basic state of affairs that underlies Gödel's first incompleteness theorem. And it seems to me that it basically comes down to this: Already due to the uncountability (or indenumerability) of P(ℕ) there are uncountably many arithmetical truths. But the...

Why is x∈V not expressible in the language of T? I was presupposing that it is expressible in that language... Doesn't it just depend on how rich we make the language of T? If that language is rich enough, x∈V should be expressible in it...

Hi everybody, Do you think the following reconstruction of Gödel's first incompleteness theorem is basically correct, or at least in the right ballpark? In my view, this incompleteness result basically turns on the mismatch between the indenumerability of the powerset of ℕ and the enumerability...

Thanks for the tips... I will get to work... Sometimes I really hate that I love mathematics while sucking at it :)

Thanks for the tip about the parentheses... The problem with the textbook I am studying is that it can be a bit lacking in explanation and illustration sometimes... It's a very comprehensive textbook, but also overly concise sometimes... Anyway, I appreciate the problem that the denominator is...

Homework Statement In the textbook I am studying one of the exercises is: lim x→0 e-x - 1 / x = ? The answer given is -1. But I can't see why this answer should follow. Homework Equations The most relevant equation here is: lim x→0 = ex - 1 / x = 1 The Attempt at a Solution So let's take...