Gödel’s Theorem says that I can construct a mathematical statement like “f(x1,x2,…,x_n)=0 has no integer solution”, where it is impossible (in a certain system of axioms) to formally prove that it’s true, and also impossible to formally prove that it’s false. I have often heard a school of thought that goes like “Well, in reality, […]

Are there any known undecidable sentences for PA are neither “self-referential” (like a sentence equivalent to its own nonprovability) nor imply consistency of PA (like in the Paris Harrington theorem)?

I realize from the answer to this post that the fallacy in my “proof” of “ZF is inconsistent” was that I was not considering that there are models with non-standard integers. However now I think I developed an actual deduction of $T \vdash \text{Cons} T$ for any sufficiently powerful theory $T$ thus implying by Godel’s […]

Is Robinson Arithmetic complete in the sense of Gödels completeness theorem? And is Robinson Arithmetic incomplete in the sense of Gödels first incompleteness theorem? If RA is both it would be a good example to explain the different notions of “completeness” used in the two theorems mentioned.

Given a φ independent of PA which is true in the standard model, will always (PA+ ¬φ) be ω-inconsistent? Does it mean that every such ¬φ can be used to prove that a Turing machine halts on a given C when it will actually never halt?

I am trying to understand Gödel’s first incompleteness theorem from his original 1931 paper. Here is a translation i am using for my studies : http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf I feel like i now have a decent intuitive understanding of it all. Now I’m trying to understand all the technical details of it. I don’t fully understand these […]

Am I right in thinking that there are countably infinitely many nonequivalent unprovable statements in ZFC due to the Gödel’s first incompleteness theorem? If not, why?

In proving Gödel’s incompleteness theorem, why does he needed the Diagonal Lemma or the Fixed Point Theorem for building a formula $\phi$ that spoke about itself? Can’t this formula be built this way: Diagonal Function. Let $D(x, y)$ be the diagonal function, such that $D$ returns the result obtained by replacing the formula with Gödel […]

I am not quite sure what really is meant when talking about “arithmetics” in context of Gödel’s incompleteness theorems. How I so far understand it: Gödel proved that every sufficiently powerful first-order theory is not both consistent and complete at the same time. This is, you pick a first-order language (so the logical symbols are […]

Like any math newbie, Godel’s Incompleteness Theorems are easy to understand in general layman’s terms, but difficult to understand beyond the typical “liar’s paradox” and “barber’s paradox” type examples. But then I started thinking, are axioms examples of the truths of mathematics that can’t be proven? For example, Peano’s Postulates: a very popular “starting point” […]

Intereting Posts

Basic Modulo Question
Sum of the series: $\sum_{n=0}^\infty\frac{1+n}{3^n}$
What is the fraction field of $R]$, the power series over some integral domain?
n people & n hats: probability that at least 1 person has his own hat
What is the angle $<(BDE,ADH)$?
The myth of no prime formula?
Density function for RV
Discrete random variable with infinite expectation
Coin flipping probability game ; 7 flips vs 8 flips
Find the distance between two lines
Proving that well ordering principle implies Zorn's Lemma.
Circular permutations with indistinguishable objects
Collatz-ish Olympiad Problem
Understanding implicit differentiation with concepts like “function” and “lambda abstraction.”
Countable unions of countable sets