4
votes
2answers
182 views

Why doesn't “$V$ is a model of $ZF$” imply consistency of ZF?

One can prove that $V$, Von Neumann's universe, satisfies all of $ZF$ axioms, so it's a model of $ZF$. But I can't see why this doesn't imply the consistency of that theorem. I'm aware that ...
1
vote
1answer
137 views

Questions on a sentence in ZFC asserting that ZFC has a model

Question 1. Let $\operatorname{con}(\mathsf{ZFC})$ be a sentence in $\mathsf{ZFC}$ asserting that $\mathsf{ZFC}$ has a model. Let $S$ be the theory $\mathsf{ZFC}+\operatorname{con}(\mathsf{ZFC})$. ...
2
votes
0answers
83 views

Minimal number of variables in a ZFC-undecidable sentence?

Let $\phi$ be a sentence of set theory. In Prenex form, $\phi$ can be written $$ {\bf Q}_1 x_1 {\bf Q}_2 x_2 \ldots {\bf Q}_n x_n \ \ \psi(x_1,x_2, \ldots ,x_n) $$ where each ${\mathbf Q}_i$ is ...
10
votes
1answer
155 views

Problem with completeness theorem and $\mathsf{Con(ZFC)}$

Using the reflection theorem we can prove in $\mathsf{ZFC}$ that for any finite $\Lambda\subseteq\mathsf{ZFC}$ there is some limit ordinal $\gamma$ such that $R(\gamma)\models\Lambda$. Then by the ...
4
votes
1answer
95 views

Incompleteness theorem and $\mathbb{L}$.

Let $\alpha > \omega$ and $u = \{\ulcorner \sigma \urcorner : \sigma \in \mathrm{Th}(\mathbb{L}_\alpha, \in)\} \subseteq \omega$, where by $\mathbb{L}_\alpha$ we denote as usual the constructible ...
5
votes
3answers
156 views

Gödel's incompleteness wrt weakend versions of ZFC

Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed. http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem ...
31
votes
6answers
2k views

Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not ...
0
votes
2answers
357 views

What is actually “relatively consistent”?

Gödel's incompleteness theorem states that: "if a system is consistent, it is not complete." And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc. However, why does this ...
7
votes
3answers
311 views

Can ZFC decide number theory?

Among the versions of the Incompleteness Theorem that I've seen are the following: Assuming the Peano axioms are consistent (which they are, if we accept the existence of the natural numbers), there ...
2
votes
1answer
140 views

Why does the existence of independent statements not prove completeness?

I've read before that, by the Principle of Explosion, if a theory is inconsistent, then absolutely any statement can be proven within it. Obviously, there are statements which are independent of ZFC ...
9
votes
2answers
263 views

How to show the existence of an infinite set of independent undecidable sentences?

How to show the existence of an infinite set of independent undecidable sentences? By "independent" I mean that no implication between any two elements is provable. A finite set that satisfies the ...