Search arXivSearch

arXiv · 1512.03667

An Intuitively Complete Analysis of Godel's Incompleteness

Abstract

A detailed and rigorous analysis of Gödel's proof of his first incompleteness theorem is presented. The purpose of this analysis is two-fold. The first is to reveal what Gödel actually proved to provide a clear and solid foundation upon which to base future research. The second is to construct a coherent explication of Gödel's proof that is not only approachable by the non-specialist, but also brings to light the core principles underlying Gödel's proof.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jason W. Steinmetz. 2020-04-28. An Intuitively Complete Analysis of Godel's Incompleteness. https://arxiv.org/abs/1512.03667

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Bluebirds and mockingbirds cannot produce a fixed-point combinator

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

math.LO

Pointwise provable equality and the failure of composition

Montagna (1989) and Di Paola--Montagna (1991) claim that the algebraic systems $S'$ and $S'_T$, respectively, are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two program indices that are pointwise provably equal in $T$ but yield inequivalent composites when each is run after the same program. Montagna's $S'$ is the case $T=\mathrm{PA}$. The failure already occurs for partial maps from $ω$ to itself. Weak totality and the proposed range assignment also depend on the choice of representatives. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $Π^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $Σ^0_1$-sound and the universal relation otherwise.

math.LO

Compactness via Consistency Properties

We will use consistency properties to characterize strongly compact cardinals, first showing an adequate Model Existence Theorem for larger fragments.

math.LO