Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.
View the Project on GitHub neugierde/cantors-attic
Quick navigation
The upper attic
The middle attic
The lower attic
The parlour
The playroom
The library
The cellar
Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine
Kripke-Platek set theory ($\text{KP}$) is a collection of axioms that is considerably weaker than ZFC. The formal language used to express each axiom is first-order with equality ($=$) together with one binary relation symbol, $\in$, intended to denote set membership.
$L_\alpha$ is a model of $\mathrm{KP}$ for admissible $\alpha$.
Sets are determined uniquely by their elements. This is expressed formally as \(\\forall x \\forall y \\big(\\forall z (z\\in x\\leftrightarrow z\\in y)\\rightarrow x=y\\big).\)
The “$\rightarrow$” can be replaced by “$\leftrightarrow$”, but the $\leftarrow$ direction is a theorem of logic.
There exists some set. In fact, there is a set which contains no members. This is expressed formally \(\\exists x \\forall y (y\\not\\in x).\)
Such an $x$ is unique by extensionality and this set is denoted by $\emptyset$.
For any two sets $x$ and $y$ (not necessarily distinct) there is a further set $z$ whose members are exactly the sets $x$ and $y$.
\[\\forall x \\forall y \\exists z \\forall w \\big(w\\in z\\leftrightarrow (w=x\\vee w=y)\\big).\]Such a $z$ is unique by extensionality and is denoted as $\{x,y\}$.
For any set $x$ there is a further set $y$ whose members are exactly all the members of the members of $x$. That is, the union of all the members of a set exists. This is expressed formally as
\[\\forall x \\exists y \\forall z \\big(z\\in y \\leftrightarrow \\exists w (w\\in x \\wedge z\\in w)\\big).\]Such a $y$ is unique by extensionality and is written as $y = \bigcup x$.
Suppose that a given property $P$ is true for some set $x$. Then there is a $\in$-minimal set for which $P$ is true. In more detail, given a formula $\varphi(x_1,\dots,x_n,x)$ the following is an instance of the induction schema: \(\\forall x\_1, \\ldots, x\_n \\big\[ \\exists x \\varphi(x\_1, \\ldots, x\_n, x) \\rightarrow \\exists y \\big( \\varphi(x\_1, \\ldots, x\_n, y) \\wedge \\forall z \\in y \\neg \\varphi(x\_1, \\ldots, x\_n, z) \\big) \\big\]\)
For any set $a$ and any $\Sigma_0$-predicate $P(x)$ written in the language of ZFC, the set $\{x\in a: P(x)\}$ exists. In more detail, given any $\Sigma_0$-formula $\varphi$ with free variables $x_1,x_2,\dots,x_n$ the following is an instance of the $\Sigma_0$-seperation schema: \(\\forall a \\forall x\_1 \\forall x\_2\\dots \\forall x\_n \\exists y \\forall z \\big(z\\in y \\leftrightarrow (z\\in a \\wedge \\varphi(x\_1,x\_2,\\dots,x\_n,z)\\big)\)
Such a $y$, unique by extensionality and is written (for fixed sets $a, x_1\dots, x_n$) $y=\{z\in a: \varphi(x_1,x_2,\dots,x_n,z)\}$.
If $a$ is a set and for all $x\in a$ there’s a some $y$ such that $(x,y)$ satisfies a given $\Sigma_0$-property, then there is some set $b$ such that for all $x \in a$ there is some $y \in b$ such that $(x,y)$ satisfies that property. In more detail, given a $\Sigma_0$-formula $\varphi(x_1,\dots,x_n,x,y)$ the following is an instance of the $\Sigma_0$-collection schema: \(\\forall a \\forall x\_1 \\dots \\forall x\_n \\big\[\\big( \\forall x\\in a \\exists y \\varphi(x\_1,\\dots,x\_n,x,y)\\big)\\rightarrow \\big(\\exists b \\forall x \\in a \\exists y \\in b \\varphi(x\_1, \\ldots, x\_n, x,y) \\big) \\big\].\)
Some authors include the axiom of infinity in Kripke-Platek set theory which states that there is an inductive set – a canonical example of an infinite set. More precisely: \(\\exists x \\big( \\emptyset \\in x \\wedge \\forall y \\in x (y \\cup \\{y \\} \\in x) \\big).\) The axiom of infinity combined with an instance of $\Sigma_0$-separation imply the axiom of null set so that it be dropped if one assumes the axiom of infinity.