I am little bite confused. I know that in a ZFC model can be defined the set of natural numbers. I know, for instance, how natural numbers are constructed in a von Neumann Universe. My question is how could I define a natural number by a sentence in the first-order lenguage of theory of sets? How could I construct natural number in arbitrary ZFC model?
How is it written "x is a natural number" in the first-order logic language of the set theory?
641 Views Asked by user178826 https://math.techqa.club/user/user178826/detail AtThere are 2 best solutions below
On
To answer a question you asked in the comment :
$n\subset m$ is an abbrevition for $\forall x, x\in n\implies x\in m$
"$x\neq \emptyset$" is an abbreviation for $\exists z, z\in x$
"$n$ is transitive" is an abbreviation for $\forall x, x\in n\implies x\subset m$
"$\in$ is a well-ordering on the transitive set $n$" is an abbreviation for "$n$ is transitive and $(\forall x, x\neq \emptyset \implies (x\subset n\implies \exists z, z\in x\land \forall y, y\in x\implies z\in y\lor z=y))\land (\forall z, z\in n\implies \neg z\in z) \land (\forall x\forall y\forall z, (x\in n\land y\in n\land z\in n)\implies ((x\in y\land y\in z)\implies x\in z))$
"$n$ is an ordinal" is an abbreviation for "$\in$ is a well-ordering on the transitive set $n$"
$x=y\cup\{y\}$ is an abbreviation for $y\subset x\land y\in x\land \forall z, z\in x\implies z=y\lor z\in y$
"$n$ is a finite ordinal" is an abbreviation for $n$ is an ordinal and $(n\neq \emptyset \implies \exists z, n=z\cup\{z\})\land (\forall y, y\in n\implies (y\neq \emptyset \implies \exists z, y=z\cup\{z\}))$
So obviously one could write down an explicit formula but: it wouldn't be very useful; it would harly be readable; it would be extremely tedious to do
Yet another approach: $x$ is a finite ordinal iff $x$ is an element of every inductive set, that is, iff $$\forall z[(\emptyset \in z\wedge\forall w(w\in z\implies w\cup\{w\}\in z))\implies x\in z].$$ That's a bit abbreviated, but hopefully it's clear how to turn it into a genuine formula in the language of set theory.