Articles of foundations

The class of all functions between classes (NBG)

Is it possible in NBG (von Neumann-Bernays-Gödel set theory) to construct the class of all functions $X \to Y$ between two (proper) classes $X,Y$? I guess that this does not work. In the special case $Y=\{0,1\}$ we would get the class of all subclasses of $X$, which does not exist. Can someone confirm this?

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I can also show that $b^{r+s} = b^r b^s$ if $r$ and $s$ are rational. If $x$ is […]

clarify the term “arithmetics” when talking about Gödel's incompleteness theorems

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 […]

Explicit example of countable transitive model of $\sf ZF$

Do we know any explicit example of a countable transitive model for $\sf ZF$ or $\sf ZFC$?

Proving that the existence of strongly inaccessible cardinals is independent from ZFC?

ZFC can’t prove that strongly inaccessible cardinals exist, or else it would prove that a model of itself exists and hence $Con(ZFC)$. So this leaves us with two options: ZFC proves there are no strongly inaccessible cardinals The existence of strongly inaccessible cardinals is independent from ZFC I’ve heard, however, that you can’t actually prove […]

Integer induction without infinity

In ZFC minus infinity (let us call this T), one can still define ordinals, and then define integers as ordinals all of whose members are zero or successor ordinals. I am looking for a formula $\psi$ such that T proves $\psi(0)$ T proves $\psi(n)\Rightarrow \psi(n+1)$ for every “integer” (in the above sense ) $n$, T […]

Ambiguity in the Natural Numbers

What I am wondering is if mathematicians know whether (assuming consistency) the natural numbers are a definite object, without ambiguity. This seems intuitively obvious, but I don’t know if its been shown. A formal question might be if two consistent axiomatic systems, each including all the axioms of the natural numbers and then some, can’t […]

Why is bounded induction stronger than open induction?

It seems to me that any formula in the language of first-order arithmetic which has only bounded quantifiers can be written as a formula without any quantifiers. For instance, “There exists an n < 1000 such that P(n)” can be written as “P(1) or P(2) or … or P(999)”, and “for all n < 100, […]

What concepts does math take for granted?

I suspect there must be some concepts that math takes for granted (there has to be a starting point). For example, after spending some time thinking about it yesterday, I wondered whether most of math could be produced from the concepts of Negation Identity Cardinality Ordinality Sethood Concepts Universality It seemed to me that other […]

On the number of countable models of complete theories of models of ZFC

This question already has an answer here: Number of Non-isomorphic models of Set Theory 2 answers