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?

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 […]

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 […]

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

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 […]

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 […]

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 […]

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, […]

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 […]

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

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