In this question, I asked about how slowly the Fibonacci terms in a algebra of elementary embeddings could grow, but I did not inquire if there was a limit to how quickly the Fibonacci terms could grow.

Define the Fibonacci terms $t_{1}(x,y)=y,t_{2}(x,y)=x$ and where $t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$ for $n\geq 1$. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with an operation $*$ defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Recall that $\mathrm{crit}(j)$ is the least ordinal where $j(\alpha)>\alpha$.

Suppose that $n$ is a natural number. Then does there exist a natural number $m$ such that if $\lambda$ is a cardinal and $X$ is an $n$-generated subalgebra of $\mathcal{E}_{\lambda}$ and $$\gamma=\min(\{\mathrm{crit}(t_{2n+1}(j,k))|j,k\in X,\mathrm{crit}(j)\leq\mathrm{crit}(k)\}),$$ then $$|\{\mathrm{crit}(j)|j\in X\}\cap\gamma|\leq m?$$ Are there any good lower or upper bounds on the size of the least such $m$?

If $\gamma<\lambda$ and $\gamma$ is a limit ordinal, then recall that $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for each $x\in V_{\gamma}$. The motivation for this question comes from the fact that a negative answer shows that the algebras $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ generalize to a purely algebraic setting in two very different ways and I still do not know if these two definitions are equivalent.

  • 1
    $\begingroup$ Is there any reason why there is a downvote to this question? $\endgroup$ – Joseph Van Name Mar 23 at 3:11
  • $\begingroup$ I am 100 percent sure that the entities who are downvoting this question are completely ignorant of the subject matter. $\endgroup$ – Joseph Van Name Mar 24 at 3:41
  • $\begingroup$ It seems like many people on this site are holding a grudge. $\endgroup$ – Joseph Van Name Mar 24 at 3:50
  • $\begingroup$ The downvoters are downvoting questions like this because they want to see these questions deleted. How sad. $\endgroup$ – Joseph Van Name yesterday

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.