All Questions
16 questions with no upvoted or accepted answers
5
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94
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The descriptive complexity and definiteness of the space of all elementary embeddings $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$
Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$.
Suppose that $(\alpha_{n})_{n}$ is an increasing cofinal sequence in $\lambda$. Give $\...
4
votes
0
answers
113
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How many compatible linear orders exist on the classical Laver tables?
Let $A_{n}$ be the unique algebra $(\{1,\dots,2^{n}\},*_{n})$ such that
$x*_{n}1=x+1\mod 2^{n}$ and
$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ for all $x,y,z$. We say that a linear ordering $\preceq$ ...
3
votes
0
answers
245
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Ordering large cardinal axioms around the level of $n$-huge by consistency strength?
So the large cardinal axioms are for the most part considered to be linearly ordered by consistency strength. For the large cardinals between extendibility and rank-into-rank (i.e. the $n$-huge ...
2
votes
0
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83
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For each $n$ is it possible to have $\mathrm{crit}(x^{[n]}*y)>\mathrm{crit}(x^{[n-1]}*y)>\dots>\mathrm{crit}(x*y)$?
Suppose that $(X,*,1)$ satisfies the following identities:
$x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$. Define the Fibonacci terms $t_{n}(x,y)$ for $n\geq 1$ by letting
$$t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=...
2
votes
0
answers
53
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Calibrating the strength of the quotients of subalgebras of the classical Laver tables
Define an algebraic structure $A_{n}$ by letting
$$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$
where $*_{n}$ is the unique operation such that $x*_{n}1=x+1\mod 2^{n}$ for $$x\in\{1,\dots,2^{n}-1,2^{n}\}$...
2
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0
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82
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Can a finitely generated algebra of rank-into-rank embeddings grow at rate $O(n\cdot\log(n))$?
Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$ is a non-trivial elementary embedding, then define $\mathrm{crit}...
1
vote
0
answers
92
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Why does $p_{n}(i,1)=1$ so often where the polynomials $p_{n}$ are obtained from the classical Laver tables
So I was doing some computer calculations with the classical Laver tables and I found polynomials $p_{n}(x,y)$ such that $p_{n}(i,1)=1$ for many $n$.
The $n$-th classical Laver table is the unique ...
1
vote
0
answers
61
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Growth rate of the critical points of the Fibonacci terms $t_{n}(x,y)$ vs $t_{n}(1,1)$ in the classical Laver tables
The classical Laver table $A_{n}$ is the unique algebra $(\{1,\dots,2^{n}\},*_{n})$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ for all $x,y,z\in A_{n}$.
Define the ...
1
vote
0
answers
75
views
Multiple roots in the classical Laver tables
The classical Laver table $A_{n}$ is the unique algebraic structure $$(\{1,\dots,2^{n}\},*_{n})$$ such that $x*_{n}1=x+1\mod 2^{n}$ and
$$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$ for all $x,y,z\in\{1,...
1
vote
0
answers
59
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Can we have $\sup\{\alpha\mid(x*x)^{\sharp}(\alpha)>x^{\sharp}(\alpha)\}=\infty$ in an algebra resembling the algebras of elementary embeddings?
A finite algebra $(X,*,1)$ is a reduced Laver-like algebra if it satisfies the identities $x*(y*z)=(x*y)*(x*z)$ and if there is a surjective function
$\mathrm{crit}:X\rightarrow n+1$ where
$\mathrm{...
1
vote
0
answers
60
views
In the classical Laver tables, do we have $o_{n}(1)<o_{n}(2)$ for any $n>8$?
The classical Laver table $A_{n}$ is the unique algebraic structure
$(\{1,\dots,2^{n}\},*_{n})$ where
$$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$$
and where $$x*_{n}1=x+1\mod 2^{n}$$ for $x,y,z\in\{1,\...
1
vote
0
answers
44
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Vastness of inverse systems of Laver-like algebras
Suppose that $(X,*,1)$ satisfies the identities
$x*(y*z)=(x*y)*(x*z),x*1=1,1*x=x$. Then we say that $(X,*,1)$ is a reduced Laver-like algebra if whenever $x_{n}\in X$ for all $n\in\omega$, there is ...
1
vote
0
answers
33
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Can we always extend a finitely generated reduced Laver-like algebra to a vast inverse system of Laver-like algebras?
An $(X,*,1)$ that satisfies the identities $x*(y*z)=(x*y)*(x*z),1*x=x,x*1=1$ is said to be a reduced Laver-like algebra if whenever $x_{n}\in X$ for $n\in\omega$, there is some $N\in\omega$ where $x_{...
1
vote
0
answers
48
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Density of different types of critical points in an algebra of elementary embeddings
Suppose that $j,k:V_{\lambda}\rightarrow V_{\lambda}$ are elementary embeddings. Let $\mathrm{crit}_{n}(j,k)$ denote the $n$-th element in $\{\mathrm{crit}(\ell)\mid\ell\in\langle j,k\rangle\}$. ...
1
vote
0
answers
42
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Density of critical points subalgebras of the algebras of elementary embeddings
Let $j:V_{\lambda}\rightarrow V_{\lambda}$ be an elementary embedding. Then $\{\mathrm{crit}(k)\mid k\in\langle j\rangle\}$ has order type $\omega$, so let $\mathrm{crit}_{n}(j)$ denote the $n$-th ...
1
vote
0
answers
76
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Which varieties are compatible with the classical Laver tables?
Let $$A_{n}=(\{1,\dots,2^{n}-1,2^{n}\},*_{n})$$ denote the $n$-th classical Laver table. The operation $*_{n}$ is the unique binary operation on
$\{1,\dots,2^{n}\}$ such that $$x*_{n}(y*_{n}z)=(x*_{n}...