As I have shown in this answer and this answer, the Laver tables (see also this chapter) contain fractal structure which resembles the Sierpinski triangle (see also this link). In fact, all of the combinatorial structure of the classical Laver tables is contained in their corresponding fractals. Induction is one of the main techniques of proving facts about Laver tables and generalizations thereof.

$\textbf{Example 1:}$ One shows that the operation $*_{n}$ on $\{1,...,2^{n}\}^{2}$ is well-defined and total by defining $x*_{n}y$ by a double induction which is descending on $x$ and for all $x$ the induction is ascending on $y$. The descending-ascending double induction is used commonly to prove facts about the Laver tables.

$\textbf{Example 2:}$ In order to prove that $*_{n}$ is self-distributive, one proceeds by induction on $n$. For each $n$ one proves that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ using a triple induction which is descending on $x$ and for each $x$ the induction is descending on $y$ and for each $y$ the induction is ascending on $z$.

$\textbf{Example 3:}$ Randall Dougherty uses induction to prove every lemma and result in this paper. In this paper, Dougherty shows that if $f(n)$ is the least natural number where $o_{f(n)}(1)\geq n$, then $f$ grows faster than the Ackermann function. The results in this paper are proven purely algebraically without necessarily resorting to large cardinals.

$\textbf{Transfinite induction:}$ Finally, since the Laver tables arise from the the algebras of elementary embeddings, one can use transfinite induction on large cardinals to prove results about finite algebras which do not have any known proofs without large cardinal hypotheses. However, since $\mathcal{E}_{\lambda}^{+}\neq\emptyset$ implies that $\text{cf}(\lambda)=\omega$ and by this answer, this sort of transfinite induction more closely resembles induction on $\mathbb{N}$ rather than the typical cases of transfinite induction.

Transfinite induction can be used to construct a linear ordering of elementary embeddings, and that linear ordering produces a compatible linear ordering on some finite algebras resembling the Laver tables.

As I have shown in this answer and this answer, the Laver tables (see also this chapter) contain fractal structure which resembles the Sierpinski triangle (see also this link). In fact, all of the combinatorial structure of the classical Laver tables is contained in their corresponding fractals. Induction is one of the main techniques of proving facts about Laver tables and generalizations thereof.

$\textbf{Example 1:}$ One shows that the operation $*_{n}$ on $\{1,...,2^{n}\}^{2}$ is well-defined and total by defining $x*_{n}y$ by a double induction which is descending on $x$ and for all $x$ the induction is ascending on $y$. The descending-ascending double induction is used commonly to prove facts about the Laver tables.

$\textbf{Example 2:}$ In order to prove that $*_{n}$ is self-distributive, one proceeds by induction on $n$. For each $n$ one proves that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ using a triple induction which is descending on $x$ and for each $x$ the induction is descending on $y$ and for each $y$ the induction is ascending on $z$.

$\textbf{Example 3:}$ Randall Dougherty uses induction to prove every lemma and result in this paper. In this paper, Dougherty shows that if $f(n)$ is the least natural number where $o_{f(n)}(1)\geq n$, then $f$ grows faster than the Ackermann function. The results in this paper are proven purely algebraically without necessarily resorting to large cardinals.

$\textbf{Transfinite induction:}$ Finally, since the Laver tables arise from the the algebras of elementary embeddings, one can use transfinite induction on large cardinals to prove results about finite algebras which do not have any known proofs without large cardinal hypotheses. However, since $\mathcal{E}_{\lambda}^{+}\neq\emptyset$ implies that $\text{cf}(\lambda)=\omega$ and by this answer, this sort of transfinite induction more closely resembles induction on $\mathbb{N}$ rather than the typical cases of transfinite induction.

Transfinite induction can be used to construct a linear ordering of elementary embeddings, and that linear ordering produces a compatible linear ordering on some finite algebras resembling the Laver tables.

As I have shown in this answer and this answer, the Laver tables (see also this chapter) contain fractal structure which resembles the Sierpinski triangle (see also this link). In fact, all of the combinatorial structure of the classical Laver tables is contained in their corresponding fractals. Induction is one of the main techniques of proving facts about Laver tables and generalizations thereof.

$\textbf{Example 1:}$ One shows that the operation $*_{n}$ on $\{1,...,2^{n}\}^{2}$ is well-defined and total by defining $x*_{n}y$ by a double induction which is descending on $x$ and for all $x$ the induction is ascending on $y$. The descending-ascending double induction is used commonly to prove facts about the Laver tables.

$\textbf{Example 2:}$ In order to prove that $*_{n}$ is self-distributive, one proceeds by induction on $n$. For each $n$ one proves that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ using a triple induction which is descending on $x$ and for each $x$ the induction is descending on $y$ and for each $y$ the induction is ascending on $z$.

$\textbf{Example 3:}$ Randall Dougherty uses induction to prove every lemma and result in this paper. In this paper, Dougherty shows that if $f(n)$ is the least natural number where $o_{f(n)}(1)\geq n$, then $f$ grows faster than the Ackermann function. The results in this paper are proven purely algebraically without necessarily resorting to large cardinals.

$\textbf{Transfinite induction:}$ Finally, since the Laver tables arise from the the algebras of elementary embeddings, one can use transfinite induction on large cardinals to prove results about finite algebras which do not have any known proofs without large cardinal hypotheses. However, since $\mathcal{E}_{\lambda}^{+}\neq\emptyset$ implies that $\text{cf}(\lambda)=\omega$ and by this answer, this sort of transfinite induction more closely resembles induction on $\mathbb{N}$ rather than the typical cases of transfinite induction.

For example, in this answer, I have mentioned about how transfinite induction can be used to construct a linear ordering of elementary embeddings in some forcing extension, and that linear ordering produces a linear ordering on some finite algebras.

As I have shown in this answer and this answer, the Laver tables (see also this chapter) contain fractal structure which resembles the Sierpinski triangle (see also this link). In fact, all of the combinatorial structure of the classical Laver tables is contained in their corresponding fractals. Induction is one of the main techniques of proving facts about Laver tables and generalizations thereof.

$\textbf{Example 1:}$ One shows that the operation $*_{n}$ on $\{1,...,2^{n}\}^{2}$ is well-defined and total by defining $x*_{n}y$ by a double induction which is descending on $x$ and for all $x$ the induction is ascending on $y$. The descending-ascending double induction is used commonly to prove facts about the Laver tables.

$\textbf{Example 2:}$ In order to prove that $*_{n}$ is self-distributive, one proceeds by induction on $n$. For each $n$ one proves that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ using a triple induction which is descending on $x$ and for each $x$ the induction is descending on $y$ and for each $y$ the induction is ascending on $z$.

$\textbf{Example 3:}$ Randall Dougherty uses induction to prove every lemma and result in this paper. In this paper, Dougherty shows that if $f(n)$ is the least natural number where $o_{f(n)}(1)\geq n$, then $f$ grows faster than the Ackermann function. The results in this paper are proven purely algebraically without necessarily resorting to large cardinals.

$\textbf{Transfinite induction:}$ Finally, since the Laver tables arise from the the algebras of elementary embeddings, one can use transfinite induction on large cardinals to prove results about finite algebras which do not have any known proofs without large cardinal hypotheses. However, since $\mathcal{E}_{\lambda}^{+}\neq\emptyset$ implies that $\text{cf}(\lambda)=\omega$ and by this answer, this sort of transfinite induction more closely resembles induction on $\mathbb{N}$ rather than the typical cases of transfinite induction.

Transfinite induction can be used to construct a linear ordering of elementary embeddings, and that linear ordering produces a compatible linear ordering on some finite algebras resembling the Laver tables.