resolver faces is the survival problem, then the adapter errors are literally
death errors, or at least pain, while the perceiver errors are just mental
previsions of what not to do. See that, if the perceiver implements the
identity analogy, as $\Re_{2}[i]$ does, then the model is good, because the
internal problem is equal to the external one. And the perceiver $\Re_{2}[i]$
is more flexible than the adapter.

Comment The perceiver is a body capable of several behaviors, a governor that
selects the current behavior, and a simulator that internalizes behaviors.

Example The perceiver governor determines what to do based upon an internal
interpretation. According to Lettvin et al. (1959), a frog is a perceiver that
uses an internal routine. Frog’s $i$ is such that any dark point that moves
rapidly in its field of vision is a fly which it will try to eat.

§5.5 Learner

§5.5.1 Definition A learner $\Re_{3}$ is any resolver that implements one
condition on the members of ${\xiibb S}\to{\xiibb S}$. We will note
$\Re_{3}[P_{F}]$ the learner that implements $P_{F}$, where $P_{F}\in(({\xiibb
S}\to{\xiibb S})\to{\xiibb B})$, that is, $\Re_{1}[P_{F}]=P_{F}\in(({\xiibb
S}\to{\xiibb S})\to{\xiibb B})$. Then the learners resources are in $({\xiibb
S}\to{\xiibb S})\to{\xiibb B}$, and $\\{\cdot\,\Re_{3}\,\cdot\\}=(({\xiibb
S}\to{\xiibb S})\to{\xiibb B})$.

Comment A learner $\Re_{3}$ implements a syntactic conditional computation.

§5.5.2 Remark Learners are to syntax as adapters are to semantics.

Comment When solutions are functions ${\xiibb S}\to{\xiibb S}$, then a learner
does what an adapter does, which is to return a predicate on solutions. That
is, learners on metaproblems are as adapters on problems. But, learners can go
further.

§5.5.3 Lemma Each learner $\Re_{3}[P_{F}]$ implements one set of members of
$({\xiibb S}\to{\xiibb S})$.

Proof Because every predicate $P_{F}:({\xiibb S}\to{\xiibb S})\to{\xiibb B}$
defines a set $F=\\{\,f\in({\xiibb S}\to{\xiibb S})\mid P_{F}(f)\,\\}\in
2^{{\xiibb S}\to{\xiibb S}}$. The condition $P_{F}$ is the characteristic
function of $F$, $\forall f\in({\xiibb S}\to{\xiibb S}),\;P_{F}(f)=[f\in F]$.
$\diamond$

Comment We will write $\Re_{3}[P_{F}]=\Re_{3}[F]=F\in 2^{{\xiibb S}\to{\xiibb
S}}$.

Comment The learner $\Re_{3}[F]$ implements $F\in 2^{{\xiibb S}\to{\xiibb
S}}$. So $\forall s\in{\xiibb S},\,\Re_{3}[F](s)=F(s)\in 2^{\xiibb S}$,
because $F(s)=\\{\,f(s)\mid f\in F\,\\}$, by the rewriting rules in §3.1.3,
and then $\Re_{3}[F](s)$ returns a set of solutions, as any well-behaved
resolution should do. Also, by the same rules,
$\Re_{3}[F](S)=F(S)=\\{\,f(s)\mid s\\!\in\\!S\times f\\!\in\\!F\,\\}\in
2^{\xiibb S}$.

§5.5.4 Lemma Every perceiver $\Re_{2}$ is a learner $\Re_{3}$, that is,
$\\{\cdot\,\Re_{2}\,\cdot\\}\subset\\{\cdot\,\Re_{3}\,\cdot\\}$.

Proof For each perceiver $\Re_{2}[f]$, which implements $f\in({\xiibb
S}\to{\xiibb S})$, there is a learner $\Re_{3}[P_{\delta_{f}}]$, see §4.3.14,
that implements the singleton $\\{f\\}\in(({\xiibb S}\to{\xiibb S})\to{\xiibb
B})$. But not every set is a singleton. Then,
$\\{\cdot\,\Re_{2}\,\cdot\\}=({\xiibb S}\to{\xiibb S})\subset(({\xiibb
S}\to{\xiibb S})\to{\xiibb B})=\\{\cdot\,\Re_{3}\,\cdot\\}$. $\diamond$

§5.5.5 Lemma $\Re_{3}[F]=\bigcup_{f\in F}\\{\Re_{2}[f]\\}$.

Proof Because $\Re_{2}[f]=f$, so $\bigcup_{f\in
F}\\{\Re_{2}[f]\\}=\bigcup_{f\in F}\\{f\\}=F=\Re_{3}[F]$. $\diamond$

Comment Again, the results are the same, but not the implementation. The union
of perceivers $\bigcup_{f\in F}\\{\Re_{2}[f]\\}$ cannot select a function to
use, so it cannot implement any meta-trial, as $\Re_{3}[R\cap\Sigma_{\Pi\pi}]$
does, see §5.5.11.

Comment These are not sets of solutions, but sets of semantic functions.

§5.5.6 Lemma $\forall f\in F$, $\forall S\in 2^{\xiibb S}$,
$\Re_{2}[f](S)\subseteq\Re_{3}[F](S)$.

Proof If $f\in F$, then $f(S)=\\{\,f(s)\mid s\in
S\,\\}\subseteq\\{\,f^{\prime}(s)\mid s\\!\in\\!S\times
f^{\prime}\\!\in\\!F\,\\}=F(S)$. $\diamond$

§5.5.7 Notation We will rewrite $\Re_{1}[\Re_{3}[F](S)\cap\Sigma_{\pi}]$ as
$\Re_{3}[F](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$.

Comment We can write
$\Re_{1}[\Re_{3}[F](S)\cap\Sigma_{\pi}]=\Re_{3}[F](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$,
because any learner can implement semantic conditions, that is, because
$\\{\cdot\,\Re_{1}\,\cdot\\}\subset\\{\cdot\,\Re_{3}\,\cdot\\}$.

§5.5.8 Lemma If $f\in F$, then $\forall S\in 2^{\xiibb S}$,
$\forall\pi\in{\xiibb P}$,
$\Xi\Re_{2}[f](S)\subseteq\Xi\Re_{3}[F](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$.

Proof Firstly see that, if $f\in F$, then $f(S)\subseteq F(S)$, by §3.1.3, so,
$f(S)\cap\Sigma_{\pi}\subseteq F(S)\cap\Sigma_{\pi}$. Secondly see that
$\forall\pi\in\Xi\Re_{2}[f](S)$, $f(S)\cap\Sigma_{\pi}=f(S)\neq\emptyset$.
This is because $\Xi\Re_{2}[f](S)=\\{\,\pi\mid f(S)\subseteq\Sigma_{\pi}\land
f(S)\neq\emptyset\,\\}$. Now, taking both together,
$\forall\pi\in\Xi\Re_{2}[f](S)$, $\emptyset\neq
f(S)=f(S)\cap\Sigma_{\pi}\subseteq F(S)\cap\Sigma_{\pi}$, so for these $\pi$,
$F(S)\cap\Sigma_{\pi}\neq\emptyset$. Then these $\pi\in\\{\,\pi\mid
F(S)\cap\Sigma_{\pi}\subseteq\Sigma_{\pi}\land
F(S)\cap\Sigma_{\pi}\neq\emptyset\,\\}=\Xi\Re_{1}[\Re_{3}[F](S)\cap\Sigma_{\pi}]$,
because $F(S)\cap\Sigma_{\pi}\subseteq\Sigma_{\pi}$ is always true. $\diamond$

Definition We will call $f\in F$ the learner condition. If the learner
condition holds, then the learner
$\Re_{3}[F](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$
solves any problem that the perceiver $\Re_{2}[f](S)$ solves.

§5.5.9 Lemma If $f\in F$, then $\forall S\in 2^{\xiibb S}$,
$\forall\pi\in{\xiibb P}$,
$\Phi\Re_{2}[f](S)\subseteq\Phi\Re_{3}[F](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$.

Proof If $f\in F$, then $f(S)\subseteq F(S)$, see §3.1.3, so,
$f(S)\cap\Sigma_{\pi}\subseteq F(S)\cap\Sigma_{\pi}$. Now
$\forall\pi\in\Phi\Re_{2}[f](S)$, $f(S)=\Sigma_{\pi}$, and then for these
$\pi$, $\Sigma_{\pi}=f(S)\cap\Sigma_{\pi}\subseteq
F(S)\cap\Sigma_{\pi}\subseteq\Sigma_{\pi}$. Therefore, for these $\pi$,
$F(S)\cap\Sigma_{\pi}=\Sigma_{\pi}$, and
$\Phi\Re_{2}[f](S)\subseteq\\{\,\pi\mid
F(S)\cap\Sigma_{\pi}=\Sigma_{\pi}\,\\}=\Phi\Re_{1}[\Re_{3}[F](S)\cap\Sigma_{\pi}]$.
$\diamond$

Comment If the learner condition holds, $f\in F$, then the learner
$\Re_{3}[F](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$
resolves any problem that the perceiver $\Re_{2}[f](S)$ resolves.

§5.5.10 Corollary In particular, if $i\in R$, then
$\Xi\Re_{2}[i](S\cap\Sigma_{\pi})\subseteq\Xi\Re_{3}[R](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$
and
$\Phi\Re_{2}[i](S\cap\Sigma_{\pi})\subseteq\Phi\Re_{3}[R](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$.

Proof By Lemmas §5.5.8 and §5.5.9. See that
$\Xi\Re_{3}[R](S\cap\Sigma_{\pi}\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})\subseteq\Xi\Re_{3}[R](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$,
and
$\Phi\Re_{3}[R](S\cap\Sigma_{\pi}\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})\subseteq\Phi\Re_{3}[R](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$,
because $\Re_{3}[R](S\cap\Sigma_{\pi})\subseteq\Re_{3}[R](S)$, so corollaries
to Lemmas §5.3.5 and §5.3.7 apply (the equal case is trivial). $\diamond$

§5.5.11 Lemma Any meta-trial resolution $T_{\Pi\pi}(R)$ can be implemented by
the learner $\Re_{3}[R\cap\Sigma_{\Pi\pi}]$, if $P_{R}$ and $P_{\Pi\pi}$ are
elementable.

Comment The diagram for the meta-trial, or trial of the metaproblem, see
§3.6.9, is:

${\pi}\mathop{\hbox
to18.3pt{\rightarrowfill}}\limits^{\hbox{$\scriptstyle\Pi$}}{\Pi\pi}\mathop{\hbox
to37.38782pt{\rightarrowfill}}\limits^{\hbox{$\scriptstyle
T_{\Pi\pi}(R)$}}{\Sigma_{\Pi\pi}}\mathop{\hbox
to21.66669pt{\rightarrowfill}}\limits^{\hbox{$\scriptstyle{\cal
T}_{\Pi}$}}{\Sigma_{\pi}}\;.$

Comment A learner solves metaproblems by trial, as an adapter solves problems
by trial. The following correlations stand: $\Re_{3}\leftrightarrow\Re_{1}$,
$\Pi\pi\leftrightarrow\pi$, $R\leftrightarrow S$, and then
$\Re_{3}[R\cap\Sigma_{\Pi\pi}]\leftrightarrow\Re_{1}[S\cap\Sigma_{\pi}]$.
Therefore, $\Re_{3}[R\cap\Sigma_{\Pi\pi}]$ compares to $\Re_{2}[i]$, where
$i\in R$, as $\Re_{1}[S\cap\Sigma_{\pi}]$ compares to $\Re_{0}[s]$, where
$s\in S$.

Proof $T_{\Pi\pi}(R)=\\{\,r\in R\mid r\in\Sigma_{\Pi\pi}\,\\}=\\{\,r\mid
r\\!\in\\!R\,\land\,r\\!\in\\!\Sigma_{\Pi\pi}\,\\}=\\{\,r\mid P_{R}(r)\land
P_{\Pi\pi}(r)\,\\}$. In the meta-trial $T_{\Pi\pi}(R)$, $R$ is a set of
resolutions, where ${\xiibb R}=({\xiibb P}\to 2^{\xiibb S})=(2^{\xiibb S}\to
2^{\xiibb S})$, that is, $P_{R}:(2^{\xiibb S}\to 2^{\xiibb S})\to{\xiibb B}$,
and the condition of the metaproblem $\Pi\pi$ is also $P_{\Pi\pi}:{\xiibb
R}\to{\xiibb B}=({\xiibb P}\to 2^{\xiibb S})\to{\xiibb B}=(2^{\xiibb S}\to
2^{\xiibb S})\to{\xiibb B}$. So if both $P_{R}$ and $P_{\Pi\pi}$ are
elementable by $\wp_{R}$ and $\wp_{\Pi\pi}$, then both of them, and its
conjunction, are implementable. Then $T_{\Pi\pi}(R)=\\{\,r\mid P_{R}(r)\land
P_{\Pi\pi}(r)\,\\}\doteq\Re_{3}[\wp_{R}\land\wp_{\Pi\pi}]=\Re_{3}[R\cap\Sigma_{\Pi\pi}]$.
$\diamond$

§5.5.12 Summary Perceiver success depends crucially on the analogy, that is,
on how much the analogy resembles the identity $i$. And a learner can adapt
the analogy to the problem it is facing, because the learner $\Re_{3}[R]$
implements a set of functions $R$ from which it can select another analogy
when the current one fails. Adapting the analogy is also known as modeling. So
a learner can apply different analogies, but a learner can also apply a
routine if it knows a solution, because the routine is more efficient, or a
trial, when the model is not good enough or too pessimistic.

Comment The learner is a body capable of several behaviors, a governor that
selects the current behavior, a simulator that internalizes behaviors, and a
modeler that adjusts the model used by the simulator.

Example Where there is modeling and simulation there is learning, because
enhancing the model prevents repeating errors. A dog is a learner.

§5.6 Subject

§5.6.1 Definition A subject $\Re_{4}$ is any resolver that implements one
transformation of the elements in ${\xiibb S}\to{\xiibb S}$ into the elements
in ${\xiibb S}\to{\xiibb S}$. We will note $\Re_{4}[{\xiifrak f}]$ the subject
that implements $\xiifrak f$, where ${\xiifrak f}\in(({\xiibb S}\to{\xiibb
S})\to({\xiibb S}\to{\xiibb S}))$, that is, $\Re_{4}[{\xiifrak f}]={\xiifrak
f}\in(({\xiibb S}\to{\xiibb S})\to({\xiibb S}\to{\xiibb S}))$. Then the
subject resources are in $({\xiibb S}\to{\xiibb S})\to({\xiibb S}\to{\xiibb
S})$, and $\\{\cdot\,\Re_{4}\,\cdot\\}=(({\xiibb S}\to{\xiibb S})\to({\xiibb
S}\to{\xiibb S}))$.

Comment A subject $\Re_{4}$ implements a syntactic functional computation.

§5.6.2 Remark Subjects are to syntax as perceivers are to semantics.

Comment When solutions are functions ${\xiibb S}\to{\xiibb S}$, then a subject
does what a perceiver does, which is to return a function on solutions to
solutions. That is, subjects on metaproblems are as perceivers on problems.
But, subjects can go further.

Comment The subject $\Re_{4}[{\xiifrak f}]$ implements function $\xiifrak f$
from ${\xiibb S}\to{\xiibb S}$ to ${\xiibb S}\to{\xiibb S}$, that is,
${\xiifrak f}:({\xiibb S}\to{\xiibb S})\to({\xiibb S}\to{\xiibb S})$. Then,
$\forall f\in({\xiibb S}\to{\xiibb S}),\;\Re_{4}[{\xiifrak f}](f)={\xiifrak
f}(f)\in({\xiibb S}\to{\xiibb S})$.

§5.6.3 Notation As resolutions return sets of elements in $\xiibb S$, to
normalize the situation of subjects, for which $\Re_{4}[{\xiifrak
f}](f)(s)\in{\xiibb S}$, we will use the rewriting rules in §3.1.3 to get
$\Re_{4}[{\xiifrak f}](F)(S)=\\{\,\Re_{4}[{\xiifrak f}](f)(s)\mid
s\\!\in\\!S\times f\\!\in\\!F\,\\}\in 2^{\xiibb S}$.

Comment Subject $\Re_{4}[{\xiifrak f}](F)(S)$ has two software levels:
semantics ($S$) and syntax ($F$).

§5.6.4 Lemma Every learner $\Re_{3}$ is a subject $\Re_{4}$, that is,
$\\{\cdot\,\Re_{3}\,\cdot\\}\subset\\{\cdot\,\Re_{4}\,\cdot\\}$.

Proof First we define set ${\xiifrak B}=\\{K_{\top},K_{\bot}\\}$, both
functions ${\xiibb S}\to{\xiibb S}$, see §5.1.8. Next we define the the
natural isomorphism between $\xiibb B$ and ${\xiifrak B}$, mapping $\top$ to
the function that always returns $\top$, which is $K_{\top}=P_{\tau}$, and
$\bot$ to the function that always returns $\bot$, which is
$K_{\bot}=P_{\bar{\tau}}$, see §3.1.11. And so ${\xiibb
B}\Leftrightarrow{\xiifrak B}:\top\leftrightarrow K_{\top},\bot\leftrightarrow
K_{\bot}$, and ${\xiibb B}\cong{\xiifrak B}\subset({\xiibb S}\to{\xiibb S})$.
Then, $(({\xiibb S}\to{\xiibb S})\to{\xiibb B})\subset(({\xiibb S}\to{\xiibb
S})\to({\xiibb S}\to{\xiibb S}))$, and therefore
$\\{\cdot\,\Re_{3}\,\cdot\\}=(({\xiibb S}\to{\xiibb S})\to{\xiibb
B})\subset(({\xiibb S}\to{\xiibb S})\to({\xiibb S}\to{\xiibb
S}))=\\{\cdot\,\Re_{4}\,\cdot\\}$. $\diamond$

Comment Each learner implements one condition $P_{F}\in(({\xiibb S}\to{\xiibb
S})\to{\xiibb B})$. And any condition on functions $P_{F}\in(({\xiibb
S}\to{\xiibb S})\to{\xiibb B})$ is also a function on functions to functions
$P_{F}\in(({\xiibb S}\to{\xiibb S})\to({\xiibb S}\to{\xiibb S}))$, because
${\xiibb B}\subset({\xiibb S}\to{\xiibb S})$. Therefore, for each learner
$\Re_{3}[P_{F}]$, which implements condition $P_{F}$, there is a subject
$\Re_{4}[P_{F}]$ that implements the function $P_{F}$, and then we write
$\Re_{3}[P_{F}]=P_{F}=\Re_{4}[P_{F}]$.

Comment Again, $\Re_{3}[P_{F}]=\Re_{4}[P_{F}]$ explains that the results are
the same, but not the implementation.

§5.6.5 Lemma $\forall F\in 2^{\xiibb S\to S}$, $\forall S\in 2^{\xiibb S}$,
$\Re_{3}[F](S)=\Re_{4}[{\xiifrak u}](F)(S)$.

Comment Function $\xiifrak u$ is the identity for programs, see §4.2.3, or
functional identity, or evaluation, see §4.3.9. Function ${\xiifrak u}$ is
syntactic because it is not restricted to semantic objects, see §5.1.10.
Syntactic function ${\xiifrak u}$ is equivalent to $\lambda$-calculus
$I=(\lambda x.x)$, see §3.4.9: $\forall f\in{\xiibb S}\to{\xiibb S}$,
${\xiifrak u}(f)=f$ and $\forall s\in{\xiibb S}$, ${\xiifrak u}(f)(s)=f(s)$.

Proof By the rewriting rules in §3.1.3, $\Re_{4}[{\xiifrak
u}](F)(S)=\\{\,\Re_{4}[{\xiifrak u}](f)(s)\mid s\\!\in\\!S\times
f\\!\in\\!F\,\\}=\\{\,{\xiifrak u}(f)(s)\mid s\\!\in\\!S\times
f\\!\in\\!F\,\\}=\\{\,f(s)\mid s\\!\in\\!S\times
f\\!\in\\!F\,\\}=F(S)=\Re_{3}[F](S)$. $\diamond$

Corollary $\forall F\in 2^{\xiibb S\to S}$, $\Re_{3}[F]=\Re_{4}[{\xiifrak
u}](F)$.

§5.6.6 Lemma $\forall F\in 2^{\xiibb S\to S}$, $\forall S\in 2^{\xiibb S}$,
$\Xi\Re_{3}[F](S)=\Xi\Re_{4}[{\xiifrak u}](F)(S)$ and
$\Phi\Re_{3}[F](S)=\Phi\Re_{4}[{\xiifrak u}](F)(S)$.

Proof Because, by Lemma §5.6.5, $\Re_{3}[F](S)=\Re_{4}[{\xiifrak u}](F)(S)$.
$\diamond$

Definition The subject condition is satisfied if it implements the functional
identity ${\xiifrak u}$.

Comment If the subject condition holds, then the subject $\Re_{4}[{\xiifrak
u}](F)(S)$ solves any problem solved by the learner $\Re_{3}[F](S)$, and also
the subject $\Re_{4}[{\xiifrak u}](F)(S)$ resolves any problem resolved by the
learner $\Re_{3}[F](S)$.

Remark Functional identity $\xiifrak u$ is the ideal for reason.

§5.6.7 Corollary In particular, $\Xi\Re_{4}[{\xiifrak
u}](R)(S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})=\Xi\Re_{3}[R](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$
and $\Phi\Re_{4}[{\xiifrak
u}](R)(S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})=\Phi\Re_{3}[R](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$.

Proof By Lemma §5.6.6. $\diamond$

Comment Subject $\Re_{4}[{\xiifrak
u}](R)(S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$
is more flexible than learner
$\Re_{3}[R](S\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$,
because $R$ is software for the subject while it is hardware in the learner,
and software is cheaper than hardware, see §5.4.3.

§5.6.8 Theorem Subject $\Re_{4}[{\xiifrak u}]$ is a full resolution machine.

Proof By Theorem §4.2.1 and Lemma §4.2.3, $\Re_{4}[{\xiifrak u}]={\xiifrak
u}={\xiifrak c}({\cal U})$, so using the program isomorphism, see §4.1.12,
$\Re_{4}[{\xiifrak u}]={\cal U}$, which is a Turing complete device, and
therefore is a full resolution machine, by Theorem §4.4.2. $\diamond$

§5.6.9 Lemma Any effectively calculable resolution $\Re$ can be implemented by
the subject $\Re_{4}[{\xiifrak u}]$.

Proof By Theorem §5.6.8. $\diamond$

Corollary Any effectively calculable meta-analogy resolution ${\cal
A}\circ{T\\!}_{{\cal A}\Pi\pi}(R)\circ{\cal T}_{\cal A}$ can be implemented by
the subject $\Re_{4}[{\xiifrak u}]$, including metaresolving, see §3.5.11.

§5.6.10 Summary The subject, by internalizing metaproblems, prevents meta-
errors, that is, the subject can test internally a resolution before executing
it. The subject is also more flexible than the learner, because subject
modeling is done in software, instead of in hardware. And subject
$\Re_{4}[{\xiifrak u}]$ can reason about any model. This means that subject
$\Re_{4}[{\xiifrak u}]$ is a resolver that can calculate solutions, but also
problems and resolutions without limits; it can represent the problem it is
facing to itself, and it can represent itself to itself. In this sense, the
subject $\Re_{4}[{\xiifrak u}]$ is conscious.

Comment The subject is a body capable of several behaviors, a governor that
selects the current behavior, a simulator that internalizes behaviors, a
modeler that adjusts the model used by the simulator, and a reason that
internalizes resolutions.

Example It seems that only our species, Homo sapiens, is Turing complete. We
deal with the evolution to Turing completeness and its relation to language in
Casares (2016b).

§5.7 Resolvers Hierarchy

§5.7.1 Theorem There is a hierarchy of resolvers:
$\\{\cdot\,\Re_{0}\,\cdot\\}\subset\\{\cdot\,\Re_{1}\,\cdot\\}\subset\\{\cdot\,\Re_{2}\,\cdot\\}\subset\\{\cdot\,\Re_{3}\,\cdot\\}\subset\\{\cdot\,\Re_{4}\,\cdot\\}$.

Proof Because ${\xiibb S}\subset({\xiibb S}\to{\xiibb B})\subset({\xiibb
S}\to{\xiibb S})\subset(({\xiibb S}\to{\xiibb S})\to{\xiibb
B})\subset(({\xiibb S}\to{\xiibb S})\to({\xiibb S}\to{\xiibb S}))$, by Lemmas
§5.3.3, §5.4.4, §5.5.4, and §5.6.4. There are not more types of resolvers,
because there is not a resolver more capable than $\Re_{4}[{\xiifrak u}]={\cal
U}$, by Theorems §5.6.8 and §4.3.4. $\diamond$

§5.7.2 Summary This table groups concepts closely related from problem theory,
as trial, computing theory, as adapter $\Re_{1}$, and set theory, as
$S\in{\xiibb S}\to{\xiibb B}$.

$\halign{\strut\hfil#\quad\vrule& \space\hfil#\hfil\space\vrule&
\space\hfil#\hfil\space&\vrule\quad#\hfil\cr\hphantom{function}&\hphantom{${\xiifrak
f} \in({\xiibb S}\to{\xiibb S})\to({\xiibb S}\to{\xiibb S})$}&&\cr&{\xiibf
Semantics}&{\xiibf Syntax}&\cr\hrule\cr&Routine&Meta-routine&\cr
one&Mechanism$\Re_0$&Perceiver$\Re_2$&element\cr&$s \in{\xiibb S}$&$f
\in({\xiibb S}\to{\xiibb S})$&\cr\hrule\cr&Trial&Meta-trial&\cr
some&Adapter$\Re_1$&Learner$\Re_3$&set\cr&$S \in{\xiibb S}\to{\xiibb B}$&$F
\in({\xiibb S}\to{\xiibb S})\to{\xiibb B}$&\cr\hrule\cr&Analogy&Meta-
analogy&\cr any&Perceiver$\Re_2$&Subject$\Re_4$&function\cr&$f \in{\xiibb
S}\to{\xiibb S}$&${\xiifrak f} \in({\xiibb S}\to{\xiibb S})\to({\xiibb
S}\to{\xiibb S})$&\cr\hrule\cr&${\xiibb S}$&${\xiibb S}\to{\xiibb
S}$&\cr&Elements&Functions&\cr}$

Comment A perceiver is a syntactic mechanism. A learner is a syntactic
adapter. A subject is a syntactic perceiver. A subject is a syntactic2
mechanism.

§5.7.3 Theorem The problem theory is complete.

Proof Aside from definitions, the problem theory posits that there are three
ways to resolve a problem: routine, trial, and analogy; see §2.3. Adding the
metaproblem of the problem, we get five ways to resolve a problem and its
metaproblem, which are the basic three plus meta-trial and meta-analogy, see
§3.6.8. For each way there is a resolver, see Lemmas §5.2.5, §5.3.9, §5.4.9,
§5.5.11, and §5.6.9, and the resources of each resolver are in a series of
mathematical objects of increasing generality that covers everything until
syntactic functions, see §5.7.1 and §5.7.2. Now, to execute meta-analogies,
$2^{2^{\xiibb S}\to 2^{\xiibb S}}\to 2^{2^{\xiibb S}\to 2^{\xiibb S}}\\!\\!$,
see §3.6.9, or at least the elementable ones, see §5.4.8, we need subjects,
which implement syntactic functions $({\xiibb S}\to{\xiibb S})\to({\xiibb
S}\to{\xiibb S})$. And there is a subject that is a Turing complete device,
$\Re_{4}[{\xiifrak u}]$, see §5.6.8, so it has the maximum computing power,
see §4.3.4, and then the maximum resolving power, see §4.4.2. This means that
there are not more resolvers beyond the subject, and therefore that the series
is complete, and then that the problem theory covers everything and is
complete. $\diamond$

Comment It also means that no more resolutions are needed, although we could
do without routine, for example, by using Theorem §3.3.7, and then reducing
routines to trials. Nevertheless, a routine is not a trial, because a semantic
element is not a semantic set, or because a mechanism implementing a routine
is not an adapter implementing a trial, see comment to Lemma §5.3.4.

Comment This theorem is true if the Turing’s thesis is true, see §4.3.1.
Conversely, if this theorem is true, then ‘what is effectively calculable to
resolve problems is computable’.

§5.7.4 Remark Provided that a bigger range means more survival opportunities,
that software is cheaper than hardware, that the adapter, the perceiver, the
learner, and the subject conditions are satisfied in some environments, and
that in each step the increasing of complexity was overcome by its fitness,
then an evolution of resolvers —mechanism to adapter to perceiver to learner
to subject— should follow.

Comment Although depending on conditions, see Lemmas §5.3.6, §5.4.6, §5.5.8,
and §5.6.6, the evolution of resolvers is directed, and its final singularity
is the Turing complete subject $\Re_{4}[{\xiifrak u}]$.

Comment In detail, the strictest evolution of resolvers is:
$\Xi\Re_{0}[s]\mathrel{\smash{\buildrel\scriptscriptstyle\\{\\!s\\!\\}\subset
S\over{\subset}}}\Xi\Re_{1}[S\cap\Sigma_{\pi}]\mathrel{\smash{\buildrel\scriptscriptstyle
S\subset
S^{\prime}\\!\\!\over{\subset}}}\Xi\Re_{2}[i](S^{\prime}\\!\cap\Sigma_{\pi})\buildrel\scriptscriptstyle\\{\\!i\\!\\}\subset
R\over{\subseteq}\Xi\Re_{3}[R](S^{\prime}\\!\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})\buildrel\scriptscriptstyle
R\subset R^{\prime}\\!\\!\over{\subseteq}\Xi\Re_{4}[{\xiifrak
u}](R^{\prime})(S^{\prime}\\!\mathbin{\ooalign{\hfil$\scriptstyle\cap$\hfil\cr$\cap$\cr}}\Sigma_{\pi})$.

§5.7.5 Thesis We are the result of an evolution of resolvers of the survival
problem.

Argument The resolvers hierarchy suggests an evolution of resolvers of the
survival problem, see §5.7.4. And lacking of better explanations, that we are
Turing complete resolvers, that is, subjects $\Re_{4}[{\xiifrak u}]={\cal U}$,
see §4.2.8, suggests that we are indeed the result of an evolution of
resolvers of the survival problem.

Comment Our species is Turing complete. Therefore we must explain the
evolution of Turing completeness.

§6 Conclusion

§6.1 Purpose

The problem theory is the union of set theory and computing theory. The
integration of the two theories is achieved by using a new vocabulary to refer
to old concepts, but mainly by giving the old theories a purpose that they did
not have: to resolve problems. For example, a set defined by intension is
named a problem, and the same set defined by extension is named its set of
solutions. While both still refer to the same set, as it is the case in set
theory, the status of each of them is now very different: one is a question
and the other is an answer. And when the problem theory states that computing
is resolving, it is calling a set resolvable if it is recursively enumerable,
but mainly it is saying that the transition from intension to extension has to
be calculated, because it is not written magically in “The Book”; someone has
to write it.

The purpose of resolving problems is not final, but the main conclusion of the
paper, the Thesis §5.7.5, is nearly ultimate: We are Turing complete subjects
because we are the result of an evolution of resolvers of the survival
problem. In other words, we resolve problems to survive. So, if survival is
indeed the ultimate purpose, then the problem theory provides purpose and
meaning to set theory and to computing theory.

The final Thesis §5.7.5 also closes a loop, because a Turing complete resolver
$\Re_{4}[{\xiifrak u}]$ can model everything, and then everything can be a
solution, as it is stated in Theorem §3.2.1. But those everythings are not
absolute, but limited to what is computable, see §4.4.11. That is, if Turing’s
thesis stands, see §4.3.1, then everything is everything that is computable.
This way a restriction of computing theory, countability, is inherited by
problem theory and transferred to set theory; see the details below in
Subsection §6.2. The other question that requires some more elaboration is the
status of the Turing’s thesis itself, which we will postpone until Subsection
§6.3.

Nevertheless, besides that main Thesis §5.7.5, the problem theory concepts
presented in this paper can be used to model, understand, and classify both
natural and artificial resolvers, because the paper provides definitions,
theorems, and taxonomies for resolvers, and also for problems. And, by the
way, the paper defines adaptation, perception, and learning, and it shows that
there are just three ways to resolve any problem: routine, trial, and analogy.

§6.2 Countability

In computing everything is countable, see §4.1.14, and the problem theory in
Turing universes inherited countability from computing theory, see §4.3.16. In
a Turing universe, see §4.3.3, the limits of calculation are the limits of
computing, and then there are only computable functions, including predicates,
see §4.1.9, and computable sets, see §4.4.3. Then the problem theory in Turing
universes is consistent if and only if computing is consistent. And computing
is consistent, as a corollary to Church-Rosser theorem in $\lambda$-calculus,
see Curry & Feys (1958) Chapter 4.

Therefore, our way to control paradoxes in set theory, and then in this paper,
is to confine ourselves to Turing universes. But don’t worry; if this is a
Turing universe, as it seems to be, then we are only excluding imaginary
universes.

For example, the mathematical theorem that states that everything is a
solution is proved, and it makes sense, see §3.2.1. But it also causes
paradoxes, because from it we derive ${\xiibb P}\subset{\xiibb S}$, but
${\xiibb P}\buildrel\scriptscriptstyle 1\over{=}({\xiibb S}\to{\xiibb
B})\buildrel\scriptscriptstyle 2\over{=}2^{\xiibb S}$, and then $|{\xiibb
P}|\buildrel\scriptscriptstyle 3\over{=}|2^{\xiibb
S}|\buildrel\scriptscriptstyle 4\over{=}2^{|\xiibb S|}>|{\xiibb S}|$, by
Cantor’s theorem. It is not a paradox in a Turing universe because the forth
equality is false in it. The second equality is false in a Turing universe
because, as we saw in Lemma §4.4.8, there are resolvable problems that are not
expressible, so $({\xiibb S}\to{\xiibb
B})\buildrel\scriptscriptstyle\ast\over{\subset}2^{\xiibb S}$. The third
equality is true, though it follows the second one! And the forth equality is
false in a Turing universe because the number of computable sets is countable,
so, if $|{\xiibb S}^{*}|={\aleph_{0}}$, then $|2^{\xiibb
S}|^{*}={\aleph_{0}}<2^{\aleph_{0}}=2^{|{\xiibb S}^{*}|}$, that is,
$|2^{\xiibb S}|\buildrel\scriptscriptstyle\ast\over{<}2^{|{\xiibb S}|}$.
Therefore, ${\xiibb P}^{*}\\!\subset{\xiibb S}^{*}$ is true, but ${\xiibb
P}^{*}$ is the set of computable predicates, that is, ${\xiibb P}^{*}={\cal
E}$ of §4.5.8, and $[2^{\xiibb S}]^{*}\subset{\xiibb S}^{*}$ is also true, but
$[2^{\xiibb S}]^{*}$ is the set of computable sets, that is, $[2^{\xiibb
S}]^{*}={\cal R}$ of §4.5.8. The conclusion is that ${\xiibb S}^{*}$, the set
of solutions, is the set of everything that is computable.

We have just rejected the uncountable case, where $|{\xiibb P}|>{\aleph_{0}}$,
but there are two other possibilities: the (infinite) countable case, where
$|{\xiibb S}^{*}|=|{\xiibb P}^{*}|=|{\xiibb R}^{*}|={\aleph_{0}}$, see
§4.3.16; and the finite case, where $|{\xiibb S}_{\Gamma}|<|{\xiibb
P}_{\Gamma}|<|{\xiibb R}_{\Gamma}|<{\aleph_{0}}$, see §5.1.7.

We are finite, so it would be natural to restrict our investigations to the
finite case, calling for finiteness instead of calling for countableness. But
the finite case is trivial, and more importantly, the difference between an
unrestricted universal computer and a finite universal computer is not
qualitative but quantitative. There is not any step of any calculation that an
unrestricted universal computer can compute and a finite universal computer
cannot compute, see §4.2.10. So in the limit, that is, without time nor memory
restrictions, we are universal computers. And note that those restrictions are
variable, and that they can be relaxed nearly as desired just spending some
more time, or building a faster computer machine, or using some more external
memory. In the case of a Turing machine, the external memory is the tape, and
the internal memory is where its processor keeps the internal state, see
§4.1.3. Note also that we can code a program to generate every natural number,
although we cannot follow the computation till its end. Summarizing: we are
better defined saying that we are qualitatively universal Turing machines, but
with some unspecified quantitative limitations, than saying that we are
qualitatively finite state automata, because finite state automata are not
expandable.

Finally, the rejections of finiteness and uncountableness imply that
countableness is the golden mean. This is Pythagorean heaven revisited,
everything is countable, but this time we have rescued the terrifying
$\sqrt{2}$, and other irrational numbers. As Kronecker said: “God made
counting numbers; all else is the work of man”.

§6.3 Intuition

Is it possible to resolve a non-computable problem? A problem is computable
if, by definition, see §4.4.3 and §4.4.4, a Turing machine can execute a valid
resolution of the problem, so the non-computable problem would not be resolved
by computing, but by other means. My answer to the question is ‘no’, because I
think that a problem is resolvable if, and only if, the problem is computable,
see §4.3.12.

Nevertheless you may think otherwise, and say that there is another way of
resolving, let us call it ‘intuition’, that is not computable. If that were
the case, then the problem theory with its mathematical formulation, as
presented in this paper, would capture the concept of ‘computable problem’,
but not the whole concept of ‘problem’. In order to see this, please consider
the following two statements:

$\circ$ Some problems are computable.

$\circ$ A universal computer can execute any computable resolution.

Even if you believe that there are resolvable problems that are not
computable, you can still decide easily that both are true; the first is a
fact, and the second is a theorem. And then everything in this paper would
still be true of computable problems, computable resolutions, and computable
solutions.

The key point in this discussion is that ‘intuition’ would refute Turing’s
thesis, see §4.3.1, because if there were ‘intuitive’ resolutions, then we
could effectively calculate what is not computable. Turing’s thesis is not a
theorem, and we follow Post (1936) in considering Turing’s thesis to be a law
of nature that states a limitation of our own species calculating capacity,
see Casares (2016a), by which we are bound to see ourselves as a final
singularity. Summarizing: If Turing’s thesis were eventually false, then this
problem theory would be about computable problems. But, while Turing’s thesis
remains valid, the problem theory is about problems, the set of effectively
calculable functions is countable (§4.3.1), universal computers are the most
capable computing devices (§4.3.4), everything is an expression (§4.3.10),
resolving is computing (§4.3.12), and the problem theory is complete (§5.7.3).

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