Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?


I have long been intrigued by the observation that much of mathematics can be defined as properties of recursive processes or Turing Machines. One can construct the arithmetic hierarchy by all finite iterations of the question: does a TM have an infinite number of outputs such that an infinite subset of these outputs are Gödel numbers of TMs with an an infinite number of outputs? One can construct the hyperarithmetic hierarchy by iterating this question up to all recursive ordinals. One can go farther by asking does a TM which accepts an arbitrarily long sequence of integer inputs halt for every possible infinite sequence?


These are all questions that are in some sense logically determined by a recursively enumerable sequence of events: all the possible paths a TM can take with any possible finite sequence of inputs. *Can one define a subset of the integers that cannot be defined in this way?* Saying that a set is the even integers if the continuum hypothesis is true and the odd integers otherwise does not count since both alternatives are definable as properties of a TM. On the other hand giving a recursively enumerable sequence of statements whose truth values cannot be encoded as a property of recursive processes would count.


Of course the notion of property of a TM is indefinitely expandable. but it must be limited to a process that depends only on a recursively enumerable sequence of events.



Is there a well defined subset of the integers that cannot be defined as a property of a recursive process or Turing Machine?
Over at [my answer to I. J. Kennedy's question about degrees of irrationality](https://mathoverflow.net/questions/53724/are-some-numbers-more-irrational-than-others/53754#53754), I described several hierarchies of definable complexity that transcend computability. I have copied my answer below. Already beginning with the low levels of the arithmetic hierarchy, one transcends the computable, while retaining a formal definability outlook. Here are a few very specific sets of natural numbers that cannot be accounted for by computation:


* the set of all true statements in the language of arithmetic. This has complexity $\Sigma\_\omega$, just beyond the arithmetic hierarchy.
* the set of all Turing machine programs that compute a well-ordered relation on $\mathbb{N}$. This set has complexity complete $\Pi^1\_1$ in the projective hiearchy, well beyond what can be computed by Turing machines.
* the set of all statements true in the realm of the hereditarily countable sets $\text{HC}=\langle H\_{\omega\_1},{\in}\rangle$. This is the complexity of projective truth, outside and above the projective hierarchy.
* the set of natural numbers $n$ such that $2^{\aleph\_n}=\aleph\_{n+1}$. This set of natural numbers is defined by a $\Sigma\_2$ assertion of set theory (not in the arithmetic hierarchy, but the Levy hierarchy). It cannot be expressed as a statement about computation, since the particular contents of the set can be changed by forcing, and forcing does not affect the nature of any computation. Perhaps you will object that it is like your CH example, but it isn't of that binary form, and it would seem hard to say that this isn't a legitimate definition of a set of natural numbers that could be defined in some other way.
* there are many others...


I notice in your question that you mention the arithmetic and hyperarithmetic hierarchies, but I disagree with your characterization of them as being statements concerning properties of computation. Although it is true that one can iterate a $\Sigma\_1$ process through the ordinals to arrive at the hyperarithmetic sets, it is the complexity of the ordinals that drives this hierarchy, rather than the comparatively trivial use of computation at each step. Similarly, although one could say that projective statements are statements about Turing machines, provided that one is allowed to quantify over integers and reals (one just writes the quantifier-free part of the assertion in terms of computation), this would seem to me to miss the point, for it is the quantification over reals that matters, and not the fact that one writes the quantifier-free part in a way involving computation. Ultimately, therefore, it seems to me that the answer to the ideas in your question consists of a careful appreciation of the nature of the various definability hierarchies.


Allow me to give a little more information by mentioning that there are several
very large, intensely studied hierarchies of complexity for
real numbers. After the initial familiar notions come
several others...


* rational
* algebraic
* computable


The computable reals are those for which we can compute
rational approximations to any desired accuracy, by Turing
machine. (A concept used in [computable
analysis](https://en.wikipedia.org/wiki/Computable_analysis).)
The computable subsets of $\mathbb{N}$ are those for which
we can compute *yes/no* answers for membership in finite
time. For example, all the numbers you mention in the
question, such as $\pi$ and $e$, are computable.


* computably enumerable


The c.e. subsets of $\mathbb{N}$ are those for which there
is a computable enumeration procedure. Equivalently, you
can compute the *yes* answers for membership in finite
time. The concept of relative (oracle) computability leads to the
[hierarchy of Turing
degrees](https://en.wikipedia.org/wiki/Turing_degrees),
which measures the comparative computable complexity of a
real.


* arithmetic


A real $x$ is *arithmetic* if its digits can be defined by
a definition involving only quantification over the natural
numbers and primitive operations. Equivalently, the
arithmetic subsets of $\mathbb{N}$ arise from the
computable subsets of $\mathbb{N}^k$ by projection and
complement. The [arithmetic
hierarchy](https://en.wikipedia.org/wiki/Arithmetic_hierarchy)
breaks naturally into levels, such as $\Sigma^0\_n$ and
$\Pi^0\_n$, corresponding to the logical complexity of these
definitions, and these levels are refined by the Turing
degrees. For example, the set of Turing machine programs
$p$ which compute total functions forms a complete
$\Pi^0\_2$ set. The relativized notion leads to the arithmetic degrees.


* hyperarithmetic


A real is *hyperarithmetic* if it can be defined by two
equivalent definitions, one involving just one universal
quantifier over the reals and another having just one
existential quantifier over the reals, and otherwise any
level of arithmetic quantifiers. This is the same as
$\Delta^1\_1$. The [hyperarithmetic
hierarchy](https://en.wikipedia.org/wiki/Hyperarithmetical_theory)
is stratified in a hierarchy of length $\omega\_1^{CK}$, a
lightface version of the Borel hierarchy, in which one uses
uniformly computable countable unions and complements. The
relativized notion leads to the hyperarithmetic degrees, a
hyperarithmetic analogue of the Turing degrees.


* projective


A real is *projective* if it can be defined by a
description that quantifies only over the set of real
numbers, plus natural number quantification and the
primitive operations. The [projective
hierarchy](https://en.wikipedia.org/wiki/Projective_hierarchy)
is stratified by considering the logical complexity of
these definitions, with levels $\Sigma^1\_n$ and $\Pi^1\_n$.
For example, the lightface analytic sets are $\Sigma^1\_1$
and co-analytic is $\Pi^1\_1$, with hyperarithmetic being
$\Delta^1\_1=\Sigma^1\_1\cap\Pi^1\_1$.


* constructible


A real is *constructible* if it exists in Gödel's
[constructible universe
$L$](https://en.wikipedia.org/wiki/Constructible_universe).
The concept of relative constructibility gives rise to the
constructibility degrees, by which $x\sim y\leftrightarrow
L[x]=L[y]$, forming a rich hierarchy.


* ordinal-definable


A real (or set) is
[*ordinal-definable*](https://en.wikipedia.org/wiki/Ordinal_definable)
if there is a definition of it in the language of set
theory, using ordinal parameters. For example, the real
whose $n^{th}$ binary digit is $1$ just in case
$2^{\aleph\_n}=\aleph\_{n+1}$ is ordinal definable. The class
HOD of all hereditarily ordinal definable sets satisfies
ZFC, but can be strictly smaller than the universe of all
sets.


* generic


A real is *generic* over $L$ (or some other fixed universe
$V$) if it exists in a forcing extension of $L$ (or $V$) by
set forcing. Of course, it is relatively consistent with
ZFC that every real is generic over $L$, since this is true
in $L$ itself, but under some large cardinal axioms, there
are reals, such as $0^\sharp$, that cannot be added by
forcing over $L$.


The higher levels of these latter hierarchies are further
developed and stratified by the enormous variety of models
of set theory arising from large cardinals, various inner
model constructions, forcing extensions and so on, so that
the hierarchy loses its linear nature, becoming instead a
dense jungle of various interacting concepts of set theory.


