#  Detectability of Solar Panels as a Technosignature

Ravi Kopparapu NASA Goddard Space Flight Center, 8800 Greenbelt Road,
Greenbelt, MD 20771, USA  SEEC, Sellers Exoplanet Environment Collaboration,
NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771,
USA  CHAMPs, Consortium on Habitability and Atmospheres of M-dwarf Planets
Vincent Kofman NASA Goddard Space Flight Center, 8800 Greenbelt Road,
Greenbelt, MD 20771, USA  SEEC, Sellers Exoplanet Environment Collaboration,
NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771,
USA  American University, 4400 Massachusetts Avenue, NW, Washington, DC,
20016, USA  Jacob Haqq-Misra Blue Marble Space Institute of Science, Seattle,
WA, USA  CHAMPs, Consortium on Habitability and Atmospheres of M-dwarf Planets
Vivaswan Kopparapu Atholton High School, Columbia, MD 21044, USA  Manasvi
Lingam Department of Aerospace, Physics and Space Sciences, Florida Institute
of Technology, Melbourne, FL 32901, USA

###### Abstract

In this work, we assess the potential detectability of solar panels made of
silicon on an Earth-like exoplanet as a potential technosignature. Silicon-
based photovoltaic cells have high reflectance in the UV-VIS and in the near-
IR, within the wavelength range of a space-based flagship mission concept like
the Habitable Worlds Observatory (HWO). Assuming that only solar energy is
used to provide the 2022 human energy needs with a land cover of $\sim 2.4\%$,
and projecting the future energy demand assuming various growth-rate
scenarios, we assess the detectability with an $8$ m HWO-like telescope.
Assuming the most favorable viewing orientation, and focusing on the strong
absorption edge in the ultraviolet-to-visible ($0.34-0.52$ $\mu$m), we find
that several 100s of hours of observation time is needed to reach a SNR of 5
for an Earth-like planet around a Sun-like star at 10pc, even with a solar
panel coverage of $\sim 23\%$ land coverage of a future Earth. We discuss the
necessity of concepts like Kardeshev Type I/II civilizations and Dyson
spheres, which would aim to harness vast amounts of energy. Even with much
larger populations than today, the total energy use of human civilization
would be orders of magnitude below the threshold for causing direct thermal
heating or reaching the scale of a Kardashev Type I civilization. Any
extraterrrestrial civilization that likewise achieves sustainable population
levels may also find a limit on its need to expand, which suggests that a
galaxy-spanning civilization as imagined in the Fermi paradox may not exist.

Exoplanet atmospheric composition, technosignatures

††journal: ApJ

##  1 Introduction

The search for extraterrestrial life has primarily focused on detecting
biosignatures, which are remote observations of atmospheric or ground-based
spectral features that indicate signs of life on an exoplanet. More recently,
“technosignatures” referring to any observational manifestations of
extraterrestrial technology that could be detected or inferred through
astronomical searches has received increased attention (Tarter, 2007). While
the search for extra-terrestrial intelligence (SETI) through radio
observations has been popular for decades, recent studies have proposed
alternate searches for technosignatures in the UV to mid-infrared part of the
spectrum: see NASA Technosignatures Workshop Participants (2018); Lingam &
Loeb (2019, 2021); Socas-Navarro et al. (2021); Haqq-Misra et al. (2022c) for
a comprehensive description.

Specifically, methods to detect technosignatures through spectral signatures
from exoplanets have been proposed as a means to utilize existing techniques
and telescope facilities. These include nitrogen dioxide (NO2) pollution
(Kopparapu et al., 2021), fluorinated compounds such as chloroflorocarbons
(CFCs, Owen 1980; Lin et al. 2014; Haqq-Misra et al. 2022b), or nitrogen
trifluoride (NF3) and sulfur hexafluoride (SF6) (Seager et al., 2023), night-
side city lights (Beatty, 2022) and agricultural signatures on exoplanets
(Haqq-Misra et al., 2022a). In this work, we focus on another potential
technosignature: silicon solar panels.

Technological civilizations may harness their host star’s radiation for their
energy needs, just like our civilization has commenced with large scale solar
photovoltaics. Most solar cells use silicon in different forms. Lingam & Loeb
(2017) outlined three primary motivations for employing silicon-based solar
panels, which might be broadly applicable. The first is the relatively high
cosmic abundance of silicon compared to the elements utilized in other types
of photovoltaics such as germanium, gallium, or arsenic. Second, the
electronic structure of silicon (specifically its band gap) is well-suited for
harnessing the radiation emitted by Sun-like stars (Rühle, 2016). Third,
silicon is also cost effective in terms of refining, processing and
manufacturing solar cells (Bazilian et al.,
2013).111https://www.energy.gov/eere/solar/solar-photovoltaic-cell-basics

Based on these arguments, Lingam & Loeb (2017) suggest that the existence of
large-scale silicon solar cells could produce artificial spectral “edges” in
some UV wavelength bands when observing the atmosphere of an exoplanet in
reflected spectroscopy because of the steep change in the reflectance of
silicon. This artificial spectral edge may be similar to the vegetation red
“edge” (VRE) seen between $0.70-0.75\,\mu$m that can be noticed in the
reflected light spectrum of Earth (Sagan et al., 1993; Arnold et al., 2002;
Woolf et al., 2002). The “edge” refers to the noticeable increase in the
reflectance of the material under consideration when a reflected light
spectrum is taken of the planet. In the VRE case, the high reflectance arises
due to the contrast between chlorophyll absorption at red wavelengths
($0.65–0.70\,\mu$m) and the scattering properties of the cellular and leaf
structures at NIR wavelengths ($0.75-1.1\,\mu$m): see, for example, Seager et
al. (2005); Turnbull et al. (2006); Schwieterman et al. (2018); O’Malley-James
& Kaltenegger (2018) for more details. Detecting VRE on an exoplanet would
provide contextual information about the type of widespread biological life
(e.g., autotrophy), and corresponding atmospheric properties relevant for
habitability. Lingam & Loeb (2017) suggest that a similar artificial edge, if
manifested, could provide some contextual information about the kind of
technological activity on a planet.

Could we detect surface reflectance features of solar panels on exoplanets as
technosignatures? While Lingam & Loeb (2017) suggested this possibility, they
did not conduct any quantitative assessment of their detectablity. In this
work, we will consider the detectability of solar panels on an Earth-like
planet around a Sun-like star with a LUVOIR-B (8 meter) class space telescope.
The paper is structured as following: §2 discusses the methods and models used
in this work. In §3, we estimate the area of land that would need to be
covered to provide human civilization with its energy needs today and in
future scenarios. The potential detectability is then assessed in §4. A
discussion section and a summary of our findings follows in §5.

##  2 Methods

The methods to assess the detectability of photovoltaics as a signature for
the presence of advanced civilizations are described here. In order to
constrain the spectral signal, the following needs to be assessed: 1) The
reflectivity of photovoltaics panels. 2) The panels need to be included in a
suitable location on an Earth-ground map 3) The spectroscopic signal from the
panels need to be compared to simulations without the panels, and the signal-
to-noise that can be attributed to the panels should be computed.

###  2.1 Silicon and the reflectivity as as spectral signature

Pure silicon is not as well-suited (as a material) to be used in a
photovoltaic cell since it is highly reflective in the ultraviolet-to-visible
range. As electric energy is generated by absorbing a photon to promote an
electron across the PN junction, any light that is reflected leads to a
reduction in efficiency. To minimize reflection of light, photovoltaic cells
are either subjected to texturing (Campbell & Green, 1987; Macdonald et al.,
2004; Kim et al., 2020) or coated in anti-reflective coatings, often TiO2 or
Si3N4 (Zhao & Green, 1991; Raut et al., 2011; Sarkın et al., 2020); the
coating results in the typical dark color seen on solar panels.

In the scenario with the above anti-reflective coating, the artificial edge is
still apparent, but less pronounced and deeper in the ultraviolet (with
respect to pure silicon), when realistic materials are assumed for
photovoltaic cells. For this work, the reflectance spectrum shown in Fig. 2 is
adopted. This explains a major source of divergence between Lingam & Loeb
(2017) and our work, because the former emphasized greater surface coverage by
pure silicon solar panels that could compensate for reduced efficiency,
whereas we consider potentially more realistic photovoltaic cells endowed with
higher efficiency, thereby warranting lower coverage.

###  2.2 Generating the surface model containing solar panels

Based on the estimated surface area required for the current energy use
discussed in §3, a ground map is generated that hosts roughly 2.4% of land
coverage. The Sahara desert was fiducially chosen to host the solar panels.
This region is both close to the equator, where a comparatively greater amount
of solar energy would be available throughout the year, and has minimal cloud
coverage. However, dust storms are also prevalent, and have been increasing in
frequency over the past four decades (see Fig.4, Varga 2020). Average events
are $\sim 20$ per year with varying severity. Such events may reduce the
available sunlight, further restricting the energy generated. We recognize and
caution that no significant area of the Earth is uninhabited, and even the
placement of solar farms in seemingly barren deserts has been contentious due
to the destruction of the extremely fragile ecosystems that may consequently
arise. However, our goal in this work is to assess the detectability of solar
panels on an exoplanet, and as such, the most “optimal” land location in term
of solar energy generation is chosen for this purpose.

The ground surface model that is used for this study is based on the data
products from the moderate resolution imaging spectroradiometer (MODIS), which
is hosted on both the Terra and AQUA satellites operated by NASA Goddard Space
Flight Center222https://modis.gsfc.nasa.gov/about/. The MODIS-MCD12C1 maps
provide yearly average coverage at high spatial resolution of 18 different
land types across the entire planet. For this paper, ground coverage is
reduced to 5 different types: ocean, snow/ice, grass, forest, and bare soil.
The spatial resolution is binned down to 2.5 by 2 degree (longitude/latitude).
The ground albedo, or the fraction of light that is reflected as a function of
wavelength for the different surface types, are adopted from the United States
Geological Survey database (Kokaly et al., 2017). Subsequently, solar panels
are added as a sixth category, which are described with the reflectivity from
RELAB (Reflectance Experiments Laboratory (Pieters & Hiroi, 2004), and placed
on the surface map at the chosen locations.

###  2.3 Assessing the contribution to the reflectivity from silicon

In order for a spectral feature to be detectable, it needs to satisfy two
conditions. It has to be sufficiently strong, and its spectral signal needs to
uniquely identifiable with the source molecule or in this case, ground
coverage. The detectability of photovoltaic panels was proposed to be in the
UV-VIS region of the spectrum, which is the range where the spectral feature
of the panels are most uniquely identifiable. The infrared region is not
suitable because the difference in the reflectivity here does not correspond
to a spectrally unique feature: the features overlap with much stronger
signals from the other surface components. This study is constructed so as to
focus on the maximum detectability of the technosignature. This does not
indicate that the signal may be fully uniquely attributed to the panels, which
would require a follow-up study utilizing retrieval methods and an exhaustive
search for possible overlapping signals (i.e. Rayleigh scattering, absorption
by O3 or hazes). The first step in assessing the detectability is finding the
potential signal strength in the most optimistic case, which is pursued here.

It should also be noted that the placement of the panels in the desert is
fortunate in terms of the detectability as well, because from the ground
coverage components shown in Fig.2, the contrast exhibited with soil is the
second highest (after snow/ice).

Figure 1:  Figure 2:  Reflectance of different surface components on a planet
as a function of wavelength. The reflectance of silicon-based photovoltaics is
shown as dashed curve. The ground model is based on the data from MODIS
dataset https://modis.gsfc.nasa.gov/about/. The solar cell reflectivity data
is from RELAB (Reflectance Experiments Laboratory (Pieters & Hiroi, 2004).

##  3 Photovoltaic Requirements for Earth

At present, the power density (i.e., power generated per unit ground area) of
solar energy is estimated to be $5.4$ W m-2(Miller & Keith, 2018). Ignoring
the effect on a habitable planet environment, if the total land area of
Earth333https://ourworldindata.org/land-use is 149 $\times 10^{6}$ km2, then
the total power that could be generated if all the land were covered by solar
panels would be $5.4\text{\,W\,m}^{-2}\times 149\times
10^{6}\text{\,km}^{2}=804$ Tera Watts, or 25,374 exajoules per year. In 2022,
the world power consumption from all primary energy sources (including
commercially-traded fuels and modern renewables used to generate electricity)
was 604 exajoules.444Page 8, “Primary energy consumption”,
https://www.energyinst.org/statistical-review Clearly, all land need not be
covered: only $\sim 2.4\%$ of land coverage by solar panels would be needed to
match the world energy consumption in 2022.

Figure 4 shows historic annual world energy use in Joules, and projected
energy usage under various growth-rate scenarios. This plot is similar to the
growth rate figure shown in Mullan & Haqq-Misra (2019), where the authors
discussed the implications of population growth as related to the energy
usage. Two data sets are shown: one from the Organization for Economic
Cooperation and Development (OECD) data555https://www.oecd-
ilibrary.org/energy/primary-energy-supply/indicator/english_1b33c15a-en from
1850–2015, and another curve from the Energy Institute666Page 8, “Primary
energy consumption” https://www.energyinst.org/statistical-review from
1965–2022. The purpose of representing both datasets is to show that the two
datasets generally agree, apart from a small systematic difference. Projected
energy usage in 2030 is based on estimates from the US Energy Information
Administration777https://www.eia.gov/outlooks/archive/ieo09/world.html
(678$\times 10^{15}$ Btu $\approx$ 715 exajoules), and the corresponding land
coverage needed ($2.8\%$) to power the world entirely on solar panels is also
shown.

Figure 3:  Figure 4: Historic and projected annual world energy use in Joules
(left axis) and in the corresponding fraction of Earth’s land coverage by
solar panels that would be required (right axis). Historic data are from the
OECD (1850–2015, grey) and Energy Institute (1965-2022, black). The historic
2022 value is shown in an open circle and a projection for 2030 from the US
Energy Information Administration is shown as a filled circle. Colored lines
show future projections with constant growth at 7% yr-1 (green), 2.6% yr-1
(yellow), 2% yr-1 (blue), and 1% yr-1 (red). A hypothetical scenario of 23%
land coverage by solar panels is shown as open squares on each of these four
future projections. An additional Toy logistic projection (violet) follows a
1% yr-1 growth rate up to the maximum energy use required to sustain a
population of 30 billion people at a high standard of living, with the maximum
value of the logistic function shown as an open triangle. Horizontal dashed
grey lines indicate threshold values for causing direct heating of Earth’s
atmosphere ($\sim 10^{23}$ and reaching the Kardashev Type I limit (i.e.
harvesting all available solar energy). Calculations assume a fixed $5.4$ W
m-2 solar panel power density.

Future projections are shown in Figure 4 by assuming scenarios of constant
growth of 7% yr-1, 2.6% yr-1, 2% yr-1, and 1% yr-1 based on the 2022 world
energy consumption estimate of 604 exajoules from the Energy Institute, and
assuming a fixed solar power density value of 5.4 W m-2. The historical period
from 1850–2022 shows an average growth rate of about 2.6% yr-1, although this
growth rate decreased to about 2% yr-1 during the more recent 1965–2022
period. A growth rate of 7% yr-1 was used by Von Hoerner (1975) in a previous
analysis of limits to growth; such a projection was consistent with the more
rapid growth rate observed from the period of about 1950–1975. These
scenarios, along with a more conservative 1% yr-1 scenario and a toy logistic
scenario, serve to illustrate the range and inherent uncertainties in making
such projections about the future.

Figure 4 also indicates the point on each projection where the energy demands
would require a solar panel coverage of 23% of Earth’s land—about the size of
Africa—with an annual energy use of 5840 exajoules. This magnitude of energy
use would occur by 2056 for the 7% yr-1 scenario, by 2110 for the 2.6% yr-1
scenario, by 2137 for the 2% yr-1 scenario, and by 2250 for the 1% yr-1
scenario. The idea of covering such a large fraction of Earth’s surface with
solar panels would inevitably have many undesirable consequences on climate
and local environments; nevertheless, this 23% land coverage limit will be
used as an upper limit in the detectability calculations in §4.

It is also worth noting that the energy output from 23% land coverage would
far surpass that required to provide all people with a high standard of
living. The analysis by Jackson et al. (2022) found that human well-being may
peak at a per capita energy use of 75 gigajoules per person,888Many other
studies have also concluded that an annual energy consumption of $\lesssim
100$ GJ per person should suffice to achieve a high standard of living (Arto
et al., 2016; Millward-Hopkins et al., 2020; Vogel et al., 2021). so the 5840
exajoules generated by 23% land coverage would be more than adequate to
sustain a population of 10 billion people (corresponding to 750 exajoules, or
3% land coverage)—which is approximately the maximum human population
predicted by most United Nations models. For the purpose of illustration in
this study, the energy use of a human population of 30 billion people (2250
exajoules, 8.9% land coverage) is used as the upper bound of the toy logistic
function in Figure 4. This toy logistic function serves to demonstrate the
possibility of a nonlinear trajectory in future energy use, which may even
reach a stable equilibrium. For human civilization today, it is encouraging to
consider the possibility that 8.9% solar panel coverage of Earth’s
land—approximately the size of China and India combined—would be more than
enough to provide a high quality of life in the future. The implications of
this possibility for technosignatures will be discussed in §5.

##  4 Detectability Requirements for Photovoltaics

We performed simulations with the Planetary Spectrum Generator (PSG:
Villanueva et al. 2018, 2022) to generate reflected light spectra, and then
calculated the required signal-to-noise ratio (SNR) to detect the signature of
solar panels in the $0.34\,\mu$m–$0.52\,\mu$m range, which overlaps with the
UV and optical spectral region for HWO. We selected this particular region
because of the following reason: Fig. 2 shows that the reflectance of Silicon
solar panel has peaks below $\sim 0.4\,\mu$m and between 1-2$\,\mu$m. However,
these signatures strongly overlap with features from other surface materials.
Any change in the coverage of solar panels will be obscured by changes in the
relative amount of ocean versus soil or vegetation coverage, as can be see in
Fig. 5. Fig. 5a shows the spectral contrast ratio versus wavelength for
different viewing angles from the point of view of an observer. The contrast
ratio is greater than 1 because we are presuming a coronagraph which blocks
most of the starlight. The changes in the spectra beyond 0.8$\,\mu$m are
predominantly changes in the reflectivity of the ground coverage. Any
contribution from the silicon is blended into the overall spectra in this
wavelength region; note that the overlap of the silicon features with
molecular absorption features is not the focus of our investigation. Our
analysis provides an upper limit to the signal that may originate from
photovoltaic panels. The potential overlap with molecular features would be
the next step in the investigation in case the signal would be detectable.

Fig. 5b shows the SNR values of different land coverage of solar panels as a
function of observation time for a 8m HWO-like telescope. The model instrument
set is similar to the LUVOIR concept study telescope, as the design of HWO
mission is currently ongoing. As both LUVOIR-B and the HWO are concepts for
off-axis telescopes, the LUVOIR-B concept is the current best approximation.
It has an internal coronagraph with the key goal of direct exoplanet
observations (Juanola-Parramon et al., 2022). It is equipped with three
channels: NUV (0.2–0.525 $\mu$m), visible (0.515–1.030 $\mu$m) and NIR
(1.0–2.0 $\mu$m). The NUV channel is capable of high-contrast imaging only,
with an effective spectral resolution of R$\sim 7$. The optical channel
contains an imaging camera and integral field spectrograph (IFS) with R=140.
The planet is kept at a quadrature phase (in our simulation, an orbital phase
angle of $270^{\circ}$). This is an Earth-like planet at an Earth-like
distance around a Sun-like star at distance of $10$ pc.

The wavelength dependent SNR is calculated as the difference between the
spectra with and without the solar panels divided by the noise simulated by
PSG for the instrument under consideration (see section 5.3 of Villanueva et
al. (2018), chapter 8 in Villanueva et al. (2022), and also the PSG
website,999https://psg.gsfc.nasa.gov/helpmodel.php where the noise model is
discussed in detail). The “net SNR” is calculated by summing the squares of
the individual SNRs at each wavelength within a given band (either NUV or
VIS), and then taking the square root. This methodology is largely robust to
SNR, as long as the feature is resolved by the spectrum. The SNR is calculated
at different longitudinal views of an Earth-like planet from observer’s view
point. Only three longitudinal views are shown for illustrative purpose. A
$0^{\circ}$ longitude represents a viewing angle where the placement of the
solar cells are only visible partially from the observer’s view point. A 315∘
longitudinal view indicates the placement of solar cells in almost the full
view of the observer. A $90^{\circ}$ longitude represents a viewing angle
where solar cells are not visible to the observer.

Fig. 5b indicates that even with the most ambitious land coverage ($\approx
23\%$), and with a favorable viewing perspective to the observer (planet
longitude of $315^{\circ}$), it would take several hundred hours of
observation time in reflected light spectra with an 8 m size telescope to
reach SNR $\sim 5$ (solid purple curve). This result can be understood from
inspecting Fig. 7, where the spectra in planet-star contrast ratio are plotted
for three cases: $2.4\%$ (blue solid), $23\%$ (red solid) and zero (dashed
green) land coverage of silicon panels. Because we have chosen the
$0.34\,\mu$m–$0.52\,\mu$m range to calculate the impact of silicon panels on
the reflectance spectra, the difference between a planet with and without
silicon is not markedly different, even with $23\%$ land cover, as can be seen
from Fig. 7. This suggests that the artificial silicon edge suggested by
Lingam & Loeb (2017) may not be detectable. The discrepancy partly stems from
the choice of photovoltaic cell properties, as indicated in Section 2.1,
because the reflectance of realistic solar cells is less pronounced than pure
silicon, the latter of which was evaluated in Lingam & Loeb (2017). A larger
coverage than $23\%$ would result in slightly lower observation times;
however, refer to Section 5 for the implications of larger coverage by
photovoltaic cells.

Figure 5: (a) Reflectance spectra (expressed in terms of the planet to star
contrast ratio) for different planetary orientations with respect to the
observer for a $23\%$ land cover of solar panels. We choose the
$0.34-0.52\mu$m range to estimate the detectability of Si-based solar panels,
as discussed in §4 and shown in Fig.2. Within this range, there is a
noticeable difference in the spectra, which contributes to higher SNR for an
observing longitude of $315^{\circ}$ (fully visible panel coverage). (b) SNR
of detecting silicon solar panels as a function of observation time with a 8m
class LUVOIR-B-like telescope for different orientations of the planet towards
the observer. A longitudinal orientation of $315^{\circ}$ yields a higher SNR
because the solar panels on the planet are better oriented towards the
observer, as compared to a longitude of $90^{\circ}$ (partially visible).
Figure 6:  Figure 7: Planet-star contrast ratio as a function of wavelength
for $2.4\%$ (blue solid), $23\%$ (red solid) and $0\%$ (green dashed) land
coverage of solar panels. The planet longitudinal orientation is
$315^{\circ}$, where the solar panels are oriented towards the observer. Only
the $23\%$ land cover displays any significant variation in the spectrum. This
trend is also manifested in the SNR curve of Fig.5, where the maximum SNR is
obtained for a $23\%$ land cover oriented at $315^{\circ}$ planet longitude.

##  5 Discussion

The results from the previous section seem to indicate that even ambitious
deployment of solar panels (constructed with silicon and hosting anti-
reflective coatings) covering a significant land area of an Earth-like
exoplanet may not be enough to be detectable with a 8 m HWO-like telescope.
These calculations presumed a fixed-present day efficiency for solar panels,
so any technological improvements in efficiency would only decrease the
required land coverage, and consequently decrease detectability.

Lingam & Loeb (2017) conjectured that tidally locked planets around M-dwarfs
might be a suitable place to start looking for photovoltaic cells.
Unfortunately, due to the close proximity of planets in the habitable zone
around M-dwarfs – which poses issue for spatially resolving the planet –
currently there is no technological pathway to directly observing these types
of planets. In the case that this setup would be possible though, the
detectability of the silicon UV edge on planets around M-dwarfs would be
further diminished, since the stellar emission in the relevant range
($0.34-0.52\,\mu$m) is significantly lower.

Note that we have performed all of the SNR calculations based on an orbital
phase angle of $270^{\circ}$ with edge-on orbital configuration (i.e.,
inclination $=90^{\circ}$). Additional calculations varying the phase angle,
varying the inclination angle, and so forth are needed to fully assess the
potential detectability. Furthermore, we have not explored the extent of the
effect of varying mirror size on the SNR, or a different stellar host spectral
type (like a K-dwarf star). Berdyugina & Kuhn (2019) generated composite maps
of orbiting photovoltaic power sources if they were situated around a planet
like Proxima Centauri b. Specific simulations and the corresponding SNR
estimates relevant for HWO from such orbiting structures needs to be
undertaken. These additional considerations are left for future work.

As Earth remains the only example of an inhabited planet with a technosphere,
future trajectories of Earth can provide insight into the constraints that
might also apply to extraterrestrial technospheres. It is worth reiterating
that only 3% land coverage by solar panels would be needed to support a
population of $10$ billion people at a high standard of living, and the toy
logistic curve shown in Figure 4 indicates that even a population of $30$
billion people at a high standard of living would require far less energy than
the power output of the 23% land coverage scenario used in these detectability
calculations. These estimates not only do not account for increased efficiency
due to technological innovations, but they also do not account for other
sources of energy. Hence, the actual solar requirements for Earth would likely
be much reduced, especially in case controlled nuclear fusion becomes viable.
For Earth, these results suggest the possibility of a future in which the
energy needs of even a larger-than-predicted population could be fully met
with known technology.

If Earth is taken as an example in the search for technosignatures, then this
raises the question: Is the large-scale deployment of solar panels on a
planetary surface ever needed? Any actual large-scale deployment would
certainly raise numerous issues regarding the feasibility of such a deployment
or the logistical challenges in global distribution of energy, but a more
fundamental unknown is whether a technological civilization would ever require
such large energy demands to justify a scenario such as 23% land coverage or
greater. Several of the projections shown in Figure 4 reach a time at which
dissipation from energy use exceeds the threshold for contributing significant
direct heating to Earth’s climate ($\sim 3\times 10^{23}$ J), which occurs in
the year 2265 for the $2.6\%$ yr-1 growth rate scenario and 2338 for the $2\%$
yr-1 growth rate scenario. Yet, it is not evident that such vast energy
requirements will ever be necessary on Earth: the energy requirements for the
global human population to afford a high standard of living fall several
orders of magnitude below the direct heating threshold. The motivation to
prevent direct heating of the atmosphere may serve as an additional deterrent
to avoid such scenarios, and further suggests that only modest-scale
deployment of solar panels would ever be needed to achieve sustainable
development goals on Earth.

Speculating even further shows that these scenario projections will reach the
limit of a Kardashev Type I civilization (Kardashev, 1964), which is a
civilization that uses all available energy on its planet (i.e., all starlight
incident at the top of the atmosphere). This Type I threshold ($\sim 5\times
10^{24}$ J) occurs in the year 2377 for the $2.6\%$ yr-1 growth rate scenario
and 2482 for the $2\%$ yr-1 growth rate scenario. The equivalent land area
required to meet such energy demands with solar power exceeds 100% in these
cases, which indicate that other sources of power—possibly including space-
based solar power—would be required in these scenarios. Such speculative
scenarios suggest images of “Dyson spheres/swarms” (e.g., Dyson, 1960; Wright,
2020) of solar collectors that expand a civilization’s ability to capture the
energy output of its host star.

Kardashev (1964) even imagined a Type II civilization as one that utilizes the
entirety of its host star’s output; however, such speculations are based
primarily on the assumption of a fixed growth rate in world energy use. But
such vast energy reserves would be unnecessary even under cases of substantial
population growth, especially if fusion and other renewable sources are
available to supplement solar energy. The concept of a Type I or Type II
civilization then becomes an exercise in imagining the possible uses that a
civilization would have for such vast energy reserves. Even activities such as
large-scale physics experiments and (relativistic) interstellar space travel
(cf. Lingam & Loeb, 2021, Chapter 10) might not be enough to explain the need
for a civilization to harness a significant fraction of its entire planetary
or stellar output. In contrast, if human civilization can meet its own energy
demands with only a modest deployment of solar panels, then this expectation
might also suggest that concepts like Dyson spheres would be rendered
unnecessary in other technospheres.

These results also have implications for the problem known as the Fermi
paradox (Webb, 2015; Ćirković, 2018; Forgan, 2019) or Great Silence (Brin,
1983): if extraterrestrial expansion through the galaxy is relatively easy,
then where are they? In the context of limits to growth, the recognition that
human civilization could reach a sustainable equilibrium in its population and
energy use that falls well below the Type I threshold suggests that
extraterrestrial civilizations may not be compelled to expand for reasons of
subsistence. This conclusion mirrors the “sustainability solution” to the
Fermi paradox (e.g., Haqq-Misra & Baum, 2009; Mullan & Haqq-Misra, 2019),
which suggests that any extant extraterrestrial civilizations only expand
proportionally with their planet’s carrying capacity. Any civilization that is
able to achieve sustainable population levels with high standard of living may
also settle on a limit on any need to expand, which may likewise constrain the
magnitude of any potential technosignatures generated by such a civilization.

Underlying this discussion are two competing philosophical positions regarding
the tendency of life to expand. The first position assumes that (a) life will
evolve to utilize the maximum energy available in its environment, while the
second assumes that (b) life will evolve to utilize as much energy as needed
in its environment to reach an optimal level of existence. To the extent that
the environment will always impose thermodynamic limits, the second option (b)
may probably be favored by life. But does life tend to expand up to such
thermodynamic limits if unopposed, or will life tend to optimize its
consumption of resources rather than maximize them? Such interdisciplinary
questions highlight the intersection of technosignature science with concepts
from ecology (e.g., Meurer et al., 2024), and further scrutiny of the proposed
general tendency of life to expand may be useful for thinking about and
enivisioning the particular tendencies for technospheres to expand.

##  6 Conclusion

We analyzed the detectability of silicon-based solar panels (with anti-
reflective coatings) as a signature of extraterrestrial technology. Assuming a
8-meter HWO-like telescope, focusing on the reflection edge in the UV-VIS, and
considering varying land coverage of solar panels on an Earth-like exoplanet
that match the present and projected energy needs, we estimate that several
hundreds of hours of observation time is needed to reach a SNR of $\sim 5$ for
a high land coverage of $\sim 23\%$.

We subsequently assessed the need for the deployment of such large-scale solar
panels, taking into consideration various projected growth rates of future
energy consumption. We find that, even with significant population growth, the
energy needs of human civilization would be several orders of magnitude below
the energy threshold for a Kardashev Type I civilization, or a Dyson
sphere/swarm which harnesses the energy of a star. This line of inquiry
reexamines the utility of such concepts, and potentially addresses one crucial
aspect of the Fermi paradox: we have not discovered any large scale
engineering yet, conceivably because advanced technologies may not need them.

J.H.M., M.L., and R.K.K. gratefully acknowledge support from the NASA
Exobiology program under grant 80NSSC22K1009. The authors also acknowledge
support from the Goddard Space Flight Center (GSFC) Sellers Exoplanet
Environments Collaboration (SEEC), which is supported by the NASA Planetary
Science Division’s Research Program. Any opinions, findings, and conclusions
or recommendations expressed in this material are those of the authors and do
not necessarily reflect the views of their employers or NASA.

## References

  * Arnold et al. (2002) Arnold, L., Gillet, S., Lardière, O., Riaud, P., & Schneider, J. 2002, Astronomy & Astrophysics, 392, 231
  * Arto et al. (2016) Arto, I., Capellán-Pérez, I., Lago, R., Bueno, G., & Bermejo, R. 2016, Energy for Sustainable Development, 33, 1
  * Bazilian et al. (2013) Bazilian, M., Onyeji, I., Liebreich, M., et al. 2013, Renewable Energy, 53, 329
  * Beatty (2022) Beatty, T. G. 2022, Monthly Notices of the Royal Astronomical Society, 513, 2652
  * Berdyugina & Kuhn (2019) Berdyugina, S., & Kuhn, J. 2019, The Astronomical Journal, 158, 246
  * Brin (1983) Brin, G. D. 1983, Quarterly Journal of the Royal Astronomical Society, Vol. 24, NO. 3, P. 283-309, 1983, 24, 283
  * Campbell & Green (1987) Campbell, P., & Green, M. A. 1987, Journal of Applied Physics, 62, 243, doi: 10.1063/1.339189
  * Ćirković (2018) Ćirković, M. M. 2018, The great silence: Science and philosophy of Fermi’s paradox (Oxford University Press)
  * Dyson (1960) Dyson, F. J. 1960, Science, 131, 1667
  * Forgan (2019) Forgan, D. H. 2019, Solving Fermi’s paradox, Vol. 10 (Cambridge University Press)
  * Haqq-Misra & Baum (2009) Haqq-Misra, J., & Baum, S. 2009, Journal of the British Interplanetary Society, 62, 47
  * Haqq-Misra et al. (2022a) Haqq-Misra, J., Fauchez, T. J., Schwieterman, E. W., & Kopparapu, R. 2022a, The Astrophysical Journal Letters, 929, L28
  * Haqq-Misra et al. (2022b) Haqq-Misra, J., Kopparapu, R., Fauchez, T. J., et al. 2022b, The Planetary Science Journal, 3, 60
  * Haqq-Misra et al. (2022c) Haqq-Misra, J., Schwieterman, E. W., Socas-Navarro, H., et al. 2022c, Acta Astronautica, 198, 194
  * Jackson et al. (2022) Jackson, R. B., Ahlström, A., Hugelius, G., et al. 2022, Ecosphere, 13, e3978
  * Juanola-Parramon et al. (2022) Juanola-Parramon, R., Zimmerman, N. T., Pueyo, L., et al. 2022, Journal of Astronomical Telescopes, Instruments, and Systems, 8, 034001, doi: 10.1117/1.JATIS.8.3.034001
  * Kardashev (1964) Kardashev, N. S. 1964, Soviet Ast., 8, 217
  * Kim et al. (2020) Kim, M. S., Lee, J. H., & Kwak, M. K. 2020, International Journal of Precision Engineering and Manufacturing, 21, 1389
  * Kokaly et al. (2017) Kokaly, R. F., Clark, R. N., Swayze, G. A., et al. 2017, USGS Spectral Library Version 7, Tech. Rep. 1035, U.S. Geological Survey, doi: 10.3133/ds1035
  * Kopparapu et al. (2021) Kopparapu, R., Arney, G., Haqq-Misra, J., Lustig-Yaeger, J., & Villanueva, G. 2021, The Astrophysical Journal, 908, 164
  * Lin et al. (2014) Lin, H. W., Gonzalez Abad, G., & Loeb, A. 2014, ApJ, 792, L7, doi: 10.1088/2041-8205/792/1/L7
  * Lingam & Loeb (2017) Lingam, M., & Loeb, A. 2017, MNRAS, 470, L82, doi: 10.1093/mnrasl/slx084
  * Lingam & Loeb (2019) —. 2019, Astrobiology, 19, 28, doi: 10.1089/ast.2018.1936
  * Lingam & Loeb (2021) —. 2021, Life in the Cosmos: From Biosignatures to Technosignatures (Cambridge: Harvard University Press)
  * Macdonald et al. (2004) Macdonald, D. H., Cuevas, A., Kerr, M. J., et al. 2004, Solar Energy, 76, 277, doi: 10.1016/j.solener.2003.08.019
  * Meurer et al. (2024) Meurer, J. C., Haqq-Misra, J., & de Souza Mendonça, M. 2024, International Journal of Astrobiology, 23, e3
  * Miller & Keith (2018) Miller, L. M., & Keith, D. W. 2018, Environmental Research Letters, 13, 104008
  * Millward-Hopkins et al. (2020) Millward-Hopkins, J., Steinberger, J. K., Rao, N. D., & Oswald, Y. 2020, Global Environmental Change, 65, 102168
  * Mullan & Haqq-Misra (2019) Mullan, B., & Haqq-Misra, J. 2019, Futures, 106, 4
  * NASA Technosignatures Workshop Participants (2018) NASA Technosignatures Workshop Participants. 2018, arXiv e-prints, arXiv:1812.08681, doi: 10.48550/arXiv.1812.08681
  * O’Malley-James & Kaltenegger (2018) O’Malley-James, J. T., & Kaltenegger, L. 2018, Astrobiology, 18, 1123
  * Owen (1980) Owen, T. 1980, in Strategies for the Search for Life in the Universe (Springer), 177–185
  * Pieters & Hiroi (2004) Pieters, C. M., & Hiroi, T. 2004, in Lunar and Planetary Science Conference, ed. S. Mackwell & E. Stansbery, Lunar and Planetary Science Conference, 1720
  * Raut et al. (2011) Raut, H. K., Ganesh, V. A., Nair, A. S., & Ramakrishna, S. 2011, Energy & Environmental Science, 4, 3779
  * Rühle (2016) Rühle, S. 2016, Solar Energy, 130, 139, doi: 10.1016/j.solener.2016.02.015
  * Sagan et al. (1993) Sagan, C., Thompson, W. R., Carlson, R., Gurnett, D., & Hord, C. 1993, Nature, 365, 715
  * Sarkın et al. (2020) Sarkın, A. S., Ekren, N., & Sağlam, Ş. 2020, Solar Energy, 199, 63, doi: 10.1016/j.solener.2020.01.084
  * Schwieterman et al. (2018) Schwieterman, E. W., Kiang, N. Y., Parenteau, M. N., et al. 2018, Astrobiology, 18, 663, doi: 10.1089/ast.2017.1729
  * Seager et al. (2023) Seager, S., Petkowski, J. J., Huang, J., et al. 2023, Scientific Reports, 13, 13576
  * Seager et al. (2005) Seager, S., Turner, E. L., Schafer, J., & Ford, E. B. 2005, Astrobiology, 5, 372, doi: 10.1089/ast.2005.5.372
  * Socas-Navarro et al. (2021) Socas-Navarro, H., Haqq-Misra, J., Wright, J. T., et al. 2021, Acta Astronaut., 182, 446, doi: 10.1016/j.actaastro.2021.02.029
  * Tarter (2007) Tarter, J. C. 2007, Highlights of Astronomy, 14, 14, doi: 10.1017/S1743921307009829
  * Turnbull et al. (2006) Turnbull, M. C., Traub, W. A., Jucks, K. W., et al. 2006, The Astrophysical Journal, 644, 551
  * Varga (2020) Varga, G. 2020, Environment international, 139, 105712
  * Villanueva et al. (2022) Villanueva, G. L., Liuzzi, G., Faggi, S., et al. 2022, Fundamentals of the Planetary Spectrum Generator (Greenbelt, Maryland: NASA Goddard Space Flight Center)
  * Villanueva et al. (2018) Villanueva, G. L., Smith, M. D., Protopapa, S., Faggi, S., & Mandell, A. M. 2018, J. Quant. Spec. Radiat. Transf., 217, 86, doi: 10.1016/j.jqsrt.2018.05.023
  * Vogel et al. (2021) Vogel, J., Steinberger, J. K., O’Neill, D. W., Lamb, W. F., & Krishnakumar, J. 2021, Global Environmental Change, 69, 102287
  * Von Hoerner (1975) Von Hoerner, S. J. 1975, Journal of the British Interplanetary Society, 28, 691
  * Webb (2015) Webb, S. 2015, If the universe is teeming with aliens… where is everybody?: seventy-five solutions to the fermi paradox and the problem of extraterrestrial life (Springer)
  * Woolf et al. (2002) Woolf, N. J., Smith, P. S., Traub, W. A., & Jucks, K. W. 2002, The Astrophysical Journal, 574, 430
  * Wright (2020) Wright, J. T. 2020, Serbian Astronomical Journal, 1
  * Zhao & Green (1991) Zhao, J., & Green, M. A. 1991, IEEE Transactions on Electron Devices, 38, 1925, doi: 10.1109/16.119035


# There are no
minimal essentially undecidable Theories

Fedor Pakhomov  Vakgroep Wiskunde: Analysis, Logic and Discrete Mathematics,
Ghent University, Krijgslaan 281, B9000 Ghent, Belgium
and Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina 8,
119991 Moscow, Russia<EMAIL_ADDRESS>,  Juvenal Murwanashyaka
Department of Mathematics, the Faculty of Mathematics and Natural Sciences,
University of Oslo, Moltke Moes vei 35, Niels Henrik Abels hus, 0851 Oslo,
Norway<EMAIL_ADDRESS>and  Albert Visser  Philosophy, Faculty of
Humanities, Utrecht University, Janskerkhof 13, 3512BL Utrecht, The
Netherlands<EMAIL_ADDRESS>
###### Abstract.

We show that there is no theory that is minimal with respect to
interpretability among recursively enumerable essentially undecidable
theories.

###### Key words and phrases:

interpretability, essential undecidability

###### 2010 Mathematics Subject Classification:

03F25,03F30,03F40,

Research of Fedor Pakhomov was supported by FWO grant G0F8421N

We thank Yong Cheng and Tim Button for comments on and their corrections to
the preprint version of this paper.

##  1\. Introduction

For any salient property of recursively enumerable theories $\mathcal{P}$, one
can ask the obvious question _is there a weakest recursively enumerable theory
satisfying $\mathcal{P}$?_ But what does _weakest_ mean here? A traditional
answer is to take a theory $T$ to be given by a recursively enumerable set of
axioms $\mathcal{A}$. _Weakest_ is then interpreted as: $T$ has $\mathcal{P}$
and no theory axiomatised by a proper subset of $\mathcal{A}$ has
$\mathcal{P}$.

A paradigmatic example of an answer to our question under this reading, for
the case where we take $\mathcal{P}$ to be _essential undecidability_ , is the
well-known result by Tarski, Mostowski and Robinson that the theory Q with its
standard axiomatisation is minimally essentially undecidable in the sense that
all theories given by a proper subset of the axioms have a decidable
extension. See [TMR53, Chapter 2, Theorem 11]. Similar results for theories of
concatenation were obtained by Juvenal Murwanashyaka. See [Mur22]. For the
theory R, a result in the same spirit is due to Cobham. See [JS83]. Only here
minimality is applied to natural _groups_ of axioms rather than to single
axioms.

The above results crucially depend on the chosen axiom set. After all, each
non-trivial finitely axiomatisable theory is axiomatisable by a single axiom.
Suppose pure predicate logic does not satisfy $\mathcal{P}$. If a theory has
property $\mathcal{P}$, then it is automatically minimal with respect to the
single-axiom axiomatisation. Along a different line, for example, in the case
of Q, it is easy to produce finitely axiomatisable strict sub-theories that
are still essentially undecidable, if one allows tampering with the axioms.
For example, we may relativise the quantifiers in the axioms for plus and
times to the class of $x$ such that ${\sf S}x\neq x$. Jones and Shepherdson,
in [JS83], provide an example of an essentially undecidable sub-theory of R
that is strictly below R in the sense that it proves strictly less theorems.
However, they do strengthen one axiom group in order to be able to drop
another.

In this paper, we zoom in on the property of _essential undecidability of
recursively enumerable theories_ and we consider an ordering of recursively
enumerable theories that only depends on the theory-qua-set-of-theorems, to
wit _interpretability_. So, we ask whether there is an interpretability-
minimal recursively enumerable essentially undecidable theory. In this paper
we show that there is no such theory. Thus, our main result is:

######  Theorem 1.1.

There is no interpretability-minimal recursively enumerable essentially
undecidable theory.

It is easy to see that, if a recursively enumerable theory is _minimally_
essentially undecidable with respect to interpretability, then it is, _ipso
facto_ , the _minimum_ recursively enumerable theory, modulo mutual
interpretability. This is because essentially undecidable recursively
enumerable theories are closed under finite infima. Thus, it is sufficient to
prove the following lemma.

######  Lemma 1.2.

There is no recursively enumerable essentially undecidable theory that is the
interpretability-minimum.

The paper is structured as follows. In Section 2 we introduce the basic
concepts needed for the paper as a whole. This section also contains the
reduction of Theorem 1.1 to Lemma 1.2. In Section 3 we give a simple proof
that there is no minimal theory with respect to interpretability among
_finitely axiomatised theories_. In Section 4 we prove Theorem 1.1. We provide
two different proofs. The first proof employs a direct diagonalisation
argument, the second reduces our question to a general recursion theoretic
result.

##  2\. Some Basic Concepts

Theories in the present paper are one-sorted recursively enumerable theories
of predicate logic in finite signature. Our results are about theories-qua-
sets-of-theorems. However, our methods sometimes demand the intensional
perspective where the axiom set is given by a formula or by a recursive index.

There is a whole range of notions of interpretability. We can or cannot have
pieces, multidimensionality, parameters, relativisation, non-preservation of
identity. Our main result is not sensitive to the specific details of the
notion of interpretation as long as we consider mixtures of these specific
features.

We want to prove that there is no interpretability-minimal recursively
enumerable essentially undecidable theory. The proof can be divided in a
preliminary step and a main step.

  1. I.

We show that, modulo mutual interpretability, if there is an interpretability-
minimal recursively enumerable essentially undecidable theory, then there is
an interpretability-minimum recursively enumerable essentially undecidable
theory. To do this we show (i) that interpretability is a lower semi-lattice
(modulo mutual interpretability), i.e., binary infima exist and (ii) that
infima preserve essential undecidability. This gives the reduction of Theorem
1.1 to Lemma 1.2.

  2. II.

We prove Lemma 1.2.

Below we will define one uniform construction on theories defining the binary
minimum operation. This construction provides a binary minimum for all notions
of interpretability that are given by combinations of the above features. So,
(I) works for all notions of interpretability we under consideration.

It is immediate that, if we have (II) for the most inclusive notion, where we
have pieces, multidimensionality, parameters, relativisation, non-preservation
of identity, then we also have it for all weaker notions. So, we only need to
prove (II) for our strongest notion of interpretability. Alternatively, one
could look at what is used in the proof of (II) and see that all notions
satisfy these assumptions.

In this paper, we will use the most inclusive notion, to wit piecewise,
multidimensional (with dimensions varying over pieces), relative, non-
identity-preserving interpretability with parameters.

We refer the reader for definitions to [Vis17]. Here we will just fix
notations and give some basic facts.

We write $U\rhd V$ for $U$ interprets $V$ and $V\lhd U$ for $V$ is
interpretable in $U$.

Given two theories $U$ and $V$ we form $W:=U\ovee V$ in the following way. The
signature of $W$ is the disjoint union of the signatures of $U$ and $V$ with
an additional fresh zero-ary predicate $P$. The theory $W$ is axiomatised by
the axioms $P\to\varphi$ if $\varphi$ is a $U$-axiom and $\neg\,P\to\psi$ if
$\psi$ is a $V$-axiom. One can show that $U\ovee V$ is the infimum of $U$ and
$V$ in the interpretability ordering $\lhd$. This result works for all choices
of our notion of interpretation.111It is a bit strange that we use a
disjunction-like notation for an infimum. This strangeness is due to the fact
that the conventional choice for the interpretability-ordering puts the
weakest theory below where in boolean algebras the strongest proposition is
the lower one. In our notation, we follow the boolean intuition and view our
operation as a kind of disjunction of theories. We note that $\ovee$ preserves
finite axiomatisability.

A theory is _essentially undecidable_ iff all its consistent extensions in the
same language are undecidable. Salient examples of essentially undecidable
theories are R and Q. See [TMR53]. We have the following basic insights.

######  Theorem 2.1.

$U$ is essentially undecidable iff every consistent $V$ such that $V\rhd U$,
is undecidable.

###### Proof.

Since any $V$ extending $U$ also interprets $U$, the “if” part is trivial. For
“only if” part we note that if $V\rhd U$ and $V$ is consistent and decidable,
then any interpretation $\iota\colon V\rhd U$ gives a decidable consistent
extension $T$ of $U$ that is axiomatized by all the sentences $\varphi$ of the
language of $U$ such that $V$ proves the $\iota$-translation of $\varphi$. ❑

######  Theorem 2.2.

We have:

  1. i.

If $U$ is essentially undecidable and $V\rhd U$, then $V$ is essentially
undecidable.

  2. ii.

If $U$ and $V$ are essentially undecidable, then so is $U\ovee V$.

###### Proof.

The first claim follows immediately from Theorem 2.1 and the transitivity of
interpretability.

To prove the second claim we assume for a contradiction that there is a
consistent decidable extension $T$ of $U\ovee V$. Either $T$ is consistent
with $P$ or with $\lnot P$. If $T$ is consistent with $P$, then $T+P$ is a
consistent decidable theory that interprets $U$, contradicting essential
undecidability of $U$. Analogously, if $T$ is consistent with $\lnot P$, then
$T+\lnot P$ is a consistent decidable theory that interprets $V$,
contradicting essential undecidability of $V$. ❑

We note that the fact that essentially undecidable theories are closed under
binary infima implies that, if there is an interpretability-minimal one, then
there is a minimum with respect to interpretability. Thus, we have reduced
Theorem 1.1 to Lemma 1.2.

This reduction still works when we restrict ourselves to finitely axiomatised
theories, since $\ovee$ also preserves finite axiomatisability..

##  3\. Finitely axiomatisable Theories

In this section, we prove the non-existence of a minimal essentially
undecidable theory with respect to interpretability for the case where we
restrict ourselves to finitely axiomatised theories.

Our result is really a triviality as soon as the required machinery is in
place. We introduce this machinery in the next subsection.

###  3.1. Theories of a Number

We need the theory TN of a number. This theory is given as follows.

  1. TN1.

$\vdash x\not<0$

  2. TN2.

$\vdash(x<y\wedge y<z)\to x<z$

  3. TN3.

$\vdash x<y\vee x=y\vee y<x$

  4. TN4.

$\vdash x=0\vee\exists y\,x={\sf S}y$

  5. TN5.

$\vdash{\sf S}x\not<x$

  6. TN6.

$\vdash x<y\to(x<{\sf S}x\wedge y\not<{\sf S}x)$

  7. TN7.

$\vdash x+0=x$

  8. TN8.

$\vdash x+{\sf S}y={\sf S}(x+y)$

  9. TN9.

$\vdash x\cdot 0=0$

  10. TN10.

$\vdash x\cdot{\sf S}y=x\cdot y+x$

We note that TN allows finite models which can be identified with the natural
numbers viewed as the finite von Neumann ordinals with the added structure of
zero, addition, multiplication and $<$. Moreover, if a model of TN has a
maximal element, the model can be viewed as a (possibly) non-standard number.
In this case we take ${\sf S}a=a$ on the maximal element and adapt plus and
times accordingly. In [Vis17], the reader may find some further discussion of
TN.

A $\Delta_{0}$-formula is _pure_ iff (i) all bounding terms are variables and
(ii) all occurrences of terms are in subformulas of the form ${\sf S}x=y$,
$x+y=z$ and $x\cdot y=z$. A $\Sigma_{1}$-sentence _pure_ if it is of the form
$\exists\vec{x}\;\sigma_{0}\vec{x}$, where $\sigma_{0}$ is a pure
$\Delta_{0}$-formula.

We can transform an arbritrary $\Sigma_{1}$-sentence $\sigma$ into a pure
$\Sigma_{1}$-sentence. See [Vis17], for a sketch of the argument. In Section
3, we will assume that all $\Sigma_{1}$-sentences $\sigma$ are rewritten in
pure form.

Let $\sigma:=\exists\vec{y}\;\sigma_{0}\vec{y}$, where $\sigma_{0}$ is a pure
$\Delta_{0}$-formula. We define:

$[\sigma]:={\sf TN}+\exists x\,\exists\vec{y}<x\,\sigma_{0}\vec{y}.$

We note that if $\sigma$ is false, then $[\sigma]$ extends R. Thus, if
$[\sigma]$ is, in addition, consistent, we find that $[\sigma]$ is essentially
undecidable.

###  3.2. The main Result for finitely axiomatisable Theories

We prove our main result for the finitely axiomatised case.

######  Theorem 3.1.

There is no interpretability minimal essentially undecidable finitely
axiomatised theory.

###### Proof.

Since the finitely axiomatisable essentially undecidable theories are closed
under interpretability-infima, it is sufficient to show that there is no
minimum theory $A^{\star}$ among finitely axiomatised essentially undecidable
theories.

Suppose there was such an $A^{\star}$. Consider any $\Sigma^{0}_{1}$-sentence
$\sigma$. If $\sigma$ is true, then $[\sigma]$ has a finite model, so,
clearly, $[\sigma]\mathrel{\not\\!\rhd}A^{\star}$. If $\sigma$ is false and
$[\sigma]$ is consistent, we have $[\sigma]$ is essentially undecidable, so
$[\sigma]\rhd A^{\star}$. If $[\sigma]$ is inconsistent, then, trivially,
$[\sigma]\rhd A^{\star}$. _Ergo_ , $\sigma$ is false iff $[\sigma]\rhd
A^{\star}$. However, this is impossible, since the set of $\sigma$ such that
$[\sigma]\rhd A^{\star}$ is recursively enumerable. ❑

We note that our proof uses very little about interpretability. So there is a
good chance that it will work for even more general notions, like forcing-
interpretability. However, we did not explore this.

##  4\. Recursively Enumerable Theories

In this section we prove our main result.

###  4.1. A Result by Janiczak

In this subsection, we present a basic result by Janiczak. See [Jan53]. This
result will be the main tool for both proofs of Lemma 1.2.

We formulate an immediate consequence of the results of Section 3 of [Jan53].
Let J be the theory in the language with one binary relationsymbol E with the
following (sets of) axioms.222Our theory differs slightly from the theory
considered by Janiczak in that we added J3\. We did this to make the
characterisation in Theorem 4.1 as simple as possible.

  1. J1.

E is an equivalence relation.

  2. J2.

There is at most one equivalence class of size precisely $n$

  3. J3.

There are at least $n$ equivalence classes with at least $n$ elements.

We define ${\sf A}_{n}$ to be the sentence: there exists an equivalence class
of size precisely $n+1$. It is immediate that the ${\sf A}_{n}$ are mutually
independent over J.

######  Theorem 4.1 (Janiczak).

Over J, every sentence is equivalent with a boolean combination of the ${\sf
A}_{n}$.

###### Proof.

A base $n\geq 1$ configuration $C(\vec{x}\,)$ of variables $\vec{x}$ consists
of two equivalence relations ${=_{C}}\subseteq\mathsf{E}_{C}$ on the set
$\\{\vec{x}\,\\}$ of all variables from $\vec{x}$ together with a set ${\sf
S}_{C}\subseteq\\{1,\ldots,n-1\\}$ and a function ${\sf
s}_{C}\colon\\{\vec{x}\\}\to{\sf S}_{C}\cup\\{n\\}$, such that we have:

  * $\bullet$

$x\mathrel{\sf E}_{C}x^{\prime}\Rightarrow{\sf s}_{C}(x)={\sf
s}_{C}(x^{\prime})$;

  * $\bullet$

for any $x\in\\{\vec{x}\,\\}$, the $\mathrel{\sf E}_{C}$-equivalence class of
$x$ is split into at most ${\sf s}_{C}(x)$-many $=_{C}$-equivalence classes;

  * $\bullet$

for any $i\in{\sf S}_{C}$, its ${\sf s}_{C}$-preimage is either empty or a
single $\mathsf{\sf E}_{C}$-equivalence class.

Working in $\mathsf{J}$, we say that $\vec{x}$ satisfies the configuration
$C(\vec{x}\,)$ iff we have the following:

  * $\bullet$

the restriction of $=$ to $\\{\vec{x}\,\\}$ is $=_{C}$ and the restriction of
E to $\\{\vec{x}\,\\}$ is ${\sf E}_{C}$ (this includes both positive and
negative information);

  * $\bullet$

for each $i\in\\{1,\ldots,n-1\\}$, we have: $i$ is in ${\sf S}_{C}$ iff there
is an $\mathsf{\sf E}$-equivalence class of the size $i$;

  * $\bullet$

for each $x\in\\{\vec{x}\,\\}$, if ${\sf s}_{C}(x)<n$, then $x$ is in the
$\mathsf{E}$-equivalence class of the size ${\sf s}_{C}(x)$;

  * $\bullet$

for each $x\in\\{\vec{x}\,\\}$, if ${\sf s}_{C}(x)=n$, then $x$ is in an
$\mathsf{E}$-equivalence class of the size $\geq n$.

We denote the set of all base $n$ configurations depending on the variables
$\vec{x}$ as $\mathfrak{C}_{n}(\vec{x})$. Clearly, we can express the fact
that $\vec{x}$ satisfies configuration $C(\vec{x})$ by a $\mathsf{J}$-formula
which we will confuse with $C(\vec{x})$.

Now observe the following facts:

  1. 1.

$\mathsf{J}\vdash\bigvee\limits_{C(\vec{x})\in\mathfrak{C}_{n}(\vec{x}\,)}C(\vec{x}\,)$.

  2. 2.

For distinct $C_{1}(\vec{x}\,),C_{2}(\vec{x}\,)\in\mathfrak{C}_{n}(\vec{x}\,)$
we have $\mathsf{J}\vdash\lnot\,(C_{1}(\vec{x}\,)\land C_{2}(\vec{x}\,))$.

  3. 3.

For each $C(\vec{x}\,)\in\mathfrak{C}_{n}(\vec{x}\,)$, we have
$\mathsf{J}\vdash\lnot\,C(\vec{x}\,)\leftrightarrow\bigvee\limits_{C^{\prime}(\vec{x}\,)\in\mathfrak{C}_{n}(\vec{x}\,)\setminus\\{C(\vec{x}\,)\\}}C^{\prime}(\vec{x}\,)$.
This follows from (1) and (2).

  4. 4.

Given an atomic $\varphi(\vec{x}\,)$ and
$C(\vec{x}\,)\in\mathfrak{C}_{n}(\vec{x})$, we have either $\mathsf{J}\vdash
C(\vec{x}\,)\to\varphi(\vec{x}\,)$ or $\mathsf{J}\vdash
C(\vec{x}\,)\to\lnot\,\varphi(\vec{x}\,)$.

  5. 5.

Given quantifier-free $\varphi(\vec{x}\,)$ and
$C(\vec{x})\in\mathfrak{C}_{n}(\vec{x}\,)$, either $\mathsf{J}\vdash
C(\vec{x}\,)\to\varphi(\vec{x}\,)$ or $\mathsf{J}\vdash
C(\vec{x}\,)\to\lnot\,\varphi(\vec{x}\,)$. This follows from (4).

  6. 6.

For any quantifier-free $\varphi(\vec{x}\,)$ and $n\geq 1$, there is
$\mathcal{C}\subseteq\mathfrak{C}_{n}(\vec{x})$, such that
$\mathsf{J}\vdash\varphi(\vec{x}\,)\leftrightarrow\bigvee\limits_{C(\vec{x}\,)\in\mathcal{C}}C(\vec{x}\,)$.
This follows from (1) and (5).

  7. 7.

For $C(\vec{x},y)\in\mathfrak{C}_{n}(\vec{x},y)$, let
$\exists_{y}C(\vec{x},y)\in\mathfrak{C}_{n}(\vec{x})$ be the configuration
$C^{\prime}$ where $S_{C^{\prime}}=S_{C}$, and
$=_{C^{\prime}},\mathsf{E}_{C^{\prime}},s_{C^{\prime}}$ are the restrictions
of $=_{C},\mathsf{E}_{C},s_{C}$, respectively, to the new set of variables. If
the size of $\vec{x}$ is at most $n-1$, then we have
$\mathsf{J}\vdash\exists_{y}C(\vec{x},y)\leftrightarrow\exists
y\,C(\vec{x},y)$.

We claim that for any formula $\varphi(\vec{x}\,)$ of the language of
$\mathsf{J}$ there is an $n$ and a
$\mathcal{C}\subseteq\mathfrak{C}_{n}(\vec{x}\,)$, such that

$\mathsf{J}\vdash\varphi(\vec{x}\,)\leftrightarrow\bigvee\limits_{C(\vec{x}\,)\in\mathcal{C}}C(\vec{x}\,).$

For this, we put $\varphi(\vec{x}\,)$ in prenex normal form
$Q_{1}y_{1},\ldots,Q_{m}y_{m}\,\psi(\vec{x},y_{1},\ldots,y_{m})$. We fix $n$
to be the sum of $m$ and the size of $\vec{x}$. We first use the fact (6) to
transform the matrix $\psi(\vec{x},\vec{y}\,)$ into the disjunction of
configurations from $\mathfrak{C}_{n}(\vec{x},y_{1},\ldots,y_{m})$. And,
secondly, using facts (3) and (7), we prove, by induction on $0\leq i<m$, that
$Q_{m-i}y_{m-i}\ldots Q_{m}y_{m}\,\psi(\vec{x},y_{1},\ldots,y_{m})$ is
equivalent to a disjunction of configurations from
$\mathfrak{C}_{n}(\vec{x},y_{1},\ldots,y_{m-i})$. In the case of $i=m-1$ we
obtain the desired representation for $\varphi(\vec{x}\,)$.

Combining the claim above with the observation that any
$C()\in\mathfrak{C}_{n}()$ is $\mathsf{J}$-provably equivalent to a formula of
the form $\bigwedge\limits_{i\in{\sf
S}_{C}}\mathsf{A}_{i-1}\wedge\bigwedge\limits_{i\in\\{1,\dots
n-1\\}\setminus{\sf S}_{C}}\neg\,\mathsf{A}_{i-1}$, we see that, over
$\mathsf{J}$, any sentence is equivalent to a boolean combination of the
$\mathsf{A}_{n}$. ❑

######  Remark 4.2.

Let $\mathsf{B}_{n}(x)$ be $\mathsf{J}$-formula expressing that $x$ is in an
equivalence class of size $>n$. The proof above, in fact shows that
$\mathsf{J}$ enjoys quantifier elimination in the signature extended by all
$\mathsf{A}_{n}$ and $\mathsf{B}_{n}(x)$. Indeed, it is easy to see that over
$\mathsf{J}$ any configuration $C(\vec{x}\,)$ (as defined in the proof of
Theorem 4.1) is equivalent to a quantifier-free formula in the extended
signature and, in this proof, we established that over $\mathsf{J}$ any
formula is equivalent to a disjunction of configurations. ❍

######  Remark 4.3.

Theorem 4.1 tells us that the Lindenbaum algebra of J is isomorphic with
respect to recursive boolean isomorphisms to the free boole algebra on
countably many generators. We note that, by the results of Pour-El & Kripke
[PEK67, Theorem 2], the Lindenbaum algebra of any consistent RE theory $U$
that interprets the Tarski-Mostowski-Robinson theory R is recursively
isomorphic to the Lindenbaum algebra of, say, Peano Arithmetic. Since this is
a countable boole algebra without atoms, it follows that this algebra is
isomorphic to the free boole algebra on countably many generators. We note
that recursive isomorphism is really about _numbered objects_ or
_numerations_. Thus, the Lindenbaum algebras of J and of, e.g., PA are
isomorphic but they cannot, _qua numbered algebras_ , be recursively
isomorphic, since J is decidable and PA is not. ❍

###  4.2. A First Proof of Theorem 1.1

We prove a theorem that provides, for every essentially undecidable RE theory,
a class of theories that do not interpret it. The members of this class can
instantiate many desirable recursion theoretic properties. For example, there
is a member in any RE degree.

Let $\mathcal{X}$ be a set of numbers. We say that $W$ is a ${\sf
J},\mathcal{X}$-theory when $W$ is axiomatised over J by boolean combinations
of sentences ${\sf A}_{s}$ for $s\in\mathcal{X}$.

######  Theorem 4.4.

Consider any essentially undecidable RE theory $U$. Then, we can effectively
find a recursive set $\mathcal{X}$ (from an index of $U$) such that no
consistent ${\sf J},\mathcal{X}$-theory interprets $U$.

###### Proof.

Let ${\sf C}_{n,0},\dots,{\sf C}_{n,2^{n}-1}$ be an enumeration of all
conjunctions of $\pm{\sf A}_{i}$, for $i<n$. Suppose $U$ is an essentially
undecidable RE theory. Let $\upsilon_{0},\upsilon_{1},\dots$ be an effective
enumeration of the theorems of $U$. Let $\tau_{0},\tau_{1},\dots$ be an
effective enumeration of all translations from the $U$-language into the
J-language.

Consider $n$, $\tau_{i}$ and ${\sf C}_{n,j}$, for $j<2^{n}$. We claim that,
for some $k$, we have ${\sf J}+{\sf C}_{n,j}\nvdash\upsilon^{\tau_{i}}_{k}$.
If not, then $\tau_{i}$ would carry an interpretation of $U$ in ${\sf J}+{\sf
C}_{n,j}$. This contradicts the fact that ${\sf J}+{\sf C}_{n,j}$ is
decidable.

Thus, we can effectively find a number $p_{n,i,j}$ as follows. We find the
first $k$ such that ${\sf J}+{\sf C}_{n,j}\nvdash\upsilon^{\tau_{i}}_{k}$.
This can be effectively done since ${\sf J}+{\sf C}_{n,j}$ is decidable. Then,
we reduce, over J, the sentence $\upsilon^{\tau_{i}}_{k}$ to a boolean
combination of ${\sf A}_{s}$. Let $p_{n,i,j}$ be the supremum of the $s+1$
such that ${\sf A}_{s}$ occurs in this boolean combination.

We define $f(n,i)$ to be the maximum of the $p_{n,i,j}$ and $n$. Let $F(0):=0$
and let $F(k+1):=f(F(k)+1,k)$. Clearly $F$ is recursive and strictly
increasing. Let $\mathcal{X}$ be the range of $F$. Clearly, $\mathcal{X}$ is
recursive.

Let $W$ be any consistent ${\sf J},\mathcal{X}$-theory. Suppose we would have
$K:W\rhd U$. Let the underlying translation of $K$ be $\tau_{n^{\ast}}$.

Clearly, there is a $j^{\ast}$ such that $W+{\sf C}_{F(n^{\ast})+1,j^{\ast}}$
is consistent. (This is a non-constructive step.)

By construction, there is a $\varphi$ with $U\vdash\varphi$ and ${\sf J}+{\sf
C}_{F(n^{\ast})+1,j^{\ast}}\nvdash\varphi^{K}$, such that ${\sf
J}\vdash\varphi^{K}\leftrightarrow\chi$, where $\chi$ is a boolean combination
of ${\sf A}_{s}$, where $s<F(n^{\ast}+1)$. We note that none of the ${\sf
A}_{s}$ with $F(n^{\ast})<s<F(n^{\ast}+1)$ occurs in the axiomatisation of
$W$. So, there is a ${\sf C}_{F(n^{\ast}+1),p}$ that extends ${\sf
C}_{F(n^{\ast})+1,j^{\ast}}$ such that ${\sf J}+{\sf
C}_{F(n^{\ast}+1),p}\vdash\neg\,\varphi^{K}$. On the other hand, ${\sf
C}_{F(n^{\ast}+1),p}$ is clearly consistent with $W$, by the mutual
independence of the ${\sf A}_{\ell}$. A contradiction. ❑

It now follows:

######  Theorem 4.5.

For every essentially undecidable RE theory $U$ there is an essentially
undecidable RE theory $W$ such that $W\mathrel{\not\\!\rhd}U$. We can find an
index for $W$ effectively from an index of $U$.

###### Proof.

Suppose $U$ is an essentially undecidable RE theory. Let $\mathcal{X}$ be the
recursive set promised for $U$ by Theorem 4.4. Let $\mathcal{Y},\mathcal{Z}$
be a pair of recursively inseparable RE sets that are subsets of
$\mathcal{X}$. We define $W:={\sf J}+\\{{\sf A}_{n}\mid
n\in\mathcal{Y}\\}+\\{\neg\,{\sf A}_{n}\mid n\in\mathcal{Z}\\}$. Then $W$ is a
consistent essentially undecidable RE theory and $W\mathrel{\not\\!\rhd}U$. ❑

Clearly, Theorem 4.5 gives us Lemma 1.2 and, thus, Theorem 1.1.

######  Remark 4.6.

It is easily seen that we can give the theory $W$ of Theorem 4.4 many extra
properties. For example, it can be Turing persistent and at the same time of
any given non-zero RE degree. See the next section for these notions. ❍

###  4.3. A Result by Shoenfield

We present some basic ideas from Shoenfield’s paper [Sho58]. We will use
Shoenfield’s result in the proof of Lemma 1.2.

We first need a purely recursion theoretic result. We write $\leq_{\sf T}$ for
Turing reducibility. Our proof is just a minor variation of Shoenfield’s
proof.

######  Theorem 4.7 (Shoenfield).

Let ${\mathcal{A}}$ be any recursively enumerable (RE) set with index $a$.
Then, we can effectively find RE-indices $b$ of a set ${\mathcal{B}}$ and $c$
of a set ${\mathcal{C}}$ from $a$, such that we have:

  1. i.

${\mathcal{B}}\leq_{\sf T}{\mathcal{A}}$ and ${\mathcal{C}}\leq_{\sf
T}{\mathcal{A}}$.

  2. ii.

${\mathcal{B}}\cap{\mathcal{C}}=\emptyset$.

  3. iii.

Suppose ${\mathcal{D}}$ is an RE set that separates ${\mathcal{B}}$ and
${\mathcal{C}}$, i.e., ${\mathcal{B}}\subseteq{\mathcal{D}}$ and
${\mathcal{C}}\cap{\mathcal{D}}=\emptyset$, then ${\mathcal{A}}\leq_{\sf
T}{\mathcal{D}}$.

We represent $x\in{\mathcal{A}}$ by the formula $\exists y\,{\sf
T}_{1}(a,x,y)$. We assume that the computation $y$ is unique when it exists.
We write, e.g., $(x)_{0}\in{\mathcal{A}}$ for $\exists y\,{\sf
T}_{1}(a,(x)_{0},y)$. We treat other indices similarly. The notation $\exists
x\,\varphi\leq\exists y\,\psi$ means $\exists x\,(\varphi\wedge\forall
y<x\,\neg\,\psi)$ and $\exists x\,\varphi<\exists y\,\psi$ means $\exists
x\,(\varphi\wedge\forall y\leq x\,\neg\,\psi)$.

###### Proof.

We define:

  * $\bullet$

$x\in{\sf Z}$ iff $\exists z\,{\sf T}_{1}((x)_{1},x,z)$.

  * $\bullet$

$x\in{\mathcal{B}}$ iff $((x)_{0}\in{\mathcal{A}})<(x\in{\sf Z})$.

  * $\bullet$

$x\in{\mathcal{B}}^{\bot}$ iff $(x\in{\sf Z})\leq((x)_{0}\in{\mathcal{A}})$.

  * $\bullet$

$x\in{\mathcal{C}}$ iff $(x)_{0}\in{\mathcal{A}}\wedge
x\in{\mathcal{B}}^{\bot}$.

Claims (i) and (ii) are trivial.333Note that ${\mathcal{B}}^{\bot}$ need not
be Turing reducible to ${\mathcal{A}}$.

Consider any ${\mathcal{D}}$ with index $d$. We note that ${\langle
w,d\rangle}\in{\mathcal{D}}$ iff ${\langle w,d\rangle}\in{\sf Z}$. Suppose
${\mathcal{D}}$ separates ${\mathcal{B}}$ and ${\mathcal{C}}$.

We first show $w\in{\mathcal{A}}$ iff ${\langle w,d\rangle}\in{\mathcal{B}}$.
From right to left is immediate. Suppose $w\in{\mathcal{A}}$. Then either
${\langle w,d\rangle}\in{\mathcal{B}}$ or ${\langle
w,d\rangle}\in{\mathcal{C}}$. In case ${\langle w,d\rangle}\in{\mathcal{C}}$,
we find ${\langle w,d\rangle}\in{\sf Z}$, by the definition of
${\mathcal{B}}^{\bot}$. Hence ${\langle w,d\rangle}\in{\mathcal{D}}$. _Quod
non_ , since ${\mathcal{C}}$ and ${\mathcal{D}}$ are disjoint. So, ${\langle
w,d\rangle}\in{\mathcal{B}}$.

In case ${\langle w,d\rangle}\in{\mathcal{D}}$, we have ${\langle
w,d\rangle}\in{\sf Z}$. So, we can effectively determine whether ${\langle
w,d\rangle}\in{\mathcal{B}}$ and, thus, whether $w\in{\mathcal{A}}$. In case
${\langle w,d\rangle}\not\in{\mathcal{D}}$, we have ${\langle
w,d\rangle}\not\in{\mathcal{B}}$, and, hence, $w\not\in{\mathcal{A}}$. ❑

Let us say that an RE theory $U$ is _Turing-persistent_ iff $U$ is consistent
and, whenever $U\subseteq V$, where $V$ is RE and consistent, we have
$U\leq_{\sf T}V$. It is easy to see that the Turing-persistence of $U$
implies: if $U\lhd V$, then $U\leq_{\sf T}V$, whenever $V$ is consistent. We
note that, if $U$ is Turing persistent and undecidable, then it is essentially
undecidable.

Consider an RE set ${\mathcal{A}}$ with index $a$. Let ${\mathcal{B}}$ and
${\mathcal{C}}$ be the sets constructed above. Let ${\sf sch}(a):={\sf
J}+\\{{\sf A}_{n}\mid n\in{\mathcal{B}}\\}+\\{\neg\,{\sf A}_{m}\mid
m\in{\mathcal{C}}\\}$. We have:

######  Theorem 4.8 (Shoenfield).

The theory ${\sf sch}(a)$ has the same Turing degree as $\mathcal{A}$.
Moreover, the theory is Turing persistent. It follows that, if $\mathcal{A}$
is undecidable, then ${\sf sch}(a)$ is essentially undecidable. Thus, there is
an essentially undecidable theory in every RE, non-recursive Turing degree.

###### Proof.

For any consistent RE extension $V$ of $\mathsf{sch}(a)$, the set $\\{i\mid
V\vdash\mathsf{A}_{i}\\}$ clearly is an RE set separating ${\mathcal{B}}$ and
${\mathcal{C}}$ and hence ${\mathcal{A}}$ is Turing reducible to $V$. It is
obvious that $\mathsf{sch}(a)$ is consistent. Thus to finish the proof we only
need to show that $\mathsf{sch}(a)$ is Turing reducible to ${\mathcal{A}}$.

We describe a ${\mathcal{A}}$-recursive procedure of checking whether given
sentence $\varphi$ of the language of J is a theorem of $\mathsf{sch}(a)$. We
can effectively find a boolean combination $\varphi^{\ast}$ of ${\sf A}_{i}$
that is equivalent over J to $\varphi$. Let the set of indices of ${\sf
A}_{i}$ occurring in $\varphi^{\ast}$ be $\mathcal{I}$. Also we can
effectively find, using ${\mathcal{A}}$ as an oracle, the conjunction $\psi$
of the ${\sf A}_{j}$ with $j\in\mathcal{B}\cap\mathcal{I}$ and the $\neg\,{\sf
A}_{s}$ with $s\in\mathcal{C}\cap\mathcal{I}$. Using the mutual independence
of the ${\sf A}_{n}$, it is easy to see that ${\sf sch}(a)\vdash\varphi$ iff
${\sf J}\vdash\psi\to\varphi^{\ast}$, and ${\sf J}\vdash\psi\to\varphi^{\ast}$
iff $\psi\to\varphi^{\ast}$ is a propositional tautology. Whether
$\psi\to\varphi^{\ast}$ is a tautology can be checked with a truth table. ❑

######  Remark 4.9.

The result that there is an essentially undecidable theory in every RE, non-
recursive Turing degree is due to Shoenfield. See [Sho58]. This result was
improved by Hanf. He proves that there is an essentially undecidable _finitely
axiomatised_ theory in every RE, non-recursive Turing degree. See [Han65]. By
a simple adaptation of the argument, Hanf’s results imply that there is a
Turing persistent finitely axiomatised theory in every RE, non-recursive
Turing degree. ❍

######  Example 4.10.

We provide an example of an essentially undecidable theory that is not Turing
persistent. Consider RE Turing degrees $d$ and $e$ with $0<d<e$. Let
$\mathcal{X}$ be an RE set of degree $d$ and let $\mathcal{W}$ be an RE set of
degree $e$. We take $\mathcal{Y}$ and $\mathcal{Z}$ to be the recursively
inseparable RE sets provided by Shoenfield’s result. Let

  * $\bullet$

$U_{0}:={\sf J}+\\{{\sf A}_{2i}\mid i\in\mathcal{Y}\\}+\\{\neg\,{\sf
A}_{2j}\mid j\in\mathcal{Z}\\}$

  * $\bullet$

$U_{1}:=U_{0}+\\{{\sf A}_{2k+1}\mid k\in\mathcal{W}\\}$.

  * $\bullet$

$U_{2}:=U_{1}+\\{{\sf A}_{2k+1}\mid k\in\omega\\}$.

Clearly $U_{0}$ and $U_{2}$ are Turing persistent and have Turing degree $d$.
The theory $U_{1}$ had degree $e$ and, hence, cannot be Turing persistent.
Yet, it is clearly essentially undecidable.

So our example shows that there are essentially undecidable theories that are
not Turing persistent. Moreover, a Turing persistent theory can have a non-
Turing persistent RE extension and a non-Turing persistent recursively
inseparable theory can have a Turing persistent RE extension. ❍

######  Open Question 4.11.

A theory is _essentially_ Turing-persistent if all its consistent RE
extensions are Turing-persistent. Clearly, any Turing-persistent theory in
degree $0^{\prime}$ is essentially Turing-persistent. Are there essentially
Turing-persistent theories of other degrees? ❍

###  4.4. A Second Proof of Theorem 1.1

In this subsection, we prove our main result for recursively enumerable
theories. We need a basic fact from recursion theory.

Let Rec be the set of indices of recursive sets. We have the following
theorem.

######  Theorem 4.12 (Rogers, Mostowski).

Rec is complete $\Sigma^{0}_{3}$.

See [Rog67, Chapter 14, Theorem XVI] or [Soa16, Corollary 4.3.6]. We also have
the following theorem. We have the following theorem that is given as exercise
4.3.14 of [Soa16, Page 91]. Let Rsep be the set of pairs of indices of
recursively separable RE theories.

######  Theorem 4.13.

Rsep is complete $\Sigma^{0}_{3}$.

This follows immediately from Theorem 4.7, since that gives a reduction of Rec
to Rsep.

We now prove Theorem 1.1.

###### Proof of Theorem 1.1.

It is sufficient to prove Lemma 1.2. Suppose that there is an essentially
undecidable recursively enumerable theory $U^{\star}$ that is the
interpretability minimum. So, we have:

$\displaystyle a\not\in{\sf Rec}$ iff $\displaystyle{\sf sch}(a)\text{ is
essentially undecidable}$ iff $\displaystyle{\sf sch}(a)\rhd U^{\star}$

Since, interpretability between recursively enumerable theories is
$\Sigma^{0}_{3}$, it would follow that Rec is $\Pi^{0}_{3}$.444In fact,
interpretability between recursively enumerable theories is complete
$\Sigma^{0}_{3}$. See [Sha97]. _Quod non_ , by Theorem 4.12. ❑

Here is a variant of the proof.

###### Variant of the Proof of Theorem 1.1.

It is sufficient to prove Lemma 1.2. Suppose that there is an essentially
undecidable recursively enumerable theory $U^{\star}$ that is the
interpretability minimum. Let ${\langle a,b\rangle}$ be a pair of indices of
RE sets and let $\mathcal{X}$ and $\mathcal{Y}$ be the sets defined by $a$,
respectively $b$.

We define:

${\sf so}(a,b):={\sf J}+\\{{\sf A}_{i}\mid i\in\mathcal{X}\\}+\\{\neg\,{\sf
A}_{j}\mid j\in\mathcal{Y}\\}.$

We have:

$\displaystyle{\langle a,b\rangle}\not\in{\sf Rsep}$ iff $\displaystyle{\sf
so}(a,b)\text{ is essentially undecidable or inconsistent}$ iff
$\displaystyle{\sf so}(a,b)\rhd U^{\star}$

Since, interpretability between recursively enumerable theories is
$\Sigma^{0}_{3}$, it would follow that Rsep is $\Pi^{0}_{3}$. _Quod non_ , by
Theorem 4.13. ❑

We note that our result is insensitive for the precise notion of
interpretability used. It could very well be that it also works for even more
general notions like forcing interpretability. However, we did not explore
this.

## References

  * [Han65] W. Hanf.  Model-theoretic methods in the study of elementary logic.  In J.W. Addison, L. Henkin, and A. Tarski, editors, The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, pages 132–145. North Holland, Amsterdam, 1965.
  * [Jan53] A. Janiczak.  Undecidability of some simple formalized theories.  Fundamenta Mathematicæ, 40:131–139, 1953.
  * [JS83] J. P. Jones and J.C. Shepherdson.  Variants of Robinson’s essentially undecidable theory R.  Archiv für Mathematische Logik und Grundlagenforschung, 23:61–64, 1983.
  * [Mur22] J. Murwanashyaka.  Weak essentially undecidable theories of concatenation.  Archive for Mathematical Logic, pages 1–38, 2022.
  * [PEK67] M.B. Pour-El and S. Kripke.  Deduction-preserving “Recursive Isomorphisms” between theories.  Fundamenta Mathematicae, 61:141–163, 1967.
  * [Rog67] H. Rogers.  Theory of recursive functions and effective computability.  McGraw-Hill Book Company, New York, 1967.
  * [Sha97] V.Yu. Shavrukov.  Interpreting reflexive theories in finitely many axioms.  Fundamenta Mathematicae, 152:99–116, 1997.
  * [Sho58] J. R. Shoenfield.  Degrees of formal systems 1.  The Journal of Symbolic Logic, 23(4):389–392, 1958.
  * [Soa16] R.I. Soare.  Turing computability: theory and applications.  Springer, Berlin, 2016.
  * [TMR53] A. Tarski, A. Mostowski, and R.M. Robinson.  Undecidable theories.  North–Holland, Amsterdam, 1953.
  * [Vis17] A. Visser.  On Q.  Soft Computing, 21(1):39–56, 2017.