Prompt: Transformer-based model designed specifically for Arabic language understanding

Output: Transformer-based model designed specifically for Arabic language understanding

Output:
Title: AraBERT: Transformer-based Model for Arabic Language Understanding

Abstract: The Arabic language is a morphologically rich language with relatively few resources and a less explored syntax compared to English. Given these limitations, Arabic Natural Language Processing (NLP) tasks like Sentiment Analysis (SA), Named Entity Recognition (NER), and Question Answering (QA), have proven to be very challenging to tackle. Recently, with the surge of transformers based models, language-specific BERT based models have proven to be very efficient at language understanding, provided they are pre-trained on a very large corpus. Such models were able to set new standards and achieve state-of-the-art results for most NLP tasks. In this paper, we pre-trained BERT specifically for the Arabic language in the pursuit of achieving the same success that BERT did for the English language. The performance of AraBERT is compared to multilingual BERT from Google and other state-of-the-art approaches. The results showed that the newly developed AraBERT achieved state-of-the-art performance on most tested Arabic NLP tasks. The pretrained araBERT models are publicly available on https://github.com/aub-mind/arabert hoping to encourage research and applications for Arabic NLP.


Prompt: do switches use ip or mac?

Output: do switches use ip or mac?

Output:
Superuser Stackexchange

Q: IP Address vs MAC addresses why do we need ip addresses if mac address is enough to identify the machine???

A: There are many reasons. For example: MAC address is bound to hardware. It means you can't or rather are not supposed to have the same MAC on different hardware. Imagine that google has to change DNS binding of "google.com" to "MAC" address each time they change some network card on front-end server.
Another reason is that MAC addresses are assigned by producers while IP addresses are assigned by network admins. This allows building hierarchy of addresses that is root of packets routing. This is because way from IP A1.A2.A3.A4 to B1.B2.B3.B4 is easily predictable and way from MAC#1 to MAC#2 isn't.
Next reason is that MAC is Ethernet-related stuff and there are other network technologies that might not use MAC at all.


Prompt: do switches use ip or mac?

Output: do switches use ip or mac?

Output:
Serverfault Stackexchange

Q: Deny Static IP Assignments I am currently using a SonicWall Pro 2040 with DHCP enabled, but only with static IPs mapped to specific MAC addresses. There is no dynamic IP scope being used. Currently, no DHCP request will resolve an IP, as desired.
However, if a server or workstation network interface is still configured with a user-defined, static IP address, it works fine. I would like potential users who assign their machines with static IPs to not work.
Is there a way to deny these types of connections whose MAC address is not approved?

A: There is no way to configure a DHCP server to deny static IP addresses. If you think about it, there is a direct path between hosts on the network that simply does not go through the SonicWall:
            SonicWall
            LAN Port
                ^
                |
                v
HostA <----> Switch <----> HostB

So if you want to filter on MAC address to stop HostA from advertising an IP address on the network, you need to do so at the switch. As an example, if your switch is a Cisco, the command to use is switchport port-security.


Prompt: can I do multithreading with fastapi

Output: can I do multithreading with fastapi

Output:
FastAPI

FastAPI is a modern web framework first released in 2018 for building HTTP-based service APIs in Python. It is used for building APIs with Python 3.8+ based on standard Python-type hints.
FastAPI is based on Pydantic and uses type hints to validate, serialize and deserialize data. It also automatically generates OpenAPI documentation for APIs built with it.
Components
Pydantic
Pydantic is a data validation library for Python. While writing code in an IDE, Pydantic provides type hints for schema validation and serialization through type annotations.
Starlette
Starlette is a lightweight ASGI framework/toolkit, to support async functionality in Python.
Uvicorn
Uvicorn is a minimal low-level server/application web server for async frameworks, following the ASGI specification. Technically, it implements a multi-process model with one main process, which is responsible for managing a pool of worker processes and distributing incoming HTTP requests to them. The number of worker processes is pre-configured, but can also be adjusted up or down at runtime.
Example
The following code shows a simple web application that displays "Hello World!" when visited:
from fastapi import FastAPI
app = FastAPI()
@app.get("/")
def read_root():


Prompt: can I do multithreading with fastapi

Output: can I do multithreading with fastapi

Output:
FastAPI

FastAPI is a modern web framework first released in 2018 for building HTTP-based service APIs in Python. It is used for building APIs with Python 3.8+ based on standard Python-type hints.
FastAPI is based on Pydantic and uses type hints to validate, serialize and deserialize data. It also automatically generates OpenAPI documentation for APIs built with it.
Components
Pydantic
Pydantic is a data validation library for Python. While writing code in an IDE, Pydantic provides type hints for schema validation and serialization through type annotations.
Starlette
Starlette is a lightweight ASGI framework/toolkit, to support async functionality in Python.
Uvicorn
Uvicorn is a minimal low-level server/application web server for async frameworks, following the ASGI specification. Technically, it implements a multi-process model with one main process, which is responsible for managing a pool of worker processes and distributing incoming HTTP requests to them. The number of worker processes is pre-configured, but can also be adjusted up or down at runtime.
Example
The following code shows a simple web application that displays "Hello World!" when visited:
from fastapi import FastAPI
app = FastAPI()
@app.get("/")
def read_root():


Prompt: How can retrieval-based methods be optimized to balance the trade-off between retrieval precision and computational efficiency in large-scale neural information retrieval systems?

Output: How can retrieval-based methods be optimized to balance the trade-off between retrieval precision and computational efficiency in large-scale neural information retrieval systems?

Output:
Evaluation measures (information retrieval)

Some authors choose to interpolate the  function to reduce the impact of "wiggles" in the curve. For example, the PASCAL Visual Object Classes challenge (a benchmark for computer vision object detection) until 2010 computed the average precision by averaging the precision over a set of evenly spaced recall levels {0, 0.1, 0.2, ... 1.0}:
where  is an interpolated precision that takes the maximum precision over all recalls greater than :
.
An alternative is to derive an analytical  function by assuming a particular parametric distribution for the underlying decision values. For example, a binormal precision-recall curve can be obtained by assuming decision values in both classes to follow a Gaussian distribution.
The minimum achievable AveP for a given classification task is given by:
Precision at k
For modern (web-scale) information retrieval, recall is no longer a meaningful metric, as many queries have thousands of relevant documents, and few users will be interested in reading all of them. Precision at k documents (P@k) is still a useful metric (e.g., P@10 or "Precision at 10" corresponds to the number of relevant results among the top 10 retrieved documents), but fails to take into account the positions of the relevant documents among the top k.  Another shortcoming is that on a query with fewer relevant results than k, even a perfect system will have a score less than 1.  It is easier to score manually since only the top k results need to be examined to determine if they are relevant or not.


Prompt: How can retrieval-based methods be optimized to balance the trade-off between retrieval precision and computational efficiency in large-scale neural information retrieval systems?

Output: How can retrieval-based methods be optimized to balance the trade-off between retrieval precision and computational efficiency in large-scale neural information retrieval systems?

Output:
Information retrieval

Information retrieval (IR) in computing and information science is the task of identifying and retrieving information system resources that are relevant to an information need.  The information need can be specified in the form of a search query. In the case of document retrieval, queries can be based on full-text or other content-based indexing. Information retrieval is the science of searching for information in a document, searching for documents themselves, and also searching for the metadata that describes data, and for databases of texts, images or sounds.
Automated information retrieval systems are used to reduce what has been called information overload. An IR system is a software system that provides access to books, journals and other documents; it also stores and manages those documents. Web search engines are the most visible IR applications.


Prompt: Impact of AI chatbots on academic integrity and plagiarism

Output: Impact of AI chatbots on academic integrity and plagiarism

Output:
Title: Will ChatGPT get you caught? Rethinking of Plagiarism Detection

Abstract: The rise of Artificial Intelligence (AI) technology and its impact on education has been a topic of growing concern in recent years. The new generation AI systems such as chatbots have become more accessible on the Internet and stronger in terms of capabilities. The use of chatbots, particularly ChatGPT, for generating academic essays at schools and colleges has sparked fears among scholars. This study aims to explore the originality of contents produced by one of the most popular AI chatbots, ChatGPT. To this end, two popular plagiarism detection tools were used to evaluate the originality of 50 essays generated by ChatGPT on various topics. Our results manifest that ChatGPT has a great potential to generate sophisticated text outputs without being well caught by the plagiarism check software. In other words, ChatGPT can create content on many topics with high originality as if they were written by someone. These findings align with the recent concerns about students using chatbots for an easy shortcut to success with minimal or no effort. Moreover, ChatGPT was asked to verify if the essays were generated by itself, as an additional measure of plagiarism check, and it showed superior performance compared to the traditional plagiarism-detection tools. The paper discusses the need for institutions to consider appropriate measures to mitigate potential plagiarism issues and advise on the ongoing debate surrounding the impact of AI technology on education. Further implications are discussed in the paper.


Prompt: Impact of AI chatbots on academic integrity and plagiarism

Output: Impact of AI chatbots on academic integrity and plagiarism

Output:
Title: Will ChatGPT get you caught? Rethinking of Plagiarism Detection

Abstract: The rise of Artificial Intelligence (AI) technology and its impact on education has been a topic of growing concern in recent years. The new generation AI systems such as chatbots have become more accessible on the Internet and stronger in terms of capabilities. The use of chatbots, particularly ChatGPT, for generating academic essays at schools and colleges has sparked fears among scholars. This study aims to explore the originality of contents produced by one of the most popular AI chatbots, ChatGPT. To this end, two popular plagiarism detection tools were used to evaluate the originality of 50 essays generated by ChatGPT on various topics. Our results manifest that ChatGPT has a great potential to generate sophisticated text outputs without being well caught by the plagiarism check software. In other words, ChatGPT can create content on many topics with high originality as if they were written by someone. These findings align with the recent concerns about students using chatbots for an easy shortcut to success with minimal or no effort. Moreover, ChatGPT was asked to verify if the essays were generated by itself, as an additional measure of plagiarism check, and it showed superior performance compared to the traditional plagiarism-detection tools. The paper discusses the need for institutions to consider appropriate measures to mitigate potential plagiarism issues and advise on the ongoing debate surrounding the impact of AI technology on education. Further implications are discussed in the paper.


Prompt: What are the main causes and effects of the Industrial Revolution?

Output: What are the main causes and effects of the Industrial Revolution?

Output:
Industrial Revolution

Primitivism Primitivism argues that the Industrial Revolution have created an un-natural frame of society and the world in which humans need to adapt to an un-natural urban landscape in which humans are perpetual cogs without personal autonomy.
Certain primitivists argue for a return to pre-industrial society, while others argue that technology such as modern medicine, and agriculture are all positive for humanity assuming they are controlled by and serve humanity and have no effect on the natural environment.
Pollution and ecological collapse
The Industrial Revolution has been criticised for leading to immense ecological and habitat destruction. It has led to immense decrease in the biodiversity of life on Earth. The Industrial revolution has been said to be inherently unsustainable and will lead to eventual collapse of society, mass hunger, starvation, and resource scarcity.
The Anthropocene
The Anthropocene is a proposed epoch or mass extinction coming from humanity (anthropo- is the Greek root for humanity). Since the start of the Industrial revolution humanity has permanently changed the Earth, such as immense decrease in biodiversity, and mass extinction caused by the Industrial revolution. The effects include permanent changes to the Earth's atmosphere and soil, forests, the mass destruction of the Industrial revolution has led to catastrophic impacts on the Earth. Most organisms are unable to adapt leading to mass extinction with the remaining undergoing evolutionary rescue, as a result of the Industrial revolution.


Prompt: What are the main causes and effects of the Industrial Revolution?

Output: What are the main causes and effects of the Industrial Revolution?

Output:
De-industrialisation of India

The fall in the hegemony of Mughals reduced the overall productivity of agriculture and reduced the supply of grains. The grain was the primary consumption good for the Indian workers and was non-tradeable. The reduction in the supply of grain resulted in the rise of its prices. This rise in prices and negative supply shock led to a rise in the nominal wages in the cotton and weaving industry. The increased competition from British cotton and rising nominal wages reduced the profitability of the cotton industry of India. Thus, the negative supply shock in agricultural production is also an important reason behind the de-industrialisation of cotton–industries.
The short run as well as long run impact on living standards and growth rate of GDP providing agriculture sector competitive advantage with strengthening of the productivity advance on the land at home or increasing openness to world in turn increases GDP in the short run. The causes of de-industrialisation are region or country specific as in the case of India in the 19th and 20th century. The colonial rule under the British led to the decline of textile and handicrafts industries through their policies and introduction of machine made goods in to the Indian market. Some of the causes of de-industrialisation in India during that period were:
Introduction of machine made goods in the Indian subcontinent at a cheaper rate, which led to the decline of the traditional textile industry of India.


Prompt: are electrical outlets on studs?

Output: are electrical outlets on studs?

Output:
Electronics Stackexchange

Q: Purpose of breadboard "studs" on base off micro breadboards? I recently bought some micro breadboards on Ebay.  Extremely useful little things.  However each have two awkward studs on their bases.  See the attached photo.   What I'm curious about is their purpose?  What might these "plugin" to or be used for?  I think I'm going to file them off.

A: They connect to a base that fixes them with regards to each other.


Prompt: are electrical outlets on studs?

Output: are electrical outlets on studs?

Output:
Math Stackexchange

Q: Write $\cos^2(x)$ as linear combination of $x \mapsto \sin(x)$ and $x \mapsto \cos(x)$
Can we write $\cos^2(x)$ as linear combination of $x \mapsto \sin(x)$ and  $x \mapsto \cos(x)$?

I know
$$
\cos^2(x)
= \frac{\cos(2x) + 1}{2}
= 1 - \sin^2(x)
= \cos(2x) + \sin^2(x)
$$
but none of these helped.
Then, I tried to solve
$$
\cos^2(x) = \alpha \sin(x) + \beta \cos(x)
$$
for the coefficients $\alpha, \beta \in \mathbb{R}$.
But when plugging in $x = 0$ I get $\beta = 1$ and for $x = \frac{\pi}{2}$ I get $\alpha = 0$. Plugging those values back in I obtain a false statement, and WolframAlpha can't do better!
This is from a numerical analysis exam and the second function is $x \mapsto \sqrt{2}\cos\left(\frac{\pi}{4} - x \right)$, which can easily be expressed in terms of $x \mapsto \sin(x)$ and $x \mapsto \cos(x)$ by the corresponding addition formula.

A: The function $f(x):=\cos^2 x$ has $f(x+\pi)\equiv f(x)$, but any linear combination $g$ of $\cos$ and $\sin$ has $g(x+\pi)\equiv -g(x)$.


Prompt: What are the differences between machine learning and deep learning?

Output: What are the differences between machine learning and deep learning?

Output:
Comparison of deep learning software

The following table compares notable software frameworks, libraries and computer programs for deep learning.
Deep-learning software by name
Comparison of compatibility of machine learning models


Prompt: What are the differences between machine learning and deep learning?

Output: What are the differences between machine learning and deep learning?

Output:
Deep learning

Overview Most modern deep learning models are based on multi-layered neural networks such as convolutional neural networks and transformers, although they can also include propositional formulas or latent variables organized layer-wise in deep generative models such as the nodes in deep belief networks and deep Boltzmann machines.
Fundamentally, deep learning refers to a class of machine learning algorithms in which a hierarchy of layers is used to transform input data into a slightly more abstract and composite representation. For example, in an image recognition model, the raw input may be an image (represented as a tensor of pixels). The first representational layer may attempt to identify basic shapes such as lines and circles, the second layer may compose and encode arrangements of edges, the third layer may encode a nose and eyes, and the fourth layer may recognize that the image contains a face.
Importantly, a deep learning process can learn which features to optimally place in which level on its own. Prior to deep learning, machine learning techniques often involved hand-crafted feature engineering to transform the data into a more suitable representation for a classification algorithm to operate upon. In the deep learning approach, features are not hand-crafted and the model discovers useful feature representations from the data automatically. This does not eliminate the need for hand-tuning; for example, varying numbers of layers and layer sizes can provide different degrees of abstraction.


Prompt: when to use an air gap for dishwasher

Output: when to use an air gap for dishwasher

Output:
Air gap (networking)

An air gap, air wall, air gapping or disconnected network is a network security measure employed on one or more computers to ensure that a secure computer network is physically isolated from unsecured networks, such as the public Internet or an unsecured local area network. It means a computer or network has no network interface controllers connected to other networks, with a physical or conceptual air gap, analogous to the air gap used in plumbing to maintain water quality.


Prompt: when to use an air gap for dishwasher

Output: when to use an air gap for dishwasher

Output:
Air gap (networking)

An air gap, air wall, air gapping or disconnected network is a network security measure employed on one or more computers to ensure that a secure computer network is physically isolated from unsecured networks, such as the public Internet or an unsecured local area network. It means a computer or network has no network interface controllers connected to other networks, with a physical or conceptual air gap, analogous to the air gap used in plumbing to maintain water quality.


Prompt: A paper on scalable and accurate graph transformers using sparse attention and expander graphs

Output: A paper on scalable and accurate graph transformers using sparse attention and expander graphs

Output:
Title: Exphormer: Sparse Transformers for Graphs

Abstract: Graph transformers have emerged as a promising architecture for a variety of graph learning and representation tasks. Despite their successes, though, it remains challenging to scale graph transformers to large graphs while maintaining accuracy competitive with message-passing networks. In this paper, we introduce Exphormer, a framework for building powerful and scalable graph transformers. Exphormer consists of a sparse attention mechanism based on two mechanisms: virtual global nodes and expander graphs, whose mathematical characteristics, such as spectral expansion, pseduorandomness, and sparsity, yield graph transformers with complexity only linear in the size of the graph, while allowing us to prove desirable theoretical properties of the resulting transformer models. We show that incorporating Exphormer into the recently-proposed GraphGPS framework produces models with competitive empirical results on a wide variety of graph datasets, including state-of-the-art results on three datasets. We also show that Exphormer can scale to datasets on larger graphs than shown in previous graph transformer architectures. Code can be found at \url{https://github.com/hamed1375/Exphormer}.


Prompt: A paper on scalable and accurate graph transformers using sparse attention and expander graphs

Output: A paper on scalable and accurate graph transformers using sparse attention and expander graphs

Output:
Title: Deformable Graph Transformer

Abstract: Transformer-based models have recently shown success in representation learning on graph-structured data beyond natural language processing and computer vision. However, the success is limited to small-scale graphs due to the drawbacks of full dot-product attention on graphs such as the quadratic complexity with respect to the number of nodes and message aggregation from enormous irrelevant nodes. To address these issues, we propose Deformable Graph Transformer (DGT) that performs sparse attention via dynamically sampled relevant nodes for efficiently handling large-scale graphs with a linear complexity in the number of nodes. Specifically, our framework first constructs multiple node sequences with various criteria to consider both structural and semantic proximity. Then, combining with our learnable Katz Positional Encodings, the sparse attention is applied to the node sequences for learning node representations with a significantly reduced computational cost. Extensive experiments demonstrate that our DGT achieves state-of-the-art performance on 7 graph benchmark datasets with 2.5 - 449 times less computational cost compared to transformer-based graph models with full attention.


Prompt: What are the key principles of quantum mechanics?

Output: What are the key principles of quantum mechanics?

Output:
Quantum mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of nature at and below the scale of atoms. It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.
Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic/microscopic) scale.
Quantum systems have bound states that are quantized to discrete values of energy, momentum, angular momentum, and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves (wave–particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).


Prompt: What are the key principles of quantum mechanics?

Output: What are the key principles of quantum mechanics?

Output:
Introduction to quantum mechanics

Many aspects of quantum mechanics are counterintuitive and can seem paradoxical because they describe behavior quite different from that seen at larger scales. In the words of quantum physicist Richard Feynman, quantum mechanics deals with "nature as She is—absurd". Features of quantum mechanics often defy simple explanations in everyday language. One example of this is the uncertainty principle: precise measurements of position cannot be combined with precise measurements of velocity. Another example is entanglement: a measurement made on one particle (such as an electron that is measured to have spin 'up') will correlate with a measurement on a second particle (an electron will be found to have spin 'down') if the two particles have a shared history. This will apply even if it is impossible for the result of the first measurement to have been transmitted to the second particle before the second measurement takes place.
Quantum mechanics helps us understand chemistry, because it explains how atoms interact with each other and form molecules. Many remarkable phenomena can be explained using quantum mechanics, like superfluidity. For example, if liquid helium cooled to a temperature near absolute zero is placed in a container, it spontaneously flows up and over the rim of its container; this is an effect which cannot be explained by classical physics.
History
James C. Maxwell's unification of the equations governing electricity, magnetism, and light in the late 19th century led to experiments on the interaction of light and matter. Some of these experiments had aspects which could not be explained until quantum mechanics emerged in the early part of the 20th century.
Evidence of quanta from the photoelectric effect


Prompt: What is the history and significance of the Silk Road?

Output: What is the history and significance of the Silk Road?

Output:
Silk Road

The Silk Road was a network of Eurasian trade routes active from the second century BCE until the mid-15th century. Spanning over , it played a central role in facilitating economic, cultural, political, and religious interactions between the Eastern and Western worlds. The name "Silk Road" was first coined in the late 19th century, but some 20th- and 21st-century historians instead prefer the term Silk Routes, on the grounds that it more accurately describes the intricate web of land and sea routes connecting Central, East, South, Southeast, and West Asia as well as East Africa and Southern Europe.


Prompt: What is the history and significance of the Silk Road?

Output: What is the history and significance of the Silk Road?

Output:
Silk Road

The Silk Road was a network of Eurasian trade routes active from the second century BCE until the mid-15th century. Spanning over , it played a central role in facilitating economic, cultural, political, and religious interactions between the Eastern and Western worlds. The name "Silk Road" was first coined in the late 19th century, but some 20th- and 21st-century historians instead prefer the term Silk Routes, on the grounds that it more accurately describes the intricate web of land and sea routes connecting Central, East, South, Southeast, and West Asia as well as East Africa and Southern Europe.


Prompt: How do black holes form and what are their main properties?

Output: How do black holes form and what are their main properties?

Output:
Black hole

Formation and evolution Given the bizarre character of black holes, it was long questioned whether such objects could actually exist in nature or whether they were merely pathological solutions to Einstein's equations. Einstein himself wrongly thought black holes would not form, because he held that the angular momentum of collapsing particles would stabilize their motion at some radius. This led the general relativity community to dismiss all results to the contrary for many years. However, a minority of relativists continued to contend that black holes were physical objects, and by the end of the 1960s, they had persuaded the majority of researchers in the field that there is no obstacle to the formation of an event horizon.
Penrose demonstrated that once an event horizon forms, general relativity without quantum mechanics requires that a singularity will form within. Shortly afterwards, Hawking showed that many cosmological solutions that describe the Big Bang have singularities without scalar fields or other exotic matter. The Kerr solution, the no-hair theorem, and the laws of black hole thermodynamics showed that the physical properties of black holes were simple and comprehensible, making them respectable subjects for research. Conventional black holes are formed by gravitational collapse of heavy objects such as stars, but they can also in theory be formed by other processes.


Prompt: How do black holes form and what are their main properties?

Output: How do black holes form and what are their main properties?

Output:
Black hole

Formation and evolution Given the bizarre character of black holes, it was long questioned whether such objects could actually exist in nature or whether they were merely pathological solutions to Einstein's equations. Einstein himself wrongly thought black holes would not form, because he held that the angular momentum of collapsing particles would stabilize their motion at some radius. This led the general relativity community to dismiss all results to the contrary for many years. However, a minority of relativists continued to contend that black holes were physical objects, and by the end of the 1960s, they had persuaded the majority of researchers in the field that there is no obstacle to the formation of an event horizon.
Penrose demonstrated that once an event horizon forms, general relativity without quantum mechanics requires that a singularity will form within. Shortly afterwards, Hawking showed that many cosmological solutions that describe the Big Bang have singularities without scalar fields or other exotic matter. The Kerr solution, the no-hair theorem, and the laws of black hole thermodynamics showed that the physical properties of black holes were simple and comprehensible, making them respectable subjects for research. Conventional black holes are formed by gravitational collapse of heavy objects such as stars, but they can also in theory be formed by other processes.


Prompt: What are the implications of Gödel’s incompleteness theorems on the foundations of mathematics and computer science?

Output: What are the implications of Gödel’s incompleteness theorems on the foundations of mathematics and computer science?

Output:
Gödel's incompleteness theorems

Observe then, that if we can prove that the system  is consistent (ie. the statement in the hypothesis of ), then we have proved that  is not provable. But this is a contradiction since by the 1st Incompleteness Theorem, this sentence (ie. what is implied in the sentence , """ is not provable") is what we construct to be unprovable. Notice that this is why we require formalizing the first Incompleteness Theorem in : to prove the 2nd Incompleteness Theorem, we obtain a contradiction with the 1st Incompleteness Theorem which can do only by showing that the theorem holds in . So we cannot prove that the system  is consistent. And the 2nd Incompleteness Theorem statement follows.
Discussion and implications
The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system of formal logic to define their principles.
Consequences for logicism and Hilbert's second problem
The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic. Bob Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply equally to first-order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.


Prompt: What are the implications of Gödel’s incompleteness theorems on the foundations of mathematics and computer science?

Output: What are the implications of Gödel’s incompleteness theorems on the foundations of mathematics and computer science?

Output:
Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of  in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers.  For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.
The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.
Formal systems: completeness, consistency, and effective axiomatization