May 2027 TOK Essay Title 3: Knowledge Built on Assumptions

1. Title decoded · 2. Hidden assumptions · 3. Knowledge questions · 4. TOK concept map

5. Three thesis positions · 6. Choosing areas of knowledge · 7. Argument map · 8. Perspectives

9. Ten verified examples, each on its own card · 10. Overused examples

11. Paragraph blueprint · 12. Word budget · 13. Implications bank · 14. Conclusion options · 15. Pitfalls

16. Top-band checklist · 17. Teacher interactions (TK/PPF) · 18. Glossary · 19. Bibliography

Sustained focus on the title

Every paragraph explains how reliability survives (or fails to survive) assumptions, not just that assumptions exist.

Sections 1, 5, 14

Linked effectively to areas of knowledge

Your two areas of knowledge are analysed as ways of producing knowledge, and compared directly.

Sections 6, 7

Arguments clear, coherent and effectively supported by specific examples

Each claim rests on a named, dated, real example that is analysed, not just described.

Sections 7, 9, 11

Implications of arguments are considered

You explain what follows if your argument is right, for knowers and for knowledge.

Section 13

Clear awareness and evaluation of different points of view

Other views are weighed, and you say why one is stronger in a given case.

Sections 7, 8

Insightful, convincing, accomplished, lucid

The IB's own list of characteristics for this band. Clarity and originality count.

Sections 2, 9, 15

“In the production of knowledge”

The focus is on how knowledge is made: methods, models, measurements and data.

Writing about how knowledge is used or taught instead of how it is produced.

“how can knowledge be reliable”

A “how” question. You must explain mechanisms that make knowledge dependable despite its foundations.

Answering “it can't” or “it can” without explaining how.

“reliable”

Dependable enough to act on. Reliable for what purpose, for whom, and within what limits? Reliability is not the same as certainty or truth.

Treating reliable as meaning certain or proven.

“built on assumptions”

Assumptions range from declared starting points (axioms, model simplifications, units, definitions) to hidden ones nobody notices (who is in the sample, what counts as “average”).

Treating all assumptions as the same, or as the same thing as bias.

“the human sciences and one other area”

The human sciences are fixed. The second area should handle assumptions differently.

Choosing a second area that behaves just like the human sciences.

Is reliability a property of knowledge itself, or of how well its limits are understood?

Whose experience is built into the assumptions of a model or measurement, and whose is left out?

Does making assumptions explicit make knowledge more reliable?

What responsibilities do knowledge producers have to disclose the assumptions behind their claims?

Can knowledge be reliable for practical purposes even if it is built on assumptions we know are false?

Justification

Methods and tools

If knowledge rests on assumptions, what justifies trusting it? Track record, testing, or the transparency of the assumptions?

Certainty

Scope

Mathematics gives certainty relative to its axioms. The human sciences rarely reach certainty; reliability may be a better goal.

Evidence

Methods and tools

Evidence tests assumptions, but only if someone looks. Examples 2 and 5 show evidence overturning results once assumptions were checked.

Objectivity

Methods and tools

Hidden assumptions can undermine objectivity without anyone intending bias (Examples 6, 7).

Perspective

Perspectives

The “average person” in a model is someone's perspective. Whose? (Examples 5, 7)

Power

Perspectives

Assumptions in official statistics shape policy and can be contested politically (Examples 3, 4).

Truth

Scope

Can a result be true within a system yet say nothing reliable about the world? (Example 9)

Explanation

Scope

Models explain by simplifying. When does simplifying become distorting? (Example 1)

Interpretation

Methods and tools

The same data, read under different assumptions, give different conclusions (Examples 2, 3, 4).

Responsibility

Ethics

Who is responsible when an unexamined assumption causes harm? (Examples 6, 8)

Values

Ethics

Choosing an assumption can be a value choice, such as where to draw a poverty line (Example 4).

Culture

Perspectives

Samples from one culture may be assumed to represent all humans (Example 5 and the WEIRD critique).

Knowledge built on assumptions can be reliable when the assumptions are stated openly, conclusions are limited to where the assumptions hold, and there are ways to test them.

It gives a clear answer to “how”, and the contrast between declared axioms in mathematics and hidden assumptions in the human sciences does the comparing for you.

Mathematics may look too easy. Use Examples 9 and 10 to show that even declared assumptions carry surprises.

Without assumptions there is no knowledge at all. Assumptions fix what we are measuring and how, which is what allows results to be checked and repeated.

A counter-intuitive thesis that reads as insightful if well supported.

You must deal with cases where assumptions caused serious harm (Examples 1, 6, 8).

Especially in the human sciences, hidden assumptions make knowledge less reliable than it appears; confidence usually comes from the track record, which can hide a flaw for decades.

Strong examples support it (Examples 1, 2, 5), and it is a real position, not a straw man.

It can slide into “we can't know anything”. Explain how the human sciences have improved by exposing assumptions.

Human sciences + mathematics

Mathematics declares its assumptions (axioms); the human sciences often discover theirs only when predictions fail. Examples 1–5, 9, 10.

Economics, Psychology, Maths AA

Human sciences + natural sciences

Natural sciences test assumptions against physical measurements; human sciences study people who react to being studied. Examples 1–8.

Economics, Psychology, Physics, Biology

Economics focus + mathematics

Economics borrows mathematical form. Does that make its assumptions more reliable, or only look more reliable? Examples 1, 2, 3, 9.

Economics HL, Maths AA HL

Assumptions are chosen for good reasons: simplicity, data availability, tractability (Examples 1, 4).

Good reasons for an assumption do not guarantee it holds. Ask how the model-builder tests it.

Often the person who exposes the assumption (Examples 2, 3, 5).

Critics make assumptions too. Example 3's critique used its own set of proxy indicators.

Bear the cost when the “average” excludes them (Examples 6, 7).

Their experience is evidence, but individual cases need systematic data to become knowledge.

Produce official knowledge and defend its assumptions (Examples 3, 4).

They may have political interests; so may their critics. Evaluate both.

Most accept useful axioms even when they give strange results (Example 9).

Some (constructivists) reject certain axioms. What does disagreement at the foundations mean for reliability?

Test assumptions against physical reality, where errors show up fast (Example 8).

Failures show up fast but can be catastrophic. Is fast feedback the key to reliability?

1

Alan Greenspan's “flaw” testimony (October 2008)

Counter

Rare

2

Reinhart and Rogoff's 90% debt threshold (2010–2013)

Both

Less common

3

The debate over India's GDP series (2015–2019)

Both

Rare

4

The World Bank's new poverty line (June 2025)

Both

Rare

5

The marshmallow test revisited (1990 and 2018)

Both

Less common

6

Pulse oximeters and skin pigmentation (2020)

Counter

Less common

7

The first average-female crash-test dummy (2022–2023)

Both

Less common

8

The Mars Climate Orbiter (1999)

Counter

Less common

9

The Banach–Tarski paradox and the axiom of choice (1924)

Both

Less common

10

Gödel's incompleteness theorems (1931)

Counter

Less common

Human sciences: economics

Alan Greenspan chaired the US Federal Reserve from 1987 to 2006 and was widely admired. After the 2008 financial crisis, he told a congressional committee he was in “shocked disbelief” that lending institutions' self-interest had failed to protect their shareholders. Asked whether his ideology had led him astray, he said he had found a flaw in the model he believed defined how the world works, after some forty years of what he saw as considerable evidence that it worked well.

A core assumption (that institutions acting in self-interest will manage risk) produced decades of apparently reliable results, then failed badly. It shows that a long track record can hide an assumption's weakness.

Counter-claim

Methods and tools, Scope. Concepts: justification, explanation, certainty.

• If forty years of evidence seemed to confirm the assumption, what kind of evidence would have exposed it earlier?

• Is a model reliable if it works most of the time but fails catastrophically when it matters most?

• Greenspan called it a “model” and an “ideology”. Is there a difference, and does it matter for reliability?

Economists still disagree about the crisis's causes. Use Greenspan's own statement, not a claim that one assumption caused everything.

Economics students studying market failure and regulation.

• Al Jazeera, October 2008

• New York Times report, 24 October 2008 (course archive copy)

Human sciences: economics

In 2010 two Harvard economists, Carmen Reinhart and Kenneth Rogoff, published a paper suggesting that economic growth falls sharply when public debt exceeds 90% of GDP. It was widely cited in debates about cutting government spending. In 2013 a University of Massachusetts graduate student, Thomas Herndon, with Michael Ash and Robert Pollin, obtained the spreadsheet. They found some data excluded, a spreadsheet formula that left out five countries, and a weighting method under which one year of New Zealand data counted as much as 19 years of UK data. Corrected, average growth above 90% was about 2.2%, not −0.1%. Reinhart and Rogoff accepted the spreadsheet error but defended their other choices.

The result depended on assumptions about which data to include and how to weight it, which were invisible until someone repeated the calculation. It shows both how assumptions distort and how open checking restores reliability.

Claim and counter-claim

Methods and tools. Concepts: evidence, interpretation, justification.

• Which of the three problems was an error, and which were assumptions reasonable people could dispute?

• Why did it take three years, and a student, to find the problem?

• Does the correction show that economics is reliable (it self-corrects) or unreliable (policy was already made)?

The original authors argue the corrected median result still shows lower growth at high debt. Present both sides.

Economics students; Maths students interested in weighted averages.

• Robert Pollin, LSE Politics and Policy blog

• New Statesman, April 2013

• World Economic Forum review, April 2013

Human sciences: economics and official statistics

In January 2015 India revised how it calculates GDP, moving the base year from 2004–05 to 2011–12 and changing data sources to align with the UN's System of National Accounts. In June 2019 former Chief Economic Adviser Arvind Subramanian published a Harvard working paper estimating that growth from 2011–12 to 2016–17 may have averaged about 4.5% rather than the official 7%, using 17 indicators such as electricity use and vehicle sales. The Ministry of Statistics rejected this, saying accepted procedures had been followed.

Both the official figures and the critique rest on assumptions: about which data sources measure output, and about which proxy indicators track growth. It shows that reliability can be contested when two sets of assumptions give different answers.

Claim and counter-claim

Methods and tools, Perspectives. Concepts: interpretation, power, evidence.

• What assumption lets electricity use or vehicle sales stand in for GDP? When might it fail?

• How can a citizen decide which set of assumptions to trust?

• Does the political importance of GDP make its assumptions harder to examine openly?

This is politically sensitive. Present both the critique and the government's response neutrally, and avoid claiming either is proven.

Strong for students in India; Economics students studying measurement of growth.

• The Week, June 2019

• Onmanorama: the government's rebuttal, June 2019

• HuffPost India explainer, June 2019

Human sciences: economics and measurement

In June 2025 the World Bank raised its international extreme poverty line from $2.15 to $3.00 a day, after updating price comparisons between countries. Overnight, its estimate of people in extreme poverty in 2022 rose from 713 million to 838 million. Sub-Saharan Africa's rate rose sharply, while South Asia's was revised down. The line is set to reflect the national poverty lines of the poorest countries, several of which had raised their own lines.

Nobody's life changed, yet 125 million people were added to the count. The knowledge depends on assumptions about prices and where to draw the line, and the World Bank states them openly, which is what makes the figures usable.

Claim and counter-claim

Methods and tools, Ethics. Concepts: values, interpretation, certainty.

• Is someone living on $3.01 a day meaningfully less poor than someone on $2.99? What does that say about the assumption of a single line?

• Because the assumptions are published, the change can be explained. Does transparency make the figures more reliable?

• Why might South Asia's rate fall while Africa's rose under the same change?

Poverty lines are tools for comparison, not claims about individual lives. The World Bank says national lines are better for national policy.

Economics and Global Politics students studying development.

• World Bank factsheet, June 2025

• World Bank Data Blog, June 2025

• Our World in Data, August 2025

Human sciences: psychology

Walter Mischel's 1960s experiments at Stanford's Bing Nursery School tested whether young children could wait for a larger treat. A 1990 follow-up linked longer waiting to higher later SAT scores, though that analysis rested on fewer than 100 parent-reported scores, and the authors urged caution. In 2018 Tyler Watts, Greg Duncan and Haonan Quan tested the idea with about 900 children from a national US study, focusing on children of mothers without college degrees. The link was half as large, and shrank by two-thirds once family background, early ability and home environment were controlled for.

The original finding assumed that a small sample from a university nursery told us about children in general, and that waiting reflected self-control rather than circumstances. Testing those assumptions changed the knowledge.

Counter-claim, then claim

Methods and tools, Perspectives. Concepts: evidence, culture, objectivity.

• Which assumption did the 2018 study change: the sample, the controls, or both?

• The original authors warned against over-reading their results. Why did the public ignore the warning?

• Does the 2018 study show psychology is unreliable, or that it is reliable because it tests itself?

A “conceptual replication” uses different methods, so it does not simply prove the original wrong.

Psychology students; students who have heard the “willpower predicts success” story.

• Watts, Duncan and Quan (2018), Psychological Science

• Association for Psychological Science summary

Natural sciences: medical technology

Pulse oximeters, clipped to a finger to estimate blood oxygen, have been in use since the 1980s and were tested mainly on people with lighter skin. US approval rules required only a small share of test subjects with dark pigmentation. In December 2020 a University of Michigan team reported in the New England Journal of Medicine that dangerously low oxygen went undetected about three times as often in Black patients (11.7%) as in white patients (3.6%). The US Food and Drug Administration issued a safety warning in 2021 and has since worked on stricter testing guidance.

A hidden assumption about who the device was calibrated on made a trusted measurement less reliable for some people, and the problem stayed hidden for decades.

Counter-claim

Methods and tools, Ethics. Concepts: objectivity, responsibility, perspective.

• The device measures light absorption. How can a physical measurement carry a social assumption?

• Why did the problem become visible during COVID-19, when oximeters were used far more widely?

• Is a measurement reliable if it is accurate on average but less accurate for a particular group?

The 2020 study was retrospective, and other factors such as blood circulation also affect readings. The FDA noted these limits.

Biology and Physics students; also a link to the natural sciences as the second area of knowledge.

• Anesthesia Patient Safety Foundation statement

• University of Michigan summary of the research

• KFF Health News on FDA guidance

Natural sciences: engineering and physics

Since the 1970s crash-test dummies have been modelled on the average man. Women were represented by a scaled-down male dummy about the size of a 12-year-old girl, and regulations required testing only with the average-male dummy. A team led by Professor Astrid Linder at Sweden's national road and transport research institute (VTI) developed SET 50F, a dummy based on an average woman: 162 cm and 62 kg, with a different neck, shoulders and hips. It was tested from late 2022 and presented publicly in 2023.

Safety knowledge rested on the assumption that the average man represents all adults. Replacing that assumption changes what counts as a safe car seat.

Counter-claim, then claim

Methods and tools, Perspectives. Concepts: objectivity, perspective, evidence.

• Was the old testing unreliable, or reliable only for the people it assumed?

• Why might an assumption like this survive for fifty years in a field that tests things physically?

• Does the new dummy remove the assumption of an “average” body, or just add a second one?

The new dummy was designed for low-speed rear impacts, so it does not settle all questions about crash safety.

Physics students studying forces and collisions; Design Technology students.

• VTI news release, June 2023

• NPR interview with Astrid Linder, November 2022

Natural sciences: engineering

NASA's Mars Climate Orbiter was lost on 23 September 1999 as it tried to enter orbit around Mars. The investigation found that software from contractor Lockheed Martin gave thruster data in pound-force seconds, while NASA's navigation team assumed the figures were in newton-seconds, as the contract specified. The spacecraft flew too close to Mars. The board also found weak checking and poor communication; warning signs in the trajectory before arrival were not acted on.

A simple unstated assumption about units made highly precise calculations unreliable. It shows that even in the most rigorous sciences, knowledge depends on shared assumptions being actually shared.

Counter-claim

Methods and tools. Concepts: certainty, responsibility, justification.

• The mathematics was correct. What exactly failed?

• Why is an assumption about units harder to notice than an assumption in a theory?

• What checking processes would make knowledge built on assumptions reliable?

This is engineering rather than discovery, so link it clearly to how natural-science knowledge is produced.

Physics students (units and dimensional analysis); Computer Science students (software interfaces).

• Analysis of the mishap investigation findings

• SimScale engineering summary

Mathematics

In 1924 Stefan Banach and Alfred Tarski proved that, if the axiom of choice is assumed, a solid ball can be cut into a finite number of pieces and reassembled into two balls each the same size as the first. Later work by Gödel (1930s) and Cohen (1963) showed the axiom can be neither proved nor disproved from the other standard axioms. Most mathematicians accept it because it makes so much other mathematics work; some reject it.

A reasonable-sounding assumption leads to a result that seems impossible, yet the proof is fully reliable given the assumption. It shows reliability relative to assumptions, and the choice mathematicians make about which assumptions to accept.

Claim and counter-claim

Scope, Methods and tools. Concepts: certainty, truth, justification.

• Is the theorem reliable? Reliable about what: the world, or the axiom system?

• Mathematicians accept the axiom for its usefulness. Is usefulness a good reason to accept an assumption?

• Rejecting the axiom leads to other strange results. What does it mean when every set of assumptions has surprising consequences?

Avoid technical detail. The key point is the relationship between an assumption and the reliability of what follows from it.

Maths AA HL students; anyone interested in infinity.

• Encyclopaedia Britannica: axiom of choice

• Stan Wagon, Cambridge University Press blog, 2017

• ThatsMaths: a readable introduction

Mathematics

In the 1920s David Hilbert hoped to place all of mathematics on a complete set of axioms and prove that it contained no contradictions. In 1931 Kurt Gödel proved that any consistent system of axioms rich enough to express arithmetic contains true statements it cannot prove, and cannot prove its own consistency.

Even mathematics cannot prove, from within, that its assumptions are safe. Its reliability rests partly on assumptions it cannot verify, which makes it a powerful comparison with the human sciences.

Counter-claim; limits of reliability

Scope. Concepts: certainty, justification, truth.

• If mathematics cannot prove its own consistency, why do mathematicians still trust it?

• Does this make mathematics more like the human sciences, or is its reliability still of a different kind?

• Is trust built on a long record without contradictions similar to Greenspan's forty years of evidence (Example 1)?

Popular accounts often exaggerate Gödel. Stick to what the theorems say, and do not claim they make mathematics unreliable.

Maths AA HL and Computer Science students.

• Encyclopedia of Mathematics

• EBSCO Research Starters summary

Homo economicus and behavioural economics

The standard example of a flawed assumption; also on our free Title 3 page.

Use Greenspan's own admission (Example 1) as a sharper, documented case.

WEIRD samples in psychology (2010)

Widely used; also on our free page.

Pair it with the marshmallow test (Example 5) to show a specific study changing.

Euclid's parallel postulate

The default mathematics example for assumptions.

Use Banach–Tarski or Gödel (Examples 9, 10) for a fresher angle.

Newton versus Einstein

Common for “assumptions later overturned”.

Ask instead why Newtonian physics is still reliable for most purposes.

“All swans are white”

A classroom cliché about induction.

Replace with a real case where an assumption about a population failed (Examples 6, 7).

1. Claim

One arguable sentence that answers the title.

In the natural sciences, group scrutiny makes knowledge more reliable than individual insight.

2. Example

A named, dated, specific case in two or three sentences.

(A named peer-review failure or success, with date and journal.)

3. Analysis

Explain how the example supports the claim. This is where most marks are.

Show which step of the group process caught (or missed) the error, and why an individual could not have.

4. Link to a knowledge question

Name the underlying question about knowledge.

What makes a method of checking claims trustworthy?

5. Evaluation

Weigh the claim against a counter-view or limitation.

But groups can share the same blind spot (a named case of consensus that was wrong).

6. Link back

Return to the title's exact words.

So groups make knowledge more reliable only when the group itself is diverse.

Introduction

150

Distinguish declared and hidden assumptions briefly; define “reliable”.

The human sciences (claim and counter-claim)

550

Explaining economic theory at length. Keep only what shows the assumption and its consequences.

Second area of knowledge (claim and counter-claim)

550

Technical mathematics or engineering detail. Keep explanations short and focus on the assumption.

Conclusion

250

Summarising each paragraph again. Use the space for your judgement and its implications.

Buffer

100

Keep it for linking sentences. The hard limit is 1,600; examiners stop reading at that point.

State the conditions for reliability → show them met in the second area and often missing in the human sciences → note what the human sciences have learned → one implication.

Show that knowledge needs assumptions to exist → concede the harmful cases → explain why the fix is better assumptions, not none → one implication.

Summarise hidden-assumption failures → acknowledge self-correction → judge whether correction comes too late to call the knowledge reliable → one implication.

Treating assumptions as bias

They overlap but are not the same. Axioms are assumptions without bias.

Define assumption early and distinguish it from bias.

Treating “reliable” as “certain”

Makes the question trivial: nothing is certain, so nothing is reliable.

Define reliable as dependable enough to act on, within limits.

Explaining the maths or economics at length

The lower bands describe work that is “largely descriptive”.

Two or three sentences of explanation, then analysis.

A second area that behaves the same way

Leaves nothing to compare.

Choose mathematics or the natural sciences to contrast with the human sciences.

Only failures

Makes the essay one-sided.

Include cases where exposing assumptions improved reliability.

Political point-scoring

Examples 1–3 can tempt students into arguing policy.

Stay with how the knowledge was produced and tested.

☐ My introduction defines “assumptions” and “reliable” in my own words.

Sustained focus on the title

☐ Every body paragraph ends by linking back to the title's exact words.

Sustained focus on the title

☐ I analyse the human sciences and my second area of knowledge in roughly equal depth.

Linked effectively to areas of knowledge

☐ I compare the two areas of knowledge directly in the body, not only in the conclusion.

Linked effectively to areas of knowledge

☐ Each argument has a named, dated, specific example.

Effectively supported by specific examples

☐ I spend more words analysing examples than describing them.

Arguments are clear, coherent

☐ Each counter-claim has its own example and is taken seriously.

Evaluation of different points of view

☐ I say why one view is stronger in a given case, not just that views differ.

Evaluation of different points of view

☐ At least one implication of my argument is developed.

Implications of arguments are considered

☐ I have used TOK concepts to sharpen arguments, not as decoration.

Insightful

☐ The title is copied exactly, with no changes.

IB rule: modified titles lose relevance

☐ The essay is no more than 1,600 words, and all sources are cited.

IB rules on word count and acknowledgement

Two candidate titles, with two possible examples for each.

Which title lets me build the strongest argument with the examples I understand best?

Your thesis position, two claims, two counter-claims, chosen examples, and the knowledge questions you will use.

Have I explained how reliability survives assumptions? Is my second area a genuine contrast?

Your full draft, within the word limit, with sources cited.

Where is my analysis thinnest? Is my evaluation convincing?

Something taken as true without proof in order to build knowledge on it.

A starting assumption in mathematics from which theorems are proved.

A simplified representation of a system, used to explain or predict.

How far knowledge can be depended on, for a given purpose and within given limits.

Something measured in place of what we really want to know, such as electricity use for economic output.

Repeating a study to see whether the result holds.

The group actually studied, from which conclusions about a larger population are drawn.

Adjusting a measuring device against a known standard.

A system is consistent if it cannot prove a contradiction.

A way of comparing incomes across countries by adjusting for price differences.

International Baccalaureate Organization. (2020). Theory of knowledge guide (first assessment 2022). IBO.

Official IB document; access through your school.

Al Jazeera. (2008, October 26). Greenspan admits “flaw” in ideology. aljazeera.com

International news.

Pollin, R. Public debt, GDP growth, and austerity: why Reinhart and Rogoff are wrong. LSE blog. blogs.lse.ac.uk

Written by a co-author of the critique.

New Statesman. (2013, April 17). Statistic cited to defend austerity partially based on Excel error. newstatesman.com

Includes the original authors' response.

The Week. (2019, June 11). India's GDP growth overestimated by 2.5%, says former CEA. theweek.in

Indian news magazine.

Onmanorama. (2019, June 12). Government rebuts Subramanian. onmanorama.com

Reports the official response.

World Bank. (2025, June 5). June 2025 update to global poverty lines. worldbank.org

Official source.

Our World in Data. (2025). $3 a day: A new poverty line. ourworldindata.org

Respected data publisher.

Watts, T. W., Duncan, G. J., and Quan, H. (2018). Revisiting the marshmallow test. Psychological Science. journals.sagepub.com

Peer-reviewed; primary source.

Anesthesia Patient Safety Foundation. Statement on pulse oximetry and skin tone. apsf.org

Professional body.

VTI. (2023, June 15). First female crash test dummy displayed at VTI. vti.se

The research institute itself.

NPR / WAMC. (2022, November 1). The first female crash test dummy has only now arrived. wamc.org

Public radio interview.

Mars Climate Orbiter: unit conversion and interface accountability. btw.media

Secondary analysis; NASA's board report is the primary source.

Encyclopaedia Britannica. Axiom of choice. britannica.com

Reference work.

Wagon, S. (2017). It is best to accept the Banach–Tarski paradox. Cambridge University Press blog. cambridgeblog.org

Written by a leading author on the topic.

Encyclopedia of Mathematics. Gödel incompleteness theorem. encyclopediaofmath.org

Specialist reference.