False Statement Quote by Melanie Mitchell
““This statement is not provable.” Think about it for a minute. It’s a strange statement, since it talks about itself—in fact, it asserts that it is not provable. Let’s call this statement “Statement A.” Now, suppose Statement A could indeed be proved. But then it would be false (since it states that it cannot be proved). That would mean a false statement could be proved—arithmetic would be inconsistent. Okay, let’s assume the opposite, that Statement A cannot be proved. That would mean that Statement A is true (because it asserts that it cannot be proved), but then there is a true statement that cannot be proved—arithmetic would be incomplete. Ergo, arithmetic is either inconsistent or incomplete.””
About This Quote
Source Lecture: Introduction to Logic and Computability, 2019
A self-referential claim that it cannot be proved leads to a paradox showing arithmetic must be either inconsistent or incomplete.
In simple terms: A statement about its own unprovability creates a logical dilemma.
Key Takeaway
Recognize limits of formal systems.
Themes
logic
paradox
foundations of mathematics
Gödel’s incompleteness
Mood
thoughtful
inquisitive
Type
philosophical
analytical
When to use this quote
- formal proof
- mathematical research
- philosophical debate
Key Concepts
self‑reference
provability
consistency
incompleteness
Questions to Reflect On
- Can we accept an incomplete but consistent system?
- How does this affect trust in mathematical foundations?
A Different Perspective
If arithmetic were inconsistent, any statement could be proved; if incomplete, some truths remain unprovable.
![“Interesting-sounding, supposed applications of the incompleteness theorem [here earlier taken explicitly to refer to the incompleteness theorems as a whole] outside mathematics, therefore, often ignore the essential condition of encompassing some basic arithmetic, and formulate the theorem…” quote by Torkel Franz](https://sms4smile.com/quotes/img-quotes/6a9/torkel-franz-131664_quote_interesting-sounding-supposed-applicatio_413379.jpg)
