An essay review of the book Godel: A Life of Logic, by John L Casti
This is a captivating tale of mathematical heroism. It follows the journey of Kurt Godel, a shy young mathematical genius who embarks on a quest to challenge the reigning mathematical giant of his time, David Hilbert.
For most of the 20th century, Hilbert held sway as the heavyweight champion of mathematics. And as part of this high honor, he was granted the privilege of promulgating fundamental questions about mathematics and logic, particularly those that could potentially undermine the very foundation of the discipline.
One such question that Hilbert posed was whether it is possible to prove the truth of every valid mathematical statement.
Hilbert envisioned a truth machine that could accept a mathematical statement and, then with a simple turn of the crank, provide an unequivocal answer either true or false, but never both.
In response to Hilbert's machine, Kurt Godel, a young unassuming member of the renowned Vienna Circle, emerged in 1928 with his incompleteness theorem.
This groundbreaking discovery revealed that the system of logic upon which arithmetic relies is inherently incomplete.
In essence, entering a valid mathematical statement into Hilbert's machine and turning the crank could not guarantee that the machine would yield answers that were only true or false. They could be both.
Godel's upbringing was shaped by the esteemed Vienna Circle, which included figures like Franz Kafka, Karl Popper, John Von Neumann, Bertrand Russell, Albert Einstein, and most notably, the eccentric yet brilliant philosopher and logician, Ludwig Wittgenstein. Godel shared a deep bond with Wittgenstein.
Wittgenstein believed in a logical mirroring between the facts of language and the facts of reality. He argued that language serves as the foundation upon which we represent the world in thought, enabling us to comprehend the nature of reality.
However, he also acknowledged that this parallel between language and reality may not encompass all aspects of reality.
Godel gave mathematical expression to Wittgensteins doubt that some aspects of reality might be left out. And this sliver of doubt proved to be all that was necessary for Godel to demonstrate, as Hilbert's question had suggested, that our system of logic might indeed possess a flaw at its very core.
Both a clear statement of the problem and a well-defined framework for its solution turned out to be the sine qua non to understanding Godel's theorem.
Numerous books have been dedicated to simplifying these aspects and making them more accessible to the general reader.
Each book has its strengths and weaknesses. For instance, of my favorite three, Rebecca Goldsteins book: Incompleteness: The Proof and Paradox of Kurt Godel, provides an excellent description of the problem but fails in a gallant attempt to make the intricate details of the solution comprehensible to the untrained mathematical mind.
On the other hand, Douglas Hofstadters award winning Godel, Escher and Bach employs cartoon characters and creative games to elucidate the intricacies of Godel's proof. However, it drowns the reader in explanatory details, and in the end neglects to adequately define and frame the problem.
Steven Dubianskys, Journey to the Edge of reason, sets the context of Godels discovery, but only vaguely outlines his theorem.
The authors of this book excel in both aspects. They provide a clear definition of the problem and explain the details of Godel's theorem. Moreover, they demonstrate the significance of Godel's theorem in various fields, including mathematics, artificial intelligence, logic, and analytic philosophy.
An outline of Godel's incompleteness theorem
In its most concise form, Godel's Theorem states that for every consistent formalization of arithmetic, there exist arithmetic truths that are not provable within that system.
On page 51, the authors present a concise outline of Godel's proof, that is difficult to improve upon.
1. Godel first develops a coding scheme using numbers to represent statements. This scheme translates every logical formula in arithmetic into a mirror image statement about the natural numbers.
2. He then replaces the concept of truth with the idea of provability.
3. Godel then shows that logical sentences have arithmetical counterparts called Godel sentences (G). He proves that the Godel sentence G must be true if the formal system is consistent.
4. Godel demonstrates that even if a new system is formed by adding new axioms, there would still be unprovable Godel sentences within that system.
5. Finally, using his coding system, Godel shows that the statement arithmetic is consistent is not provable, thus demonstrating that arithmetic as a formal system is insufficient to prove its own consistency. QED. Five stars
(Article changed on Oct 08, 2025 at 3:40 PM EDT)
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