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Gödel Without Tears, slowly, 7

Next, we have the shortest chapter so far, on Quantifier Complexity, which introduces the notions of and wffs. But there is an intriguing little result in this chapter. If the consistent theory T which includes Robinson Arithmetic Q proves a … Continue reading → The post Gödel Without Tears, slowly, 7 appeared first on Logic [More]

Protagoras

[New Entry by Mauro Bonazzi on September 8, 2020.] Protagoras (490 - 420 BCE ca) was one of the most important sophists and exerted considerable influence in fifth-century intellectual debates. His teaching had a practical and concrete goal, and many of the surviving testimonies and fragments suggest that it was mainly devoted to the development of argumentative techniques. But some of his views also raise important philosophical problems, which were going to be discussed in details by Plato, Aristotle, and many other philosophers. His famous thesis according to which "man [More]

Gödel Without Tears, slowly, 6

We get back to proving the First Incompleteness Theorem next week. In this week’s five chapters, we are reading into the record some necessary arithmetical background. Everything in today’s chapter will very probably be quite familiar; but we do need … Continue reading → The post Gödel Without Tears, slowly, 6 appeared first on Logic [More]

Boethius of Dacia

[New Entry by Sten Ebbesen on September 7, 2020.] Boethius of Dacia was a leading philosopher at the faculty of arts in Paris about 1270 - 1275. He developed the Aristotelian idea of the autonomy of each domain of knowledge in a way that could justify Aristotelian-style natural science and ethics in spite of disagreements with revealed truth. He also made a major contribution to the linguistic theory now known as [More]

The lure of imperial dreams: What are our leaders going to do to us?

Donald Trump is often represented wearing some kind of imperial garb. Actually, his presidency may have been less imperial than that of his predecessors. Yet, his style as president is very much "imperial" and his winning slogan in the 2016 elections, "MAGA," (make America great again) has a deep imperial ring to it. Earlier on, Benito Mussolini, the leader of the Italian government between the two world wars, was destroyed (and with him Italy and not just Italy) by his Imperial ambitions.  When things get tough, people seem to think that they need tough leaders and this is a clear trend in the world, nowadays. It is a deadly mechanism that tends to bring dangerous psychotic personalities to the top government positions. I already noted in a previous post how imperial ambitions coupled with incompetence (both common conditions in high-level leaders) can destroy entire countries. Here, let me examine an interesting feature of how Benito Mussolini (1883 -1945) ruled Italy. Despite his warlike rhetoric, during the first phase of his government he pursued a moderate foreign policy, avoiding wars. Then, the second phase of his rule was characterized by a series of disastrous wars that led to the destruction of Italy (and not just of Italy) and to the downfall of Mussolini himself. Whether this story can tell us something about a possible second term for Donald Trump as president, is left to the readers to decide. Benito Mussolini and the Italian Empire: How [More]

Gödel Without Tears, slowly, 5

Where have we got to in this slow introduction to the incompleteness theorems? Theorem  8 from Chapter 4 tells us that if a consistent, effectively axiomatized, theory is “sufficiently strong”, then it must be negation incomplete. And we announced that … Continue reading → The post Gödel Without Tears, slowly, 5 appeared first on Logic [More]