Logic
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Set Theory
The “no self-defeating object” argument « What’s new
A fundamental tool in any mathematician’s toolkit is that of reductio ad absurdum : showing that a statement is false by assuming first that is true, and showing that this leads to a logical contradiction. A particulary pure example of reductio ad absurdum occurs when establishing the non-existence of a hypothetically overpowered object or structure , by showing that ‘s powers are “self-defeating”: the very existence of and its powers can be used (by some clever trick) to construct a counterexample to that power. Perhaps the most well-known example of a self-defeating object comes from the omnipotence paradox in philosophy (“Can an omnipotent being create a rock so heavy that He cannot lift it?”); more generally, a large number of other paradoxes in logic or philosophy can be reinterpreted as a proof that a certain overpowered object or structure does not exist.
The “no self-defeating object” argument, and the vagueness paradox « What’s new
This is the third in a series of posts on the “no self-defeating object” argument in mathematics – a powerful and useful argument based on formalising the observation that any object or structure that is so powerful that it can “defeat” even itself, cannot actually exist. This argument is used to establish many basic impossibility results in mathematics, such as Gödel’s theorem that it is impossible for any sufficiently sophisticated formal axiom system to prove its own consistency, Turing’s theorem that it is impossible for any sufficiently sophisticated programming language to solve its own halting problem, or Cantor’s theorem that it is impossible for any set to enumerate its own power set (and as a corollary, the natural numbers cannot enumerate the real numbers).One notable feature of mathematical reasoning is the reliance on counterfactual thinking – taking a hypothesis (or set of hypotheses) which may or may not be true, and following it (or them) to its logical conclusion. For instance, most propositions in mathematics start with a set of hypotheses (e.g. “Let be a natural number such that …”), which may or may not apply to the particular value of one may have in mind.
The “no self-defeating object” argument, revisited « What’s new
Kurt Gödel is best known to mathematicians and the general public for his celebrated incompleteness theorems. Physicists also know his famous cosmological model in which time-like lines close back on themselves so that the distance past and the distant future are one and the same. What is less well known is the fact that Gödel has sketched a revised version of Anselm's traditional ontological argument for the existence of God. How does a mathematician get mixed up in the God-business? Gödel was a mystic, whose mathematical research exemplified a philosophical stance akin to the Neo-Platonics. In this respect, Gödel had as much in common with the medieval theologians and philosophers as the twentieth-century mathematicians who pioneered the theory of computation and modern computer science.
Kurt Gödel's Ontological Argument
Intuitionistic mathematics for physics « Mathematics and Computation
At MSFP 2008 in Iceland I chatted with Dan Piponi about physics and intuitionistic mathematics, and he encouraged me to write down some of the ideas. I have little, if anything, original to say, so this seems like an excellent opportunity for a blog post. So let me explain why I think intuitionistic mathematics is good for physics . Intuitionistic mathematics, whose main proponent was L.E.J. Brouwer , is largely misunderstood by mathematicians. Consequently, physicists have strange ideas about it, too.How to solve the hardest logic puzzle ever in two questions
Rabern and Rabern ( 2008 ) have noted the need to modify ‘the hardest logic puzzle ever’ as presented in Boolos 1996 in order to avoid trivialization. Their paper ends with a two-question solution to the original puzzle, which does not carry over to the amended puzzle. The purpose of this note is to offer a two-question solution to the latter puzzle, which is, after all, the one with a claim to being the hardest logic puzzle ever. Recall, first, Boolos's statement of the puzzle: Three gods A, B and C are called, in some order, True, False and Random.
In 1992, the philosopher George Boolos gave what he called the “Hardest Logic Puzzle Ever” , which he attributed to Raymond Smullyan. In 2008, a clever paper by two graduate students, Brian Rabern and Landon Rabern , appeared in the philosophical journal “Analysis” which gave a simpler solution to the puzzle than Boolos gave—and furthermore claimed that a solution to a stronger puzzle was possible! As its name implies, the “Hardest Logic Puzzle Ever” has a number of complicating factors which will be irrelevant for this discussion. Instead, consider the following much simpler puzzle which will do just as well. You are on an island populated by knights and knaves. Knights always tell the truth; knaves always lie.
