Maverick Philosopher

Footnotes to Plato from the foothills of the Superstition Mountains

Category: Set Theory

  • Notes on Infinite Series

    The resident nominalist writes, Your post generated a lot of interest. What I have to say now is better put as a separate post, rather than a long comment. Feel free to post. 1) Plural reference provides a means of dealing with numbers-of-things without introducing extra unwanted entities such as sets. Even realists agree that…

  • On Sets: A Response to Brightly

    David Brightly in a comment far below writes: Bill says, at 03:21 PM. …a family cannot be reduced to a number of persons; it is not a mere manifold, or a mere many… The same is true for a pair (of shoes) and other collective terms. They all imply relations of some sort between their…

  • Nominalism Presupposes What it Denies

    What makes a pair of shoes a pair and not just two physical artifacts? Nominalist answer: nothing in reality. Our resident nominalist tells us that it is our use of 'a pair' that imports a unity, conventional and linguistic in nature, a unity that does not exist in reality apart from our conventional importation. We…

  • Why Not be a Nominalist about Sets?

    The resident nominalist comments: Nominalists say that the conception of an actual infinity of natural numbers depends on there being a set of all such numbers. But Ockhamists do not believe in sets. They say that the term ‘a pair of shoes’ is a collective noun which deceives by the singular expression ‘a pair’. Deceives, because it…

  • Infinity and Mathematics Education

    Time for a re-post. This first appeared in these pages on 18 August 2010. ……………………. A reader writes, Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie); on the other hand, I…

  • Can a Mereological Sum Change its Parts?

    This post is an attempt to understand and evaluate Peter van Inwagen's "Can Mereological Sums Change Their Parts," J. Phil. (December 2006), 614-630.  A preprint is available online here. The Wise Pig and the Brick House: My Take On Tuesday the Wise Pig  takes delivery of 10,000 bricks.  On the following Friday he completes construction…

  • Infinity and Mathematics Education

    A reader writes, Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie); on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to…

  • Kline on Cantor on the Square Root of 2

    Morris Kline, Mathematics: The Loss of Certainty, Oxford 1980, p. 200: . . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the…

  • Does Potential Infinity Rule Out Mathematical Induction?

    In an earlier thread David Brightly states that "The position on potential infinity that he [BV] is defending is equivalent to the denial of the principle of mathematical induction."  Well, let's see. 1.  To avoid lupine controversy over 'potential' and 'actual,' let us see if we can avoid these words.  And to keep it simple, let's…

  • On Potential and Actual Infinity

    Some remarks of Peter Lupu in an earlier thread suggest that he does not understand the notions of potential and actual infinity.  Peter writes: (ii) An acorn has the potential to become an oak tree. But the acorn has this potential only because there are actual oak trees . . . .  If for some…

  • Collective Inconsistency and Plural Predication

    We often say things like 1. The propositions p, q, r are inconsistent. Suppose, to keep things simple, that each of the three propositions is self-consistent.  It will then be false that each proposition is self-inconsistent. (1), then, is a plural predication that cannot be given a distributive paraphrase.  What (1) says is that the three…

  • Sets and the Number of Objects: An Antilogism

    Commenter Jan, the Polish physicist, gave me the idea for the following post. An antilogism is an aporetic triad, an array of exactly three propositions which are individually plausible but collectively inconsistent.  For every antilogism, there are three corresponding syllogisms, where a syllogism is a deductive argument with exactly two premises and one conclusion.  Here is…

  • The Hatfields and the McCoys

    Whether or not it is true, the following  has a clear sense: 1. The Hatfields outnumber the McCoys. (1) says that the number of Hatfields is strictly greater than the number of McCoys.  It obviously does not say, of each Hatfield, that he outnumbers some McCoy.  If Gomer is a Hatfield and Goober a McCoy, it…

  • I Need to Study Plural Predication

    Here is a beautiful aphorism from Nicolás Gómez Dávila (1913-1994), in Escolios a un Texto Implicito (1977), II, 80, tr. Gilleland:  Stupid ideas are immortal. Each new generation invents them anew. Clearly this does not mean: 1. Each stupid idea is immortal and is invented by each new generation anew. So we try: 2. The…

  • The Axiom of Infinity as Easy Way Out?

    I posed the question, Can one prove that there are infinite sets?  Researching this question, I consulted the text I studied when I took a course in set theory in a mathematics department quite a few years ago. The text is Karl Hrbacek and Thomas Jech, Introduction to Set Theory (Marcel Dekker, 1978). On pp.…