{"id":1500,"date":"2023-04-20T15:42:47","date_gmt":"2023-04-20T15:42:47","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2023\/04\/20\/on-potential-and-actual-infinity-and-a-puzzle\/"},"modified":"2023-04-20T15:42:47","modified_gmt":"2023-04-20T15:42:47","slug":"on-potential-and-actual-infinity-and-a-puzzle","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2023\/04\/20\/on-potential-and-actual-infinity-and-a-puzzle\/","title":{"rendered":"On Potential and Actual Infinity, and a Puzzle"},"content":{"rendered":"

Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ).  If these numbers form a set, call it N, then N will of course be actually infinite.  This because a set in the sense of set theory is a single, definite object, a one-over-many, distinct from each of its members and from all of them.  N must be actually infinite because there is no greatest natural number, and because N contains all the natural numbers. <\/span><\/p>\n

It is worth noting that 'actually infinite set' is a pleonastic expression. It suffices to say 'infinite set.'  This is because the phrase 'potentially infinite set' is nonsense. It is nonsense (conceptually incoherent) because a set is a definite object whose definiteness derives from its having exactly the members it has.  A set cannot gain or lose members, and a set cannot have a membership other than the membership it actually has. Add a member to a set and the result is a numerically different set. In the case of the natural numbers, if<\/em> they form a set, then that set will be an actually infinite set with a definite transfinite cardinality. Georg Cantor refers to that cardinality as aleph-zero or aleph-nought.<\/span><\/p>\n

I grant, however, that it is not obvious that the natural numbers form a set.  Suppose they don't.  Then the natural number series, though infinite, will be merely potentially infinite.  What 'potentially infinite' means here is that one can go on adding endlessly without ever reaching an upper bound of the series.  No matter how large the number counted up to, one can add 1 to reach a still higher number. The numbers are thus created by the counting, not labeled by the counting.  The numbers are not 'out there' in Plato's topos ouranios<\/em> waiting to be counted; they are created by the counting.  In that sense, their infinity is merely potential.  But if the naturals are an actual infinity, then  they are not created but labeled.<\/span><\/p>\n

Moving now from arithmetic to geometry, consider a line segment in a plane.  One can bisect it, i.e., divide or cut it into two smaller segments of equal length.  Thus the segment AB whose end points are A and B splits into the congruent sub-segments AC and CB, where C is the point of bisection. The operation of bisection is indefinitely ('infinitely') iterable in principle<\/em>.  The term 'in principle' needs a bit of commentary. <\/span><\/p>\n

\"Salami\"<\/a>Suppose I am slicing a salami using a state-of-the-art meat slicer. I cannot go on slicing thinner and thinner indefinitely.  The operation of bisecting a salami is not indefinitely iterable in principle.  The operation is iterable only up to a point, and this for the reason that a slice must have a certain minimal thickness T such that if the slice were thinner than T it would no longer be a slice.   But if we consider the space the salami occupies — assuming that space is something like a container that can be occupied — then  a longitudinal (non-transversal) line segment running from one end of the salami to the other is bisectable indefinitely in principle.<\/span><\/p>\n

For each bisecting of a line segment, there is a point of bisection. The question can now be put as follows: Are these points of bisection only potentially infinite, or are they actually infinite?  <\/span><\/p>\n

A Puzzle<\/span><\/em><\/p>\n

I want to say that from the mere fact that the operation of bisecting a line segment is indefinitely ('infinitely') iterable in principle, it does not<\/em> follow<\/em> that the line segment is composed of an actual infinity of points. That is, it is logically consistent to maintain all three of the following:  (i) one can always make  another cut; (ii) the number of actual cuts will always be finite; and that therefore (iii) the number of points in a line will always be finite, and therefore 'infinite' only in the sense that there is no finite cardinal n<\/em> such that n<\/em> is the upper bound of the number of cuts. <\/span><\/p>\n

At this 'point,' however, I fall into perplexity which, according to Plato, is the characteristic state of the philosopher. If one can always make another cut, then the number of possible<\/em> cuts cannot be finite. For if the number of possible<\/em> cuts is finite, then it can longer be said that the line segment has a potentially infinite number of points of bisection.  It seems that a potential infinity of actual cuts logically requires an actual infinity of possible<\/em> cuts.<\/span><\/p>\n

But then actual infinity, kicked out the front door, returns through the back door.<\/span><\/p>\n

I have just posed a problem for those who are friends of the potentially infinite but foes of the actual infinite. How might they respond?<\/span><\/p>\n

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Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ).  If these numbers form a set, call it N, then N will of course be actually infinite.  This because a set in the sense of set theory is a single, definite object, a one-over-many, distinct from each … Continue reading “On Potential and Actual Infinity, and a Puzzle”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,475],"tags":[],"class_list":["post-1500","post","type-post","status-publish","format-standard","hentry","category-aporetics","category-infinity"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/1500","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/comments?post=1500"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/1500\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=1500"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=1500"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=1500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}