Re: How many different unit fractions are lessorequal than all unit fractions?

On 05.10.2024 23:57, Alan Mackenzie wrote:
 > WM <wolfgang.mueckenheim@tha.de> wrote:
 >> On 05.10.2024 15:57, Alan Mackenzie wrote:
 >
 >>> Yes!  At least, sort of.  My understanding of "doesn't exist" is either
 >>> the concept is not (yet?) developed mathematically, or it leads to
 >>> contradictions.  WM's "dark numbers" certainly fall into the first
 >>> category, and possibly the second, too.
 >
 >> Definition: A natural number is "named" or "addressed" or "identified"
 >> or "(individually) defined" or "instantiated" ....
 >
 > That's five terms for the same thing.  Four of them (at least) are thus
 > redundant.  It is unmathematical to have such redundancy.
 The vocabulary of my readers is very different. So the c hance of understanding is increased.
 >
 >> .... if it can be communicated, necessarily by a finite amount of
 >> information, in the sense of Poincaré, such that sender and receiver
 >> understand the same and can link it by a finite initial segment (1, 2,
 >> 3, ..., n) of natural numbers to the origin 0.
 >
 > This is ridiculous!  It is so far removed from the austere simplicity of,
 > for example, Peano's axioms as to be thoroughly unmathematical.
 Peano's axioms are invalid fpr large numbers.

 >
 >> .... All other natural numbers are called dark natural numbers.  Dark
 >> numbers are numbers that cannot be chosen as individuals.
 >
 > Is "chosen" a sixth redundant word for "named", "addressed", ....?
 Yes, all these words have the same meaning, but not all readers know all words.
 >
 > "Chosen as individuals" isn't a mathemtical concept.  This phrase, as it
 > is written, makes it sound like the choice is being made by a conscious
 > individual person, according to something unspecified.  That doesn't
 > belong in mathematics.
 It does not belong to the simple but inconsistent present mathematics.
 >
 >> Communication can occur
 >> - by direct description in the unary system like ||||||| or as many
 >> beeps, raps, or flashes,
 >> - by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7),
 >> - as n-ary representation, for instance binary 111 or decimal 7,
 >> - by indirect description like "the number of colours of the rainbow",
 >> - by other words known to sender and receiver like "seven".
 >
 > The existence of natural numbers is independent of their communication by
 > people.
 You are a believer in God having created them?
  >> Only when a number n is identified we can use it in mathematical
 >> discourse ....
 >
 > This is something you haven't proved.
 Try to use another number.
  > Besides, mathematicians routinely use "unidentified" numbers in
 > discourse.  For example "If p is a prime number of the form 4m + 1, it is
 > the sum of two squares.".  That is a statement about an infinite number
 > of numbers, none of which are "identified".
 It is not a statement about an individual number but about a set, some of which can be defined.
 >
 >> .... and can determine the trichotomy properties of n and of every
 >> multiple k*n or power n^k or power tower k_^n with respect to every
 >> identified number k. ℕdef contains all defined natural numbers as
 >> elements – and nothing else. ℕdef is a potentially infinite set;
 >> therefore henceforth it will be called a collection.
 >
 > All natural numbers are "defined" in your sense of that word.
 No, after every identified number almost all are following, almost all of them are dark.
     As a
 > proof, we only need note that every non-empty subset of N has a least
 > member.  Suppose there is a non-empty set of "undefined" natural numbers.
 > Then there is a least such number.
 No. The identified numbers are a potentially infinite collection. Therefore there is no least dark number.
  >>> I first came across the terms "potential infinity" and "actual infinity"
 >>> on this newsgroup, not in my degree course a few decades ago.
 >
 >> It is carefully avoided because closer inspection shows contradictions.
 >
 > There are no such contradictions.
 There are many. For instance: All unit fractions are separate points on the positive real axis, but there are infinitely many for every x > 0. That can only hold for definable x, not for all.
 >
 >> Therefore set theorists use just what they can defend. If actual
 >> infinity is shown self contradictory (without dark numbers), then they
 >> evade to potential infinity temporarily which has no completed sets and
 >> cannot complete bijections.
 >
 > Not really.  There is simply no need for "actual" and "potential"
 > infinity.
 You have not learnt about it during study and obviously not afterwards.
  > They are relics from the past, from before the time when
 > mathematicians understood infinity as they do today.
 You cannot judge because you don't know that topic and as fellow traveler can only parrot the words of matheologians who are either too stupid to recognize or too dishonest to confess the truth.
 >
 >> "You use terms like completed versus potential infinity, which are not
 >> part of the modern vernacular." [P.L. Clark in "Physicists can be
 >> wrong", tea.MathOverflow (2 Jul 2010)] This is the typical reproach to
 >> be expected when the different kinds of infinity are analyzed and taught.
 >
 >> Here the difference is clearly stated:
 >> "Should we briefly characterize the new view of the infinite introduced
 >> by Cantor, we could certainly say: In analysis we have to deal only with
 >> the infinitely small and the infinitely large as a limit-notion, as
 >> something becoming, emerging, produced, i.e., as we put it, with the
 >> potential infinite. But this is not the proper infinite. That we have
 >> for instance when we consider the entirety of the numbers 1, 2, 3, 4,
 >> ... itself as a completed unit, or the points of a line as an entirety
 >> of things which is completely available. That sort of infinity is named
 >> actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische
 >> Annalen 95 (1925) p. 167]
 >
 > That's from 1925.  It is not a modern understanding of the infinite.
 But a correct one. The modern understanding is pure deceit.
 >
 > If these terms had any significance, they would still be taught in
 > mathematics degree courses.
No, the teachers of such courses are too stupid or too dishonest.
  >  Otherwise, bright students would become
 > aware of them and catch out their teachers in inconsistencies.
 They do. But every publishing is intercepted by the leading liars.
 Regards, WM