Path: ...!local-3.nntp.ord.giganews.com!Xl.tags.giganews.com!local-2.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Tue, 07 Jan 2025 19:07:18 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Newsgroups: sci.math References: <8e95dfce-05e7-4d31-b8f0-43bede36dc9b@att.net> <53d93728-3442-4198-be92-5c9abe8a0a72@att.net> <9c18a839-9ab4-4778-84f2-481c77444254@att.net> <8ef20494f573dc131234363177017bf9d6b647ee@i2pn2.org> From: Ross Finlayson Date: Tue, 7 Jan 2025 11:06:23 -0800 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: <4cOdnTXL0qP75uD6nZ2dnZfqnPWdnZ2d@giganews.com> Lines: 206 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-92eJA2njbvbUTJRjfDw68nWxNeQbLx+9KertSoZrbHhjENnZJgUCxpsxyxywedoojAY6rOwoTZopQZh!RWdu7m8EZzQXNIybiA0y316A72KPZeHRikToN9UKM3M6eZyfuNnkUFZegrUtPKbZUks6H1i/a+s= X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 11882 On 01/07/2025 03:36 AM, Alan Mackenzie wrote: > Moebius wrote: >> Am 05.01.2025 um 12:28 schrieb Alan Mackenzie: > >>>> "We introduce numbers for counting. This does not at all imply the >>>> infinity of numbers. For, in what way should we ever arrive at >>>> infinitely-many countable things? [...] In philosophical terminology we >>>> say that the infinite of the number sequence is only potential, i.e., >>>> existing only as a possibility." [P. Lorenzen: "Das Aktual-Unendliche in >>>> der Mathematik", Philosophia naturalis 4 (1957) p. 4f] > >>> Philosopy. > >> Sure, though P. Lorenzen was an eminent mathematician who developed a >> form of constructive mathematics (constructive analysis) and dialogical >> logic. > >> See: https://en.wikipedia.org/wiki/Paul_Lorenzen >> and: https://en.wikipedia.org/wiki/Dialogical_logic > >> Note that we can't be sure if Mückenheim's translation is accurate. > >>>> "Until then, no one envisioned the possibility that infinities come in >>>> different sizes, and moreover, mathematicians had no use for 'actual >>>> infinity'. The arguments using infinity, including the Differential >>>> Calculus of Newton and Leibniz, do not require the use of infinite sets. >>>> [...] Cantor observed that many infinite sets of numbers are countable: >>>> the set of all integers, the set of all rational numbers, and also the >>>> set of all algebraic numbers. Then he gave his ingenious diagonal >>>> argument that proves, by contradiction, that the set of all real numbers >>>> is not countable. A consequence of this is that there exists a multitude >>>> of transcendental numbers, even though the proof, by contradiction, does >>>> not produce a single specific example." [T. Jech: "Set theory", Stanford >>>> Encyclopedia of Philosophy (2002)] > >>> Also philosophy. > >> Sure. But T. Jech is a leading set theorist. > >> https://en.wikipedia.org/wiki/Thomas_Jech > >>>> "Numerals constitute a potential infinity. Given any numeral, we can >>>> construct a new numeral by prefixing it with S. Now imagine this >>>> potential infinity to be completed. Imagine the inexhaustible process of >>>> constructing numerals somehow to have been finished, and call the result >>>> the set of all numbers, denoted by . Thus  is thought to be an actual >>>> infinity or a completed infinity. This is curious terminology, since the >>>> etymology of 'infinite' is 'not finished'." [E. Nelson: "Hilbert's >>>> mistake" (2007) p. 3] > >>> E. Nelson is clearly not a mathematician. > >> Holy shit! > >> "Edward Nelson (May 4, 1932 – September 10, 2014) was an American >> mathematician. He was professor in the Mathematics Department at >> Princeton University. He was known for his work on mathematical physics >> and mathematical logic. In mathematical logic, he was noted especially >> for his internal set theory, and views on ultrafinitism and the >> consistency of arithmetic. In philosophy of mathematics he advocated the >> view of formalism rather than platonism or intuitionism." > >> https://en.wikipedia.org/wiki/Edward_Nelson > >> See: https://en.wikipedia.org/wiki/Internal_set_theory > >>>> According to (Gödel's) Platonism, objects of mathematics have the same >>>> status of reality as physical objects. "Views to the effect that >>>> Platonism is correct but only for certain relatively 'concrete' >>>> mathematical 'objects'. Other mathematical 'objects' are man made, and >>>> are not part of an external reality. Under such a view, what is to be >>>> made of the part of mathematics that lies outside the scope of >>>> Platonism? An obvious response is to reject it as utterly meaningless." >>>> [H.M. Friedman: "Philosophical problems in logic" (2002) p. 9] >>>> Possibly philosophy, more likely complete nonsense. > >> *sigh* > >> "Harvey Friedman (born 23 September 1948)[1] is an American mathematical >> logician at Ohio State University in Columbus, Ohio. He has worked on >> reverse mathematics, a project intended to derive the axioms of >> mathematics from the theorems considered to be necessary. In recent >> years, this has advanced to a study of Boolean relation theory, which >> attempts to justify large cardinal axioms by demonstrating their >> necessity for deriving certain propositions considered "concrete"." > >> https://en.wikipedia.org/wiki/Harvey_Friedman > >> "This chapter focuses on the work of mathematical logician Harvey >> Friedman, who was recently awarded the National Science Foundation's >> annual Waterman Prize, honoring the most outstanding American scientist >> under thirty-five years of age in all fields of science and engineering. >> Friedman's contributions span all branches of mathematical logic >> (recursion theory, proof theory, model theory, set theory, and theory of >> computation). He is a generalist in an age of specialization, yet his >> theorems often require extraordinary technical virtuosity, of which only >> a few selected highlights are discussed. Friedman's ideas have yielded >> radically new kinds of independence results. The kinds of statements >> that were proved to be independent before Friedman were mostly disguised >> properties of formal systems (such as Gödel's theorem on unprovability >> of consistency) or assertions about abstract sets (such as the continuum >> hypothesis or Souslin's hypothesis). In contrast, Friedman's >> independence results are about questions of a more concrete nature >> involving, for example, Borel functions or the Hilbert cube." > >> Source: >> https://www.sciencedirect.com/science/article/abs/pii/S0049237X09701545 >> (1985) > >>>> "A potential infinity is a quantity which is finite but indefinitely >>>> large. For instance, when we enumerate the natural numbers as 0, 1, 2, >>>> ..., n, n+1, ..., the enumeration is finite at any point in time, but it >>>> grows indefinitely and without bound. [...] An actual infinity is a >>>> completed infinite totality. Examples: , , C[0, 1], L2[0, 1], etc. >>>> Other examples: gods, devils, etc." [S.G. Simpson: "Potential versus >>>> actual infinity: Insights from reverse mathematics" (2015)] > >>> Another philosopher? > >> "Stephen George Simpson (born September 8, 1945) is an American >> mathematician whose research concerns the foundations of mathematics, >> including work in mathematical logic, recursion theory, and Ramsey >> theory. He is known for his extensive development of the field of >> reverse mathematics founded by Harvey Friedman, in which the goal is to >> determine which axioms are needed to prove certain mathematical >> theorems.[1] He has also argued for the benefits of finitistic >> mathematical systems, such as primitive recursive arithmetic, which do >> not include actual infinity." > >> Source: https://en.wikipedia.org/wiki/Steve_Simpson_(mathematician) > >>>> "Potential infinity refers to a procedure that gets closer and closer >>>> to, but never quite reaches, an infinite end. For instance, the sequence >>>> of numbers 1, 2, 3, 4, ... gets higher and higher, but it has no end; it >>>> never gets to infinity. Infinity is just an indication of a direction – >>>> it's 'somewhere off in the distance'. Chasing this kind of infinity is >>>> like chasing a rainbow or trying to sail to the edge of the world – you >>>> may think you see it in the distance, but when you get to where you >>>> thought it was, you see it is still further away. Geometrically, imagine >>>> an infinitely long straight line; then 'infinity' is off at the 'end' of >>>> the line. Analogous procedures are given by limits in calculus, whether >>>> they use infinity or not. For example, limx0(sinx)/x = 1. This means >>>> that when we choose values of x that are closer and closer to zero, but >>>> never quite equal to zero, then (sinx)/x gets closer and closer to one." >>>> [E. Schechter: "Potential versus completed infinity: Its history and >>>> controversy" (5 Dec 2009)] > >>> There may be a history to it, but there is no controversy, at least not >>> in mathematical circles. > >> You are not familiar with "foundations of mathematics", right? > > Moderately so. > >> Of course: "In practice, most mathematicians either do not work from >> axiomatic systems, or if they do, do not doubt the consistency of ZFC, >> generally their preferred axiomatic system." > >> Source: https://en.wikipedia.org/wiki/Foundations_of_mathematics > >>> There are no "actual" and "potential" infinity in mathematics. > >> If you say so. :-P > > None of the above extracts is about mathematics. If there were such > things as "potential" and "actual" infinity in maths, then they would > make a difference to some mathematical result. There would be some > theorem provable given the existence of PI and AI which would be false or ========== REMAINDER OF ARTICLE TRUNCATED ==========