| Deutsch English Français Italiano |
|
<rtycne8gb83t9Bn7nZ2dnZfqn_idnZ2d@giganews.com> View for Bookmarking (what is this?) Look up another Usenet article |
Path: ...!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Tue, 02 Jul 2024 20:56:16 +0000 Subject: Re: Does the number of nines increase? (axiomatizing completeness) Newsgroups: sci.math References: <tJf9P9dALSN4l2XH5vdqPbXSA7o@jntp> <eZZGYbe53s6yBDBqGuTMM_Z1y7A@jntp> <v5lspm$1bs52$2@i2pn2.org> <HlcnRXFQ42qMfnJgEw40TN7tXjI@jntp> <v5mriv$1d3t3$7@i2pn2.org> <v5msu8$3ena6$1@dont-email.me> <7kWvquYAIwnpmJER42BML2v7650@jntp> <v5n8c3$3gth7$1@dont-email.me> <OZWcnZ95r_FOwOL7nZ2dnZfqn_gAAAAA@giganews.com> <v5okju$3s2o5$1@dont-email.me> <FT8a4DhRbwojKFSQC8A_96QDrbs@jntp> <v5ra2m$fcfm$1@dont-email.me> <bvScnc3yo6e97Rz7nZ2dnZfqn_adnZ2d@giganews.com> <f037b499-dc8c-4e12-9a8e-6f8b511766b4@att.net> <6yqdnR5LuPA9VRz7nZ2dnZfqnPadnZ2d@giganews.com> <aa09d54d-3e75-499f-a404-6928e648ad6e@att.net> <ibacnZBxO6vBvh_7nZ2dnZfqnPidnZ2d@giganews.com> <d0731868-9467-4a96-9c0a-105f5e30ab57@att.net> <hXydnbCF9Zpgox77nZ2dnZfqnPSdnZ2d@giganews.com> <38ed6d4d-9b12-4cb7-9b14-8e7cfa831266@att.net> <EKKdncFjgr4v7B77nZ2dnZfqnPadnZ2d@giganews.com> <15777576-8d71-4a0e-bce8-354b5615b5d6@att.net> <OVudneS-1sl8_hn7nZ2dnZfqn_GdnZ2d@giganews.com> <wrCdnYV2iqAP-Bn7nZ2dnZfqnPqdnZ2d@giganews.com> From: Ross Finlayson <ross.a.finlayson@gmail.com> Date: Tue, 2 Jul 2024 13:56:20 -0700 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: <wrCdnYV2iqAP-Bn7nZ2dnZfqnPqdnZ2d@giganews.com> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: <rtycne8gb83t9Bn7nZ2dnZfqn_idnZ2d@giganews.com> Lines: 146 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-IYC0xCgmkJWgS0DaUUUaz3m6AQhm75PV1lQdxmY23S2esIb//SKa26dubA4uQe91fDjg3WPjelcp/LS!2+H1xJCtgqheHCqg3Ri2DSglBdfhdxMvnv2l3Dp3w74Hm1CbCAZRbiQQ/qDPTYX/ZmGa6cPuFQ== 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: 6662 On 07/02/2024 01:39 PM, Ross Finlayson wrote: > On 07/02/2024 01:32 PM, Ross Finlayson wrote: >> On 07/02/2024 05:07 AM, Jim Burns wrote: >>> On 7/1/2024 11:19 PM, Ross Finlayson wrote: >>>> On 07/01/2024 06:10 PM, Jim Burns wrote: >>> >>>> What that means is that >>>> I think >>>> theory is a strong mathematical platonism, >>>> it matters what is _attained_ to, >>>> or that to which we _attain_, >>>> the "true" objects of >>>> a universe of mathematical objects, >>>> "a" universe, then with regards to descriptions >>> >>> "descriptions" >>> >>>> there's that >>>> the "extent density completeness measure" provide >>>> "extent density completeness" >>>> which you would agree that >>>> "extent density completeness" makes for >>>> satisfying the IVT. >>>> >>>> Then, as I mentioned, >>>> there's a theory, >>>> in all the universe of theories, >>>> all the abstract and contingent and fanciful and >>>> practical and otherwise, >>>> one of which is "the true theory", >>>> that among those, >>>> there's one where it appears that >>>> "it is so" is an axiom. >>>> >>>> So, given that >>>> you won't accept that via inspection, >>> >>> "inspection" >>> >>>> that a least-upper-bound is given and >>>> also a sigma algebra is given, >>>> given that extent and density are givens, >>>> then, >>>> given that it's axiomatic, >>> >>> "axiomatic" >>> >>>> and, doesn't contradict the ordinary >>>> because >>>> it just makes for the "only-diagonal" contra >>>> the "anti-diagonal", >>>> then, how's that. >>>> >>>> Good sir, .... >>> >>> Do I understand you correctly? >>> You have declined my invitation to say >>> what your symbol.string n/d: 0≤n≤d: d → ∞ means >>> because >>> you consider what you've told me to have answered me: >>> n/d: 0≤n≤d: d → ∞ satisfies IVT >>> n/d: 0≤n≤d: d → ∞ is countable >>> etc. >>> >>> You have defined it so. >>> Do you realize that? >>> >>> Definitions are two.edged swords. >>> They grant you unrestricted power, >>> but only inside the area of what.you.mean >>> and outside of that, no power. >>> >>> If what.you.mean by n/d: 0≤n≤d: d → ∞ is that >>> n/d: 0≤n≤d: d → ∞ satisfies IVT >>> n/d: 0≤n≤d: d → ∞ is countable >>> etc. >>> then, okay, you can define it so, but >>> defining it so doesn't mean it exists. >>> >>> Those proofs which >>> you think n/d: 0≤n≤d: d → ∞ disproves >>> prove that n/d: 0≤n≤d: d → ∞ doesn't exist. >>> >>> >> >> These things are demonstrated, then there's at least >> one putative theory in the theory of all theories >> where "axiom of iota-value truth" or "IVT axiom", >> is so. >> >> I don't claim any disproofs at all, except insofar as >> the integers are non-standard, rather, it's another >> proof in usual set theory's usual descriptive milieu >> that there are "non-Cartesian" functions at least >> this "only-diagonal" and it's arrived at about >> the same way as there are "non-countable" domains >> for what's the "anti-diagonal". Pick one: get both. >> >> Cantor-Schroeder-Bernstein theorem gets built a little barrier >> about the transitive quality of cardinality, and a bit >> of book-keeping results that quite standard looking results >> combine to abound. >> >> I'm glad you've arrived at "exists", though, >> mathematical objects, defined by relation. >> >> >> Definitions are only as good as they're sound. >> >> And axioms: that they're not.ultimately.untrue. >> >> >> You know, some have that least-upper-bound is >> only provided to Eudoxus/Dedekind/Cauchy real field >> via axiom. >> >> And, that "non-countable" is a non-constructivist result. >> >> >> Anyways, this putative countable domain via its construction >> as a range of continuum limit of functions isn't contradicted >> by the anti-diagonal and so on, nor by being a Cartesian function, >> as a model of a unit line segment of the linear continuum. >> >> "Real-Valued" >> >> > > > https://www.youtube.com/watch?v=NJ1D2pgTZb0&t=810s > > Here's a brief account of the "extent density completeness measure". > > "Oh, you mean Hilbert's style of continuity postulate is necessary for analytic geometry's formal foundations?" Well, yeah, that's what he said. Here instead it's just built as a limit of functions, only the simplest function that relates 0, 1, and infinity. Then an axiom of that is redundant, yet.