Path: ...!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Mon, 30 Dec 2024 02:11:32 +0000 Subject: Re: Replacement of Cardinality (infinite middle) Newsgroups: sci.logic,sci.math References: <29fc2200-8ddc-43fe-9130-ea49301d3c5d@att.net> <1c5a8e0d-db33-4254-b456-8bb8e266c295@att.net> <822a53d2-7503-47d6-b632-6ebaa3ca4a92@att.net> <97d738be-af48-4e3c-b107-d49f4053f9eb@att.net> <5-ScnQ9Ks5z8ykv7nZ2dnZfqn_SdnZ2d@giganews.com> <884e5b13-5d91-4430-ba18-5f4208e283f2@att.net> <0-CcnWR3j-ONjEf7nZ2dnZfqnPWdnZ2d@giganews.com> From: Ross Finlayson Date: Sun, 29 Dec 2024 18:11:45 -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: <0-CcnWR3j-ONjEf7nZ2dnZfqnPWdnZ2d@giganews.com> Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 173 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-JUYU06ObxiO+RWpHiQXIy6LMX14nrUrFL19OkJd/HQauxUjxVCCGTW0+ZgbSpHwXbse7/SrEQ2Vs7uq!USop0FAw5Dv46L3/XBFmdg/4PfU5Ia0OuuAPisdPsLvGTFS0b/coyv1OT9UPyL7PD7SeNVW/QJk= 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: 9115 On 09/05/2024 01:14 PM, Ross Finlayson wrote: > On 09/05/2024 12:57 PM, Ross Finlayson wrote: >> On 09/03/2024 01:50 PM, Jim Burns wrote: >>> On 9/2/2024 8:25 PM, Ross Finlayson wrote: >>>> On 09/02/2024 02:46 PM, Jim Burns wrote: >>>>> On 9/1/2024 2:44 PM, Ross Finlayson wrote: >>> >>>>>> Then the point that induction lets out is >>>>>> at the Sorites or heap, >>>>>> for that Burns' "not.first.false", means >>>>>> "never failing induction first thus >>>>>> being disqualified arbitrarily forever", >>>>> >>>>> Not.first.false is about formulas which >>>>> are not necessarily about induction. >>>>> >>>>> A first.false formula is false _and_ >>>>> all (of these totally ordered formulas) >>>>> preceding formulas are true. >>>>> >>>>> A not.first.false formula is not.that. >>>>> >>>>> not.first.false Fₖ ⇔ >>>>> ¬(¬Fₖ ∧ ∀j>>>> Fₖ ∨ ∃j>>>> ∀j>>>> >>>>> A finite formula.sequence S = {Fᵢ:i∈⟨1…n⟩} has >>>>> a possibly.empty sub.sequence {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} >>>>> of false formulas. >>>>> >>>>> If {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} is not empty, >>>>> it holds a first false formula, >>>>> because {Fᵢ:i∈⟨1…n⟩} is finite. >>>>> >>>>> If each Fₖ ∈ {Fᵢ:i∈⟨1…n⟩} is not.first.false, >>>>> {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} does not hold a first.false, and >>>>> {Fᵢ:i∈⟨1…n⟩∧¬Fᵢ} is empty, and >>>>> each formula in {Fᵢ:i∈⟨1…n⟩} is true. >>>>> >>>>> And that is why I go on about not.first.false. >>> >>>> Then about not.first.false >>>> thanks for writing that up a bit more, >>>> then that also you can see what I make of it. >>> >>> What I find poetic about not.first.false and all that >>> is that our finiteness isn't only _permitted_ >>> It is _incorporated into_ our logic. _Required_ >>> >>> A finite linear order _must be_ well.ordered >>> (must be, both ways) >>> ∀γ:T(γ) ⇐ ∀β:(T(β) ⇐ ∀α<β:T(α)) >>> ∀α:T(α) ⇐ ∀β:(T(β) ⇐ ∀γ>β:T(γ)) >>> >>> We are finite. >>> The formulas we write are finitely.many. >>> In a linear order, they must be in a well.order. >>> >>> In a well.order, >>> if each formula Φ[β] is not.first.false >>> ∀β:¬(¬T(Φ[β] ∧ ∀α<β:T(Φ[α]) >>> ∀β:(T(Φ[β]) ⇐ ∀α<β:T(Φ[α])) >>> then each formula is not.false. >>> ∀γ:T(Φ[γ]) >>> >>> ...because well.order (because finite). >>> ∀γ:T(Φ[γ]) ⇐ ∀β:(T(Φ[β]) ⇐ ∀α<β:T(Φ[α])) >>> >>>> Not.ultimately.untrue, ..., has that >>>> F, bears the value for all F_alpha parameterized by ordinals >>>> (which suffice, large enough, to totally order things), >>>> of true, and that, >>>> there are classes of formulas F, >>>> for example self-referential or differential formulas, >>>> defined for example according to >>>> "when F_alpha is not also as for an ordinal less than omega", >>>> at least making a trivial clear example of >>>> a definition that is for classes of these sorts formulas >>>> where "not.ultimately.untrue" is not held by all classes >>>> for formulas "not.first.false". >>> >>> "Not.ultimately.untrue" sounds to me vaguely like "ω-consistent". >>> But I don't really know what you are talking about. >>> I usually don't know what you are talking about. >>> It is what it is. >>> >>> >> >> That "points do not make lines" and "lines do not make points" >> yet "any two points define a line" and "any two intersecting lines >> define a point", are of course quite fundamental and elementary >> since for most of time that Euclid's Elements is the second-most >> published book in the world. >> >> (Euclid is a panel.) >> >> https://en.wikipedia.org/wiki/Transfer_principle >> >> I have pretty much no use for the hyper-reals as merely >> a "conservative" (i.e., saying nothing) extension of the >> usual Archimedean field, while, something like Nelson's >> Internal Set Theory and that it's co-consistent with ZFC, >> with regards to either "both or neither", much like the >> "both or neither" of "the anti-diagonal and the only-diagonal", >> have that there are "conservative non-standard" extensions >> saying _nothing_ and "non-conservative non-standard" extensions >> saying _something_. >> >> When Hilbert _added_ a postulate of continuity to Euclid's axioms, >> so to establish that a point-set topology could be a thing at >> all, it's quite a non-conservative non-standard axiom, as it were, >> itself, though of course for "axiomless geometry" it already >> exists from there being a prototype continuum as elementary >> in a theory, co-consistent this theory of geometry "points >> and spaces" with the usual theory of words (algebra's, >> set theory's, ...), that, more-than-less you might as >> well start reading the most-published book in the world, >> or just the first few items "in the beginning ..." there >> was space then from the middle "in the beginning ..." there >> was the word, of an example of a necessary sort of ontological >> commitment with regards to nominalism, and its weaker forms >> fictionalism, fallibilism, and anti-realism. >> >> I.e., as a strong mathematical platonist with a stronger >> logicist positivism, my model philosopher's model physicist's >> model philosophy's model physics, easily encompasses the >> tiny, weaker, hereditarily-finite fragment what's conservative >> off ZFC. >> >> https://en.wikipedia.org/wiki/Eleatics >> >> "The Eleatics have traditionally been seen as advocating a strict >> metaphysical view of monism in response to the materialist monism >> advocated by their predecessors, the Ionian school." >> >> >> >> >> It certainly is what it is, .... >> >> >> >> > > Back in the 80's and 90's it was Nelson's Internal Set Theory > where it was figured that the avenue toward true non-standard > real analysis was to result. > > I.e., not-a-real-functions with real analytical character, > like Dirac's delta function or here for example the Natural/Unit > Equivalency Function, it is expected that "foundations" _does_ > formalize them, and that what doesn't, simply, isn't, respectively. > > Not saying much, .... > ========== REMAINDER OF ARTICLE TRUNCATED ==========