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NNTP-Posting-Date: Fri, 03 Jan 2025 19:22:43 +0000
Subject: Re: Replacement of Cardinality (ubiquitous ordinals, integer
 continuum, linear continuum, continuity)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Fri, 3 Jan 2025 11:22:24 -0800
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On 01/02/2025 11:14 AM, Ross Finlayson wrote:
> On 01/02/2025 11:07 AM, Ross Finlayson wrote:
>> On 01/02/2025 09:35 AM, Jim Burns wrote:
>>> On 1/1/2025 6:50 PM, Ross Finlayson wrote:
>>>> On 01/01/2025 01:14 PM, Jim Burns wrote:
>>>>>> On 07/29/2024 12:46 PM, Ross Finlayson wrote:
>>>
>>>>>>> Or, you know, "infinity plus one".
>>>>>
>>>>> Consider the definition of a finite.cardinal as
>>>>> the cardinal #A of a set A
>>>>> smaller.by.one than sets fuller.by.one
>>>>> #A ∈ ⟦0,ℵ₀⦆  :⇔  (#A < #(A∪{a}) ⇐ A ≠ A∪{a})
>>>>>
>>>>> If,
>>>>> as might be expected,
>>>>> infinity.plus.one is different from simple.infinity,
>>>>> then,
>>>>> under that definition,
>>>>> infinity is finite.
>>>>
>>>> It's "well-ordering the universe".
>>>
>>> Please complete this sentence:
>>> ⎛ In "It's 'well-ordering the universe'",
>>> ⎜ "it" refers to
>>> ⎝
>>>
>>>> Yeah, I know,
>>>> you don't have a universe in your theory,
>>>> as you say that
>>>> there's no meta-theory your theory,
>>>> yet, what's that then, all one theory?
>>>
>>> I think that a universeᴿꟳ and a universeⁿᵒᵗᐧᴿꟳ
>>> are different.
>>>
>>> ⎛ In the formal sciences, the domain of discourse,
>>> ⎜ also called the universe of discourse, universal set,
>>> ⎜ or simply universe,
>>> ⎜ is the set of entities over which
>>> ⎜ certain variables of interest in some formal treatment
>>> ⎝ may range.
>>> https://en.wikipedia.org/wiki/Domain_of_discourse
>>>
>>> I have universesⁿᵒᵗᐧᴿꟳ for my theories, as I must
>>> wherever there are variables, and
>>> there are lots and lots of variables in 'my' theories.
>>>
>>> I take your universeᴿꟳ to be
>>> a unique, all.inclusive universeⁿᵒᵗᐧᴿꟳ.and.domain.
>>>
>>> The logic (FOL) of variables and universesⁿᵒᵗᐧᴿꟳ
>>> does not require an all.inclusive universeᴿꟳ
>>> We only need to be able to talk about
>>> what we are talking about, the current universeⁿᵒᵗᐧᴿꟳ,
>>> whichever that is.
>>>
>>>
>>> There are pragmatic motivations for talking about
>>> an all.inclusive universeᴿꟳ.
>>>
>>> There are also pragmatic motivations for talking about
>>> only what we are talking about, the current universesⁿᵒᵗᐧᴿꟳ.
>>>
>>> For example, if someone denies the existence of infinities,
>>> a good place to start might be the universeⁿᵒᵗᐧᴿꟳ of finites,
>>> which is itself not finite, and
>>> which can disobey rules designed for finites.
>>>
>>>> there's no meta-theory your theory,
>>>> yet, what's that then, all one theory?
>>>
>>> In these discussions, my bottom.floor logic is typically FOL,
>>> the logic of variables and their universesⁿᵒᵗᐧᴿꟳ.
>>>
>>> My meta.theory of FOL is the theory of
>>> finite sequences of claims, each claim of which is
>>> true.or.not.first.false.
>>> I think that I've mentioned that.
>>>
>>>
>>
>> So, is it, "not.ultimately.untrue"?
>>
>> See, here there's a meta-theory of all that,
>> yet it's all in the theory, a heno-theory,
>> otherwise being both formalist and platonic
>> all the time.
>>
>> "The Logic", ?
>>
>>
>> Is it, "not.ultimately.untrue"?  You don't say, ....
>>
>>
>
> https://www.youtube.com/watch?v=zwX9Y2oEtHs&list=PLb7rLSBiE7F4_E-POURNmVLwp-dyzjYr-&index=35
>
>
> "Logos 2000:  paradox-free reason", Ross Finlayson
>
>

https://www.youtube.com/watch?v=tODnCZvVtLg&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=5

"Moment and Motion:  points and space inversion"