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NNTP-Posting-Date: Thu, 02 Jan 2025 19:14:08 +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: Thu, 2 Jan 2025 11:14:08 -0800
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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