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NNTP-Posting-Date: Thu, 02 Jan 2025 19:07:43 +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: Thu, 2 Jan 2025 11:07:43 -0800
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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, ....