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NNTP-Posting-Date: Sun, 09 Mar 2025 17:41:02 +0000
Subject: Re: The set of necessary FISONs
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 9 Mar 2025 10:41:05 -0700
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On 03/09/2025 09:26 AM, Jim Burns wrote:
> On 3/9/2025 5:55 AM, WM wrote:
>> On 08.03.2025 20:27, Jim Burns wrote:
>>> On 3/8/2025 9:09 AM, WM wrote:
>>>> On 08.03.2025 12:58, Jim Burns wrote:
>
>>>>> you should have said somewhere
>>>>> what a finite ordinal is,
>>>>
>>>> That is not under discussion here.
>>>
>>> That has been under discussion for decades.
>>>
>>> I think that these decades of discussion
>>> have been, in large part, you assigning
>>> different meanings to 'finite', etc.
>>> and matheologians (among whom I place myself)
>>> trying to discern what your meanings are.
>>>
>>> Here's my best guess:
>>> definableᵂᴹ  ==  finite
>
> Compare what you quote to what I wrote:
> definableᵂᴹ  ==  finiteⁿᵒᵗᐧᵂᴹ  ==  #A<#Aᣕᵇ
>
>> [...]
>>
>>> darkᵂᴹ  ==  finite  ==  big
>
> Compare what you quote to what I wrote:
> darkᵂᴹ  ==  finiteⁿᵒᵗᐧᵂᴹ  ==  big and #A<#Aᣕᵇ
> matheologicalᵂᴹ  ==  infiniteⁿᵒᵗᐧᵂᴹ  ==  #A=#Aᣕᵇ
>
>> [...]
>>
>>>> n is usually denoting
>>>> a natural number.
>>>
>>> Do we mean the same by 'natural number'?
>>
>> There are two different meanings:
>> All positive integers having FISONs or all positive integers.
>
> The distinction which you (WM)
> have been dodging for decades
> is between
> sets which have
> fuller.by.one and emptier.by.one counterparts
> _which are a different size_ (finiteⁿᵒᵗᐧᵂᴹ)
> and
> sets which have
> fuller.by.one and emptier.by.one counterparts
> _which are the same size_ (infiniteⁿᵒᵗᐧᵂᴹ)
>
>>>> FISONs are finite by definition.
>>>
>>> Do we mean the same by 'finite'?
>>
>> A natural number n is finite.
>> It is an integer between 0 and ω: 0 < n < ω.
>> Sometimes 0 is included, never ω is included.
>
> You erased the distinction I made and
> you haven't given any replacement.
> (Your ωᵂᴹ is merely a big finite.)
>
> #A size of set A
> #Aᣕᵇ size of fuller.by.one set, Aᣕᵇ = A∪{b} ≠ A
>
> #A < #Aᣕᵇ larger fuller.by.one set: A finite
>
> #Y = #Yᣕᶻ same.size fuller.by.one set: Y infinite
>
> The distinction infinite.v.finite is useful
> because
> not all sets are finite,
> and the distinction, for fuller.by.one sets,
> of same.v.different sizes leads to
> very different properties.
>
> A notable example of an infinite (ie, #Y = #Yᣕᶻ) set
> is the set of finite set.sizes {#A:#A<#Aᣕᵇ}
>
> ⎛ For each finite.set.size ξ in {#A:#A<#Aᣕᵇ},
> ⎜ there is a larger subset #{#A<ξ+1:#A<#Aᣕᵇ} = ξ+1
> ⎜
> ⎜ {#A:#A<#Aᣕᵇ} doesn't contain
> ⎜ a subset larger than {#A:#A<#Aᣕᵇ}
> ⎜
> ⎜ For each finite.set.size ξ in {#A:#A<#Aᣕᵇ},
> ⎝ ξ isn't the size of {#A:#A<#Aᣕᵇ}
>
> {#A:#A<#Aᣕᵇ} is what we mean by ℕ,
> what you mean by ℕ_def.
>
> ----
>>>> My description is definite.
>>>
>>> Your description is only definite because
>>> ℕ ∋ n is definite.
>>
>> No, my description is definite
>> because
>> every n can be obtained by addition of 1's
>> (or of curly brackets).
>
> Finitely.many 1's or curly.brackets.
> How many is that?
>
> Your decades.long argument has been about
> how many that is --
> but you avoid saying what you mean.
> You even avoid hearing what we mean.
>
>

Seems kind of futile doesn't it.