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  • Re: Replacement of Cardinality (ubiquitous ordinals, integer continuum,

    From Jim Burns@21:1/5 to Ross Finlayson on Wed Jan 1 16:14:56 2025
    XPost: sci.math

    On 1/1/2025 1:10 PM, Ross Finlayson wrote:
    On 07/29/2024 12:46 PM, Ross Finlayson wrote:
    On 07/29/2024 12:44 PM, Ross Finlayson wrote:
    On 07/29/2024 05:32 AM, Jim Burns wrote:
    On 7/28/2024 7:42 PM, Ross Finlayson wrote:

    about ubiquitous ordinals

    What are ubiquitous ordinal?

    The "ubiquitous ordinals", sort of recalls Kronecker's
    "G-d made the integers, the rest is the work of Man",

    I think Kronecker's integers are the finite ordinals.
    Are you saying that 'ubiquitous' means 'finite'?

    So, "ubiquitous ordinals" is exactly what it says.

    It's surprising how unhelpful some answers can be.

    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.

    --- SoupGate-Win32 v1.05
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  • From Jim Burns@21:1/5 to Ross Finlayson on Thu Jan 2 11:47:38 2025
    XPost: sci.math

    On 1/1/2025 7:28 PM, Ross Finlayson wrote:
    On 01/01/2025 01:14 PM, Jim Burns wrote:
    On 07/29/2024 12:44 PM, Ross Finlayson wrote:
    On 07/29/2024 05:32 AM, Jim Burns wrote:
    On 7/28/2024 7:42 PM, Ross Finlayson wrote:

    about ubiquitous ordinals

    What are ubiquitous ordinal?

    So, "ubiquitous ordinals"
    is exactly what it says.

    It's surprising
    how unhelpful some answers can be.

    I'd so hope to be helpful,
    it's categorized under virtue
    somewhere between
    thrifty and reverent or reverent and clean,
    so I'd wonder,
    is it, "un-helpful", in,
    "can't think it", or,
    "keep thinking it", or,
    "keeps rebooting".

    'Unhelpful' as in
    "empty of new information".

    -- It says what it says.
    Yes, and ... ?

    unhelpful.
    ⎛ I yam what's I yam
    ⎝ an' dat's all that I yam.

    helpful.
    ⎛ I'm strong to the finich
    ⎜ 'cause I eats me spinach.
    ⎜ I'm Popeye the Sailor Man
    ⎝ <toot toot>.

    ----
    it's categorized under virtue
    somewhere between
    thrifty and reverent or reverent and clean,

    I'm thinking more of the Gricean maxim
    "Be informative".
    https://en.wikipedia.org/wiki/Cooperative_principle

    Philosopher Paul Grice describes
    the behavior which those in a conversation
    engage in and assume others engage in.

    The problem I have with calling that a virtue
    is that,
    for a too.large.fraction of people,
    that is an argument against being helpful,
    virtue having been redefined to be
    in opposition to virility.

    --- SoupGate-Win32 v1.05
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  • From Jim Burns@21:1/5 to Ross Finlayson on Thu Jan 2 12:35:42 2025
    XPost: sci.math

    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.

    --- SoupGate-Win32 v1.05
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  • From Jim Burns@21:1/5 to Ross Finlayson on Thu Jan 2 16:08:03 2025
    XPost: sci.math

    On 1/2/2025 2:04 PM, Ross Finlayson wrote:
    On 01/02/2025 08:47 AM, Jim Burns wrote:

    [...]

    "Well-Order the Universe" is
    what "ubiquitous ordinals" is.

    I read that as
    "Each set can be well.ordered".

    ( It'd be great if you (RF) were to indicate
    how close what I read comes to what you write.

    Which, you snipped?

    You're welcome.

    I snip a lot of what you send down the thread,
    in a possibly.forlorn attempt to
    make what you write comprehensible.
    I do it for love of comprehensibility, at no extra charge.

    If I snip context needed for understanding,
    it would be great if you (RF) took that as
    as sign for you to explain more fully,
    upon which sign, you (RF) explained more fully.

    It makes for a model of set theory where
    "powerset is successor is order type",

    You (RF) probably don't mean
    powerset x = {y: y⊆x}
    successor x = x∪{x}
    because, in general, {y: y⊆x} = x∪{x} is wrong.

    It would be great if you (RF) said what
    the other thing you mean by 'powerset' and 'successor'.

    thusly where the Powerset Theorem
    holds different, as it were.

    ⎛ ¬∃F onto: A → 𝒫(A)
    ⎜
    ⎜ because
    ⎝ ¬∃a′ ∈ A: F(a′) = {a∈A: a∉F(a)} ∈ 𝒫(A)

    Is that what you (RF) mean by the Powerset Theorem?
    Where does it hold different?
    Why does it hold different there?

    --- SoupGate-Win32 v1.05
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  • From Jim Burns@21:1/5 to Ross Finlayson on Thu Jan 2 16:55:54 2025
    XPost: sci.math

    On 1/2/2025 2:07 PM, Ross Finlayson wrote:
    On 01/02/2025 09:35 AM, Jim Burns wrote:

    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.

    "The Logic", ?

    First.order.logic, AKA predicate.logic
    variables which range over some non.unique universeⁿᵒᵗᐧᴿꟳ
    of what we are talking about.

    Claims true.without.exception for
    members of the universeⁿᵒᵗᐧᴿꟳ ('axioms')

    Finite sequences of claims, each claim of which
    is true.or.not.first.false (axiom.or.inference).

    Given all that, each claim in such a sequence
    is true.without.exception in that universeⁿᵒᵗᐧᴿꟳ

    A wider, all.inclusive universeᴿꟳ plays no role.

    Is it, "not.ultimately.untrue"?

    ⎛ Student:
    ⎜ Does a dog have Buddha.nature?
    ⎜ Zhaozhou:
    ⎝ Mu.

    I don't see how to make sense of
    "not.ultimately.untrue" in the context of
    true.or.not.first.false claims.
    Following a revered (though different) tradition,
    I un.ask the question and answer: Mu.

    There is no ultimate.possible not.first.false claim.
    So, to be perfectly literal,
    no claim can be ultimately.untrue.
    Or ultimately.true.

    Each claim is not.ultimately.untrue and
    each claim is not.ultimately.true.

    I doubt that my answer actually answers
    anything even a little bit like
    what you intend, so I un.ask the question.

    --- SoupGate-Win32 v1.05
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