Re: The set of necessary FISONs

On 02/03/2025 11:48 AM, Jim Burns wrote:

On 2/3/2025 1:36 PM, WM wrote:

On 03.02.2025 19:06, Jim Burns wrote:

On 2/3/2025 7:41 AM, WM wrote:

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How can Peano create the complete set by induction?

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Peano describes a set with induction.

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Without axioms
nothing must be used oin formal mathematics.

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Axioms describe the domain.
Describe a finite ordinal:
⎛ Sets of them are minimummed or empty.
⎜ Each has an immediate predecessor or is zero,
⎜ and each of the priors of each
⎝ has an immediate predecessor of is zero.
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That description is the axioms of the finite ordinals.
(There are other ways to describe them.)
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Therefore Peano, Zermelo, or v. Neumann
create ℕ as well as the set of all FISONs
for use in set theory.

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Axioms describe.
Magic spells create.
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It is a complete set which is described.
(We don't use any other, "incomplete" sets.)

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Therefore all FISONs can be removed from
the set of all FISONs.

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We can describe the removal of all of them, sic: {}
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All natural numbers can be added by induction to a set A.

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Either all natural numbers are in A,
or they aren't all in A.
Those are all the choices.
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A set with all natural numbers in it
has certain properties.
⎛ For example, enough in.set swaps
⎜ can leave Bob not.in the set,
⎝ even though no swap is out.of.set.
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1 is added to A, and
if n is added to A, then n+1 is added to A.

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Describing A:
⎛ 1 is in A.
⎝ If n is in A, then n+1 is in A
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ℕ is the unique set subset to each such set A
ℕ is the minimal inductive set.
⎛ If you deny that,
⎜ you're only absconding with the name 'ℕ'
⎜ Everything I claim about ℕ remains true of
⎝ the minimal inductive set, whatever it's called.
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All FISONs can be subtracted from the set of all FISONs
by the same procedure.
F(1) is subtracted.
If F(n) is subtracted, then F(n+1) is subtracted.

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For each FISON,
there is a larger FISON not larger than U{FISON}
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U{FISON} is larger than any FISON.
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The sum of any two FISONs is a FISON.
U{FISON} is larger than the sum of any two FISONs.
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Each end.segment U{FISON}\{1,...,j}
is larger than any FISON.
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Such behavior is unlike that of finite sets.
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We could decide that  that isn't their behavior,
but, if we decide that, everything turns to gibberish.
The reasoning is implacable, and does not disappear
because we have decided against it.
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We could decide that these sets aren't finite.
What I mean here by 'finite' is what we mean.
You (WM) mean something else.
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There exists a general preference to avoid gibberish.
This preference is what you (WM) call "matheology".
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But the claims are silent about what wasn't described.
Peano describes _the elements_ of ⋃{FISON}
⋃{FISON} isn't an element of ⋃{FISON}

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Peano creates

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...describes...
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the set of all natural numbers as well as
the set of all FISONs.

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Axioms that are not _false_, define a domain.
Others define a mere contingent contrivance.