Re: The set of necessary FISONs

WM brought next idea :

On 19.02.2025 19:36, FromTheRafters wrote:

WM has brought this to us :

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Induction has been invented for infinite sets.

 Transfinite Induction has been...

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No.
1) Induction covers all elements of an infinite inductive set.
F(1) ∈ F und F(n) ∈ F ==> F(n+1) ∈ F describes the infinite inductive set F of FISONs.
2) Subtraction all FISONs {1, 2, 3, ..., n} satisfying |ℕ \ {1, 2, 3, ..., n}| = ℵo from F leaves the empty set.
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These two well-established arguments prove my case:
UF = ℕ  ==> Ø = ℕ.
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Regards, WM

AI Overview
While both are proof techniques in mathematics, "induction" typically refers to mathematical induction, which applies only to natural numbers, whereas "transfinite induction" is an extension of that concept, allowing proofs to be made across all ordinal numbers, including infinite ones, essentially enabling proofs on "transfinite" sets which are larger than the natural numbers.
Key Differences:
Scope:
Standard induction is used to prove statements about natural numbers (0, 1, 2, 3, ...), while transfinite induction can be used to prove statements about all ordinal numbers, including infinite ones.
Well-ordering:
Transfinite induction relies on the well-ordered nature of ordinal numbers, meaning every non-empty set of ordinals has a least element.
Inductive step:
In standard induction, you typically assume a statement is true for a number "k" and then prove it's true for "k+1". In transfinite induction, you might need to consider additional cases like limit ordinals (not just successor ordinals) when making the inductive step.
Example:
Standard induction: Proving that the sum of the first "n" natural numbers is n(n+1)/2.
Transfinite induction: Proving that every well-ordered set can be put into a one-to-one correspondence with a unique ordinal number.
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