Re: The set of necessary FISONs

On 2/4/25 12:04 PM, WM wrote:

On 04.02.2025 13:17, Richard Damon wrote:

On 2/4/25 5:50 AM, WM wrote:

 

You prove that no individual F(n) is needed.

>
F(1) can be subtracted, and if F(n) can be subtracted, then F(n+1) can be subtracted. That is the needed induction.

>
Which just proves that all F(n) are in the set that can be individually not needed.

 Then they all F(n) together can be individually discarded. Just like all natural numbers together can be discarded by subtracting them.

>

Note also, Peano doesn't "create" the Naturals with induction,

>
He does. 1 or 0 ∈ ℕ, and if n is there, then n' is there.

>
Which doesn't CREATE the set N, it shows that some other set is N.

 If there is another set, then we could use it without need of Peano.
 Regards, WM

????
There are many ways to create the set of Natual Numbers.
Peano is one.
It isn't the induction axiom that does it though.
The set of Natural numbers (in Peano) are created by the OTHER axioms, like
The Value 0 is in the Set.
There is a operators S, such that if n is in the set, then Sn is in the set.
If Sn = Sm, then it must be that n = m
If n is in the set, and is not the value 0, then there must be an m in the set such that Sm = n
There is no number m in the set such that Sm = 0
THOSE are the axioms that CREATE the set of Natual Numbers.
The induction axiom just provides a way to see if another set (typically defined by a condition) is equal to the set of Natural Numbers.
You are just showing that you still don't understand what you are talking about and that you keep on confusing the Set with its Elements, and are too stupid to understand your error.