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Anonymous Sun 31 Oct 00:24:09 2021 No.13805204 View Reply Original Report

Quoted By: >>13805244

retard question.
if you can propose the test of 1 to infinity
can you also reverse that? having infinity be the start position and 1 the "end"?

Anonymous

Anonymous Sun 31 Oct 2021 00:30:19 No.13805213 Report

Quoted By:

>propose the test
?

In mathematical induction alla Peano arithmetic, i.e. induction on the class of natural numbers, what you end up with is a universal statements _for all_ numbers. But you never "reached infinity".
In transfinite induction, as e.g. implied by epsilon induction, you do reach infinities and yes there's then also transfinite recursion. (Recursion as in the way of computing n! via n!:="0 if n=0 else (n-1)!*n")

https://en.wikipedia.org/wiki/Mathematical_induction
https://en.wikipedia.org/wiki/Transfinite_induction
https://en.wikipedia.org/wiki/Epsilon-induction
https://en.wikipedia.org/wiki/Induction,_bounding_and_least_number_principles

Anonymous

Anonymous Sun 31 Oct 2021 00:31:45 No.13805218 Report

Quoted By:

>propose the test
?

In mathematical induction alla Peano arithmetic, i.e. induction on the class of natural numbers, what you end up with is a universal statements _for all_ numbers. But you never "reached infinity".

In transfinite induction, as e.g. implied by epsilon induction, you do reach infinities and yes there's then also transfinite recursion.
(Recursion as in the way of computing n! via n!:="1 if n=1 else (n-1)!*n")

https://en.wikipedia.org/wiki/Mathematical_induction
https://en.wikipedia.org/wiki/Transfinite_induction
https://en.wikipedia.org/wiki/Epsilon-induction
https://en.wikipedia.org/wiki/Induction,_bounding_and_least_number_principles

Anonymous

Anonymous Sun 31 Oct 2021 00:48:16 No.13805244 Report

Quoted By: >>13805270

>>13805204
What number would come after infinity?

Anonymous

Anonymous Sun 31 Oct 2021 00:49:53 No.13805247 Report

Quoted By:

yes, just reverse your idea of convergence.

Anonymous

Anonymous Sun 31 Oct 2021 01:02:29 No.13805270 Report

Quoted By: >>13805847

>>13805244
If $\alpha$ is an infinite number, then the successor is just denoted $\alpha+1$ .
Already EFA lets you characterize infinite well-orders up to $\omega^3$ .

Robinson arithmetic, which is like Peano but without induction, has a model where $\omega'$
is itself a "natural number" (in the sense that it's a term next to all standard natural numbers, but not reachable from them) and $\omega'+1=\omega'$

https://en.wikipedia.org/wiki/Elementary_function_arithmetic
https://en.wikipedia.org/wiki/Robinson_arithmetic

Anonymous

Anonymous Sun 31 Oct 2021 08:18:27 No.13805847 Report

Quoted By: >>13805872

>>13805270
I mean in OP's reverse enumeration down to 1.
It doesn't make sense because the reverse order type of omega is not a well-order.

Anonymous

Anonymous Sun 31 Oct 2021 08:32:53 No.13805872 Report

Quoted By: >>13807815

>>13805847
It's not clear what OP is really asking.
Why would a well-order even necessary - as opposite to just a total order.

Anonymous

Anonymous Sun 31 Oct 2021 22:21:07 No.13807815 Report

Quoted By:

>>13805872
How can one count down *from* infinity?

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