Re: The set of necessary FISONs

On 25.02.2025 19:39, Jim Burns wrote:

On 2/25/2025 3:58 AM, WM wrote:

On 24.02.2025 19:00, Jim Burns wrote:

On 2/24/2025 12:17 PM, WM wrote:

 

It is possible
but not useful to express this ponderously
by fixed sets.

>
Mathematical sets and
mathematical objects in general
do not change,
(although they can represent change in various ways).

Unchanging functions can represent changes.

Changing sets are functions of time or of other properties.

Addition and subtraction of sets are
a common techniques.
Thereby sets are changed.

>
Thereby a relationship between unchanging sets
is described.

>
That is the clumsy description.

 It makes prose more readable to say "This set changes".
As long as that's understood to mean the other,
I don't see any great crime being committed.

So it is.

Does doing things that way make clumsy prose?
Perhaps.
Many would consider clumsy prose to be a small cost
for the ability to reason reliably about
infinitely.many.

That is also possible with changing sets.

 

In order to be existing sets must be created
by men or by God.

 No activity by men or gods is required
in order for a thing to satisfy a description.

If the description is the first, then it constructs the set.

 No agreement, no disagreement, no activity by men or gods
permits or prevents,
in a finite.sequences of claims in which
each claim is true.or.not.first.false,
each claim being true.

The claims first must be made.
Zermelo for instance made such claims:
Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch des folgenden, seinem wesentlichen Inhalte von Herrn Dedekind herrührenden Axioms. ... Der Bereich enthält mindestens eine Menge Z, welche die Nullmenge als Element enthält und so beschaffen ist, daß jedem ihrer Elemente a ein weiteres Element der Form {a} entspricht ... Die Menge Z_0 enthält die Elemente 0, {0}, {{0}}, usw. und möge als "Zahlenreihe" bezeichnet werden, ...  Sie bildet das einfachste Beispiel einer "abzählbar unendlichen" Menge. [E. Zermelo: Untersuchungen über die Grundlagen der Mengenlehre I, Mathematische Annalen (1908), S. 266]
Regards, WM