On the set of all unique elements

[dennis.hamilton at acm.org]

-----Original Message-----
From: FOM Serguei Mokhov
Sent: Wednesday, January 4, 2023 07:06
Subject: Re: On the set of all unique elements

On Mon, Jan 2, 2023 at 10:51 PM I.V. Serov <i.v.serov at chf.nu> wrote:
>
> Consider, as a metaphysical example, the set of all unique "days",
> where each day (such as today) has both a unique predecessor (such as
> yesterday for today) and a unique successor (such as tomorrow for
> today) in the set, so that the days are naturally ordered one after
> another, and there is no first day, and all days are elements of the set.
>
> Substitute, if you wish, the word "day" with "thing", "event",
> "state", "thought" or "element".
>
> Is there a known formal mathematical object/structure/logic that seeks
> to describe this set?
>
> I.V. Serov
Intensional Logic? Possible worlds semantics by Kripke?
https://plato.stanford.edu/entries/logic-intensional/
--
Serguei Mokhov
[orcmid]
In practical affairs, it may be unnecessary to ponder such metalogical 
questions.

There are ordinary pragmatic interpretations that rely on such a structure. 
Consider the integral domain on the Naturals, with 0 fixed at today (or any 
specific calendar date), and the predecessors and successors progress as 
required when interpreted as ordinal days relative to the chosen 0-day.

The integral domain  supports more arithmetic than matters for a calendar-day 
enumeration, but that's not unusual or particularly disturbing when number is 
interpreted as a measure in nature, taking "dimension" into account.

In computation, this is rather commonplace, apart from the truncation of the 
sequence to something like -2^n, ..., 0, ... 2^n-1, with n rather large. 
Spreadsheet software typically represent dates in this manner, except 
rationals are used so that the time within a day is expressed as a fraction of 
a whole day in the most widely-used spreadsheet programs. (Where greater 
precision is required, the interpretation is in exact seconds rather than days 
or even some shorter instances that are technically determinable.)