On the set of all unique elements

[Thomas Klimpel]

> 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?

Not sure which sort of set you want to have. Especially, do you want
to insist that the "distance" between any two elements should be
finite?

On Fri, Jan 6, 2023 at 4:24 PM I.V. Serov wrote:
> Are the set X in the metaphysical example and the set Z of all integers
> isomorphic to each other?

I see, you wonder in which ways the metaphysical example could be
non-isomorphic to the set Z. You identified the designation of the
element 0 as one such way (below, not quoted).

If you designated two adjacent elements in your metaphysical example,
then it would at least no longer be canonically isomorphic to the set
Z: Either one of those two elements could be mapped to zero, and there
are no canonical criteria to prefer one or the other.

Of course, you don't need to designate any special elements in your
metaphysical example, since you can simply formalize it in terms of
two unary functions "succ" and "pred" with axioms like

succ( pred( x ) ) = x
pred( succ( x ) ) = x
forall x, y we have x = y or x = pred( ... pred( y ) ) or x = succ(
... succ( y ) )

The last axiom excludes structures like Z x Z or Z x Z x Z with
lexicographical order. You might not like the "..." in that axiom.
That is the metaphysical part of the formalization. You cannot even
use an axiom scheme to formalize it, because of the "or" occuring in
it. Its negation (where "or" turns into "and") can be formalized by an
axiom scheme. But this justs shifts the non-definability of the
natural numbers one metaphysical level deeper (in a certain sense, as
https://gentzen.wordpress.com/2022/09/09/getting-started-without-a-pre-existing-understanding-of-non-standard-natural-numbers/
tries to illustrate).

In case you are more interested in having an operation similar to
addition and subtraction without (implicitly) designating also the
element 0, you could define a tenary operation [x,y,z] with axioms
like [[a,b,c],d,e] = [a,[c,d,b],e] = [a,b,[c,d,e]] that would be
intended to capture
[x,y,z] := x - y + z (or for general multiplicative groups [x,y,z] := x y^-1 z)

https://en.wikipedia.org/wiki/Heap_%28mathematics%29
https://math.stackexchange.com/a/979972/12490


To summarize, I see two metaphysical questions here: (1) the role of
the "finite distance" / "the natural numbers". This question is known
to be unsolvable in a certain sense. (2) the question of how to avoid
breaking the translation symmetry of the structure that seems to
happen because any concrete specification of a model of the structure
seems to canonically designate at least some special elements. This
could be indeed an interesting question, especially due to its
"anti-foundation nature" in this example. (Its "anti-foundation
nature" makes it a bit different from the more typical problem of this
sort, like that the complex numbers i and -i cannot be distinguished
"internally".)

Regards,
Thomas