On the set of all unique elements

[Monroe Eskew]

I think the abstract structure you’re looking for is a “discrete linear order without endpoints.”  The order-structure of the integers (with all algebraic operations removed) is the prime example. There are other examples, but the integers are “prime”— the one that can be embedded into any other. They can also be characterized by saying additionally that there are only finitely many elements between any two.  (But finiteness is not first-order definable.)

Also note that the structure has many automorphisms, by “shifting” over any number of elements left or right. 

> On 06.01.2023, at 16:25, I.V. Serov <i.v.serov at chf.nu> wrote:
> Are the set X in the metaphysical example and the set Z of all integers isomorphic to each other?
> 
> To answer this question, it is necessary to clarify what a set X is.
> 
> X is defined as an infinite set in which each element x has a unique predecessor 'x and a unique successor x' and such
> that there is no binary operation on the set X with a neutral element in X.
> 
> Consider a unary numerical system, where the intuitive notion of a typical element x in X is described by a countable left-infinite sequence of tally marks.
> 
> The successor element x' in X is obtained from x by adding one tally mark at the right end.
> The predecessor element 'x in X is obtained from x by removing one tally mark at the right end.
> Imagine that infinities are aligned on the left edge, so that the set X may look like:
> 
> X = {...,
> ...|||||,
> ...||||||,
> ...|||||||,
> ...||||||||,
> ...}
> 
> The construction is similar to "Hilbert's hotel", with the difference that instead of moving all the guests,
> each new guest arrives in a new room, which is constructed for the guest by the induction master at the right end of a countable left-infinite sequence of rooms.
> 
> The hotel before the guest's arrival is associated with element x of X.
> The hotel after the guest's arrival is associated with element x' of X.
> This metaphysical process has never begun and will never end: guests have arrived, are arriving, and will continue to arrive. X is the chronicle of the hotel.
> 
> Make a notice:
> x in X is described by a countable sequence of tally marks;
> x' in X is described by the same sequence of tally marks plus one extra tally mark.
> One can think of x as part of x' and conclude that "the part is smaller than the whole."
> David Hilbert claims differently: ""The part is smaller than the whole" is no longer valid."
> 
> Any given x from X can be matched with any integer z from Z and vice versa, so that
> the set Z in its natural order {...,z-1,z,z+1,...}
> and
> the set X in its natural order {...,x-1,x,x+1,...}, where x' is denoted as x+1 and 'x is denoted as x-1,
> are in bijection.
> 
> Unlike on the set Z, there is no binary operation with a neutral element on the set X; X has at most one automorphism - the trivial one.
> The set Z, considered as a infinite monogenous group under addition, has a single non-trivial automorphism: taking the opposite sign.
> Integers, the elements of Z, including their neutral elements such as zero and one, are not elements of the set X.
> There is no isomorphism between the sets X and Z.
> 
> Let us give the elements of the set X a name and call them genuine numbers.
> Well, are the elements of X numbers?
> 
> According to Euclid:
> "An unit is that by virtue of which each of the things that exist is called one".
> "A number is a multitude composed of units".
> 
> A genuine number is that by virtue of which everything is called one and which is the multitude composed of another genuine number, called the preceding genuine number, by adding a natural unit to the latter.
> A natural unit is that by virtue of which a genuine number differs from the preceding genuine number.
> 
> Any integer z can be commutatively added to any genuine number x to obtain a genuine number x+z.
> Genuine numbers are all infinite by anti-foundational composition, while integers are all finite.
> 
> The conclusion: the set X of genuine numbers is not the same as the set Z of the integers.
> 
> The general task is to formulate a set of axioms for the set X, which determines a unique and an unambiguous law of composition and describes the structure of genuine numbers.
> The set of axioms must guarantee the existence, uniqueness and infiniteness of the set X as well as intrinsic non-neutrality of its elements.
> 
> I.V. Serov