Re: A new way for the foundation of set theory
Date: Thu, 24 Dec 2015 15:16:07 -0800 (PST)
Message-ID: <f3de3d0f-4251-4e22-bd13-69e768582d6c_at_googlegroups.com>
On Monday, December 21, 2015 at 5:00:03 PM UTC+1, vldm10 wrote:
> Elements of a set are names.
> A set is a plurality of names which is named with one name.
>
> The first three constructs of this set theory are as follows:
>
> 1. Construct1: Frege's definition of concept.
> 2. Construct2: Frege's assumption of extensions of concepts
> 3. My theory of identification.
>
> Definition:
> Two sets A and B have the same cardinality if there exists an algorithm that
> can show that every element of A is paired with exactly one element of B,
> and every element of B is paired with exactly one element of A. There are no
> unpaired elements.
>
> Example:
> We can form two columns. In the first column, we can store the names from
> set A, and in the second column are stored the names from set B. The first
> column we'll call "First name", the second column we'll call "Last name". If
> at some point in these two columns exist gaps, then the two set do not have
> the same cardinality.
> Note that here we have an algorithm, rather than the 1-1 mapping. In fact,
> some machine can "see" a gap in the columns, from this example.
>
> This set theory is based on the objects, which are defined in my papers. In
> these papers are also introduced the atomic structure (atomic objects,
> atomic propositions, atomic predicates, atomic concepts, atomic facts and
> atomic sets).
> My papers also define the following states of objects and relationships:
> current states, past states and future states.
>
> Vladimir Odrljin
Often, sets are defined by using term "object". For example: "A set is a
collection of distinct objects." or "Set theory is the mathematical theory
of well-determined collections, called sets, of objects that are called
members, or elements, of the set."
I have criticized this approach. In my thread "The original version" in my
post from January 4, 2011, I criticized the "Anchor modeling". The authors
of this "model" presented their basic term as follows: "An anchor
represents a set of entities, such as a set of actors or events."
Of course, these sets do not exist. The elements of set are not physical
objects.
Note that from from first two constructs (Frege's concepts and extensions) we can get the two axioms of set theory: comprehension and extensionality. For example in the works of John P. Burgess it's done. From Construct 3 (that is identification) we can get Zermelo's axiom of separation. Received on Fri Dec 25 2015 - 00:16:07 CET