Re: A new way for the foundation of set theory
Date: Tue, 28 Jun 2016 13:13:49 -0700 (PDT)
Message-ID: <21bdfc81-49c5-4d09-b7c3-2b7d0a15b205_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
I think that the following construct abstraction:
For every plurality of objects, there exist only one abstraction that can construct this plurality of objects as one object.
is better than Construction2 in my post from 21.12.2015.
Now, the first three constructs of this set theory, would be as follows,
Construct1: Frege's concept Construct2: Abstraction Construct3: My theory of identification
In my post of 21.12.2015. in this thread is shown as follows: from Frege's
definition of the concept and the extension of the concept, we can get two
axioms from classical set theory: extensionality and comprehension.
From Construct3 we can get axiom of separation(specification) from Zermelo–
Fraenkel's Set theory.
The idea is that in this way we can get the first three axioms of set theory
and eliminate Russell's paradox.
In today's set theory, primitives are „element" and "set", while primitives in Frege's theory are the concepts and extensions of concepts. So it is necessary to show that if we start from the "concepts" and "extensions of concepts," then we can get today's set theory, which is based on the "sets" and "elements of sets." This move is done by John Burggess. Moreover, Burgess in his book "Fixing Frege" is completed Frege's idea that the whole mathematics can be expressed through logic. This Frege's idea is probably the biggest project in the history of mathematics. This idea Burgess was implemented only for arithmetic.
In this thread I tried to complete Frege's idea about building set theory by using concepts and extensions. I think that Russell's paradox is resolved by my theory of identification of objects.
In my opinion, concepts, abstractions and the identification of objects are „stronger“ than sets and elements.
Vladimir Odrljin Received on Tue Jun 28 2016 - 22:13:49 CEST